M.G.Goman and A.V.Khramtsovsky (1993) - Textbook for KRIT Toolbox users

Methodology of the qualitative investigation�

Theory and numerical methods�

Application to Aircraft Flight Dynamics�

textbook

M�G�Goman A�V�Khramtsovsky

Contents

� Qualitative analysis of nonlinear systems �

�� Local qualitative analysis of critical elements � � � � � � � � � � � � � � � � �� Equilibrium points � � � � � � � � � � � � � � � � � � � � � � � � � � �� Periodic solutions � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Poincar�e mapping � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Bifurcational analysis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Bifurcations of the equilibrium points � � � � � � � � � � � � � � � � �� Bifurcations of the periodic trajectories � � � � � � � � � � � � � � � �

�� How to determine the phase portrait � � � � � � � � � � � � � � � � � � � � �� Stability regions � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Continuation method � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Basic numerical methods ��

�� Mathematical simulation � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Investigation of equilibrium points � � � � � � � � � � � � � � � � � � � � � � ��

�� Newton�like and gradient methods � � � � � � � � � � � � � � � � � �� The continuation method � � � � � � � � � � � � � � � � � � � � � � �� Search for set of solutions � � � � � � � � � � � � � � � � � � � � � � � �� Local stability and phase portrait � � � � � � � � � � � � � � � � � � �� Manifolds of equilibrium solutions � � � � � � � � � � � � � � � � � �

�� Investigation of the periodic solutions � � � � � � � � � � � � � � � � � � � � �� Time�advance mapping� � � � � � � � � � � � � � � � � � � � � � � � �� Poincar�e mapping� � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Investigation of the stability regions � � � � � � � � � � � � � � � � � � � � � ��

� Application to aircraft �ight dynamics ��

�� Problems to be solved � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Equations of airplane spatial motion� � � � � � � � � � � � � � � � � � � � � �� Various scalar forms of the motion equations� � � � � � � � � � � � � � � � �� Critical �ight regimes� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Approximate stall criteria� � � � � � � � � � � � � � � � � � � � � � � � � � � �� Steady�state spiral motion � � � � � � � � � � � � � � � � � � � � � � � � � � �� Aircraft maneuvers with rapid rotation � � � � � � � � � � � � � � � � � � � � �� Computing maneuver limits� � � � � � � � � � � � � � � � � � � � � � � � � � �� Analysis of spin characteristics � � � � � � � � � � � � � � � � � � � � � � � � ��

�

� CONTENTS

Appendix A� Matching of US and USSR notation ��

Chapter �

Qualitative analysis

of nonlinear dynamical systems

In recent years nonlinear dynamical systems became very popular� Nonlinear worldattracts specialists of all branches of science� There are many publications in suchdisciplines as aerodynamics� economy� biology� chemical kinetics etc� devoted to variousnonlinear phenomena� The interest in the theory of nonlinear systems and in tools forcomputer�aided research rapidly grows�

Nonlinear systems describing the behaviour of the objects of the real world usually aretoo complex to be thoroughly studied using analytical methods� That�s why nowadaysthe advances in understanding essential features of nonlinear objects are closely coupledwith the development of numerical methods� Today a researcher has a number of e�ectiveand reliable numerical methods at his disposal intended for analysis and classi�cation ofnonlinear phenomena� And there is a need for more and more powerful algorithms andmethods�

There is also a need for software packages eliminating the necessity for a researcherto write too many lines of code for his computer� This manual is closely connected withthe development of Krit package for studying nonlinear systems of ordinary di�erentialequations�

Qualitative theory of dynamical systems and theory of bifurcations constitute thetheoretical basis for the construction of new mathematical methods ��

Qualitative theory allows to de�ne on the qualitative level the structure of the statespace of the object concerned i�e� to �nd out all possible types of motion� To do soit is necessary to �nd steady�state �equilibrium and periodic� regimes of motion and toknow the domains of attraction �stability regions� of the stable steady�state regimes�The results of bifurcation theory are useful for investigation of changes in the system�sbehaviour due to the variations of parameters�

Relying on the theoretical results and also on the experience in numerical studies�we o�er the following general scheme for analysis of nonlinear dynamical systems�

�� System is studied when the parameters are �xed� The general structure of thestate space is investigated�

� a set of equilibrium and periodic solutions and their local stability are deter�

�

� CHAPTER �� QUALITATIVE ANALYSIS OF NONLINEAR SYSTEMS

mined�

� stability regions are found by means of building two�dimensional cross�secti�ons of the stability regions� boundaries�

�� The changes in the behaviour of the system due to variations of parameters arestudied�

� all possible types of the phase portraits of the dynamical system are deter�mined using continuation method�

� the research program � is performed for the every known type of the statespace structure� with parameters set to some typical values�

We assume that dynamical system is described by a set of nonlinear ordinary di�er�ential equations of the �rst order

dx

dt� F �x� c� � x � Rn� c �M � Rm�

where x is a state vector� and c is a vector of parameters�

�� Local qualitative analysis of critical elements

This section is devoted to the basic concepts and ideas of the qualitative theory ofdynamical systems which will be useful for the �nonlinear� aircraft �ight dynamicsinvestigations ��

Consider a nonlinear autonomous dynamical system depending on parameters� de�scribed by the di�erential equations

dx

dt� F �x� c� � x � Rn� c �M � Rm� ��

where F is a smooth vector function� The vector �eld F de�nes a map Rn�m � Rm� Thesystem �� satis�es to the conditions of the existence and uniqueness of the solutionx�t�xo� with initial condition x� �xo� � xo� The solution �t�xo� � x�t�xo� is called atrajectory or the �ow of the dynamical system � The set of all the trajectories constitutethe phase portrait of the dynamical system �

General theory of smooth dynamical systems is not yet �nished� We shall considerthe so�called Morse�Smale structurally stable systems�

The success of qualitative analysis of a dynamical system is closely coupled with asuccess in �nding singular trajectories or critical elements � It is also extremely importantto understand the structure of the state space in the neighbourhood of the criticalelements� The data gathered then allow to understand the global structure of the statespace of the system ��

The manifolds of di�erent dimension may be the images of the critical elements�For example� the stable state of equilibrium is the attracting point� the stable periodicsolution is the attracting closed orbit in state space� If an order of dynamical system

�� LOCAL QUALITATIVE ANALYSIS OF CRITICAL ELEMENTS

Figure �� Critical elements of the phase portrait


n � �� then more complex attractors can exist� There may be invariant tori or �strangeattractor��type structures �see �g��

The following notation will be used ��O � equilibrium point�� closed orbit or phase trajectory �i�e� periodic motion ��T � toroidal manifold of trajectories in the state space�Complex behaviour of the nonlinear dynamical system is often due to the existence

of a number of isolated attracting sets� each attractor has its own region of attraction�All that brings about the strong dependence of motion on initial conditions and on thesequence of variations of parameters�

Qualitative analysis pays special attention to the simplest attractors �i�e� equilibriumpoints and closed orbits� and their dependence on the parameters�

Stable equilibrium points and periodic motions are the simplest attractors determin�ing steady�state regimes of motion of the system� Saddle�type solutions and its stableand unstable invariant manifolds of trajectories are the decisive factors in forming globalstructure of the state space since they de�ne the boundaries of the regions of attraction��

�� Equilibrium points

Equilibrium points �or else equilibrium solutions� x � xe of the system �� are thesolutions of the system of equations

F�xe� c� � � x � Rn� c �M � Rm� ��

At any given values of the parameters c one can have a set of di�erent isolatedequilibrium points� the number of equilibrium points may change while the values of theparameters c are varied�

On linearizing in the neighbourhood of the equilibrium point xe� the system ��takes the form

dx

dt� Ax ��

where

x � x� xe � vector of deviations with respect toequilibrium xe�

A � �F�x

��x�xe

� Jacoby matrix�

If all the eigenvalues of matrix A of the linearized system are di�erent� the generalsolution of �� is

x � xe � a�e��t�� ane

�nt�n ��

where

�� n � eigenvectors of A matrix�� n � eigenvalues of A matrix�a�� an � constants depending on initial condition xo�

�� LOCAL QUALITATIVE ANALYSIS OF CRITICAL ELEMENTS �

If there are complex pairs among eigenvalues f�igi��n with the indexes � and � ��

�c�� i��c�� i��

then solution of �� can be rewritten as

x � xe � a�e��t�� a� cos��t� a�� sin��t�e��t��

�a� sin ��t� a�� cos��t�e��t�� ane�nt�n

��

The values of a�� an are de�ned by the initial deviation from the equilibriumpoint xe�

a�� a�� an�n � xo � xe ��

or� using a matrix Q in which eigenvectors are the columns

Q � jj�� njj

one may express the vector a � �a�� a�� an� as follows

a � Q��xo � xe� ��

Using matrix notation� the solution of �� is

x � xe �Q��t�Q��xo � xe� ��

where ��t� is the matrix of fundamental solutions in the eigenvectors� basis� ��t� isa solution of matrix equation

d�

dt� � �� E

where E is a unit diagonal matrix� � Q��AQ is a Jordan block�diagonal matrixof the form

�

��

If real parts of all the eigenvalues are negative Ref�igi��n � then the trajectory�t�xo� will go towards the equilibrium point xe for any xo belonging to small neigh�bourhood of the equilibrium� The equilibrium point becomes unstable if any of theeigenvalues has a positive real part�


Stable and unstable invariant manifolds of trajectories W s W u�

The placement of the eigenvalues on the complex plane de�nes the structure of the statespace in the vicinity of the equilibrium point xe�

Suppose that q eigenvalues lie to the right of the imaginary axis and p ones lie to theleft� the sum p � q � n equals to the dimension of the state space� Then two integralmanifolds of trajectories W s

p and W uq connected with the equilibrium point xe can be

de�ned as follows �W s

p � fx � �t�x�� xe as t� �gW u

q � fx � �t�x�� xe as t� �gThe stable p�dimensional manifold W s

p comprises all the trajectories in the statespace that go to xe while t� �� Similarly� the unstable q�dimensional manifold W u

q

comprises all the trajectories in the state space that go to xe while t� ��For linear approximation �� the manifolds W s

p and Wuq lie in the hyperplanes Lp

and Lq� The hyperplane Lp is determined by p stable eigenvectors f�igi�k��kp� whileLq is determined by q unstable ones f�igi�l��lp�

Lp � span ��k� ��k� � � � � ��kp�

Lq � span ��l��l�� lq�

When nonlinear terms are taken into account� then the surfaces W sp and W u

q willdeviate from hyperplanes Lp and Lq �the farther from the equilibrium� the more thedeviations�� But the surfaces remain tangent to hyperplanes Lp and Lq at xe �see�g��

The notation Op�q will be used for such an equilibrium point ��

�� Periodic solutions

As for equilibrium points� one can write the equations for deviations with respect to anarbitrary trajectory �t�xo�

dxdt � F�x� c�� x� � � xo

d�dt � A�t�� E

��

where

A�t� � �F�x jx�t��t�xo� � Jacoby matrix calculated along the trajectory

�t�xo��t� � matrix of fundamental solutions of the linearized

system� It de�nes the evolution of the small per�turbations�

Consider closed trajectory or periodic solution with the period � The trajectory isclosed when the following condition is satis�ed

�� x�� x�

�� LOCAL QUALITATIVE ANALYSIS OF CRITICAL ELEMENTS

Figure �� Invariant manifolds W sp � W

uq

where x� is a point belonging to the closed trajectory ��

If xo in �� is such that xo � �� then the matrix A�t� is one with periodic coe!�cients� The value of ��t� at time de�nes stability properties of the periodic trajectory�The matrix �� is called a monodromy matrix � The eigenvalues of �� are calledcharacteristic multipliers�

The periodic solution is stable if all the multipliers lie inside the unit circle on thecomplex plane �note that one multiplier always equals to ��

�� Poincar�e mapping

Poincar�e mapping technique is an e�ective tool for investigation of the state space inthe vicinity of the closed trajectory � or the integral invariant manifolds of the higherdimension�

The equivalent n� ��dimensional system with discrete time may be studied insteadof original dynamical system� Poincar�e mapping can be set using a n � ��dimensionalhyperplane " transversal to the closed trajectory � at the point x�� The trajectoriesin the neighbourhood of � also cross "� Thus every point xk � " �belonging to someneighbourhood of x�� can be mapped into some other point of the hyperplane xk�� "which corresponds to the second intersection of the trajectory �t�xk� with hyperplane"�

The type of the mapping P depends on the intersection condition� The mappingcan be one�sided �as above� or two�sided when all the intersection points are taken intoaccount� Poincar�e mapping generates the sequence

xk�� P�xk� ��


where xk��xk � "�Fixed point x� of the mapping P

P�x�� x�

corresponds to the periodic trajectory � �see �g��

Figure �� Closed orbits � Poincar�e mapping

The behaviour of the mapping P near x� is described by discrete linear systemderived from ��

xk � Axk��

where

A � �P�x

��x�x�

� constant matrix�

x � x� x� � deviation vector with respect to �xed point of the map�ping

The orbit of the mapping P in the small neighbourhood of the �xed point O withcoordinates x� is determined from the relationship

xk � x� � a��k�� an��

kn��n��

where f�igi��n�� and f�igi��n�� are the eigenvalues and eigenvectors of the Jacobymatrix A calculated at the �xed point of the mapping�

The parameters faigi��n�� are determined using initial condition xoa�� a�� an��n�� xo � x�

�� BIFURCATIONAL ANALYSIS ��

The eigenvalues �i are the characteristic multipliers for the �xed point� The mappingis compressing in the direction of the eigenvector �i if j�ij � �

If there are complex pairs ��e�i�� among eigenvalues then there will be a rotation ofthe points belonging to the orbit of the mapping in the plane de�ned by the eigenvectors�� and �� The equation �� will look like that�

xk � x� � a��k�� a� cos k�� a�� sin k��k��

� �a� sin k�� a�� cos k��k�� an��kn��n��

��

The multipliers f�igi��n�� do not depend on the orientation of the secant hyperplane" as far as it remains transversal to the trajectory� The multipliers coincide with theeigenvectors of the monodromy matrix�

The behaviour of the trajectories in the state space near the closed orbit � is inone�to�one relation with the behaviour of the mapping �� in the vicinity of the �xedpoint O� The type of the �xed point can be determined provided the placement of thecharacteristic multiplies with respect to the unit circle in the complex plane is known�As for an equilibrium solution �� for �xed point� one can �nd the p � dimensionalstable and q � dimensional unstable invariant manifolds W s

p and Wuq of mapping points�

Here p is a number of multipliers inside the unit circle� and q is the number of multipliersoutside unit circle�

The stable manifold W sp is composed of the points being the results of successive

mappings� each sequence of these points converges to the �xed point O� This manifoldcorresponds to the stable multipliers j�ij ��

The unstable manifold W uq is also composed of the points being the results of succes�

sive mappings� but each sequence of these points converges to the �xed point O usinginverse mappings� The manifold corresponds to the unstable multipliers j�ij � ��

Fixed point O corresponds to the closed trajectory � in the state space� Invariantmanifolds of mapping pointsW s

p andWuq for the �xed point O correspond to the invariant

manifolds of the trajectories W sp�� and W u

q�� the di�erence in the dimension of themanifolds is �� That�s why the notation �p��q�� will be used to describe the type ofthe closed trajectory�

Trajectories from W sp�� manifold go closer to � while t�� and those from W u

q doso while t � �� Other trajectories from the neighbourhood of � �rst go closer to �along W s

p�� and then go away of it along the Wuq�� manifold ��g� ��

�� Bifurcational analysis

Investigation of the possible changes in the structure of the state space due to the changesof parameters c is a vital part in the methodology of the qualitative analysis�

There are critical or bifurcational values of parameters cb when qualitative typeof the state space structure changes� The most common is the so�called one�parameterbifurcation when in the parameter space near the point cb one can �nd dynamical systemswith only two di�erent types of state space�

�� CHAPTER �� QUALITATIVE ANALYSIS OF NONLINEAR SYSTEMS

Figure �� Invariant manifolds W sp � W

uq for closed orbit

Global changes of the state space structure are the result of successive local bifur�cations in the vicinity of the singular trajectories or critical elements �i�e� equilibriumpoints� closed orbits etc�� and bifurcations with the invariant manifolds W s

p and Wuq �

�� Bifurcations of the equilibrium points

There is a condition that allows to distinguish bifurcational situations for the equilibriumpoint O of �� with coordinates x � xe� The point c �in the space of parameters� is nota bifurcational one for the equilibrium O provided there are no eigenvalues �computedat the equilibrium point� on the imaginary axis in the complex plane� In that case�the dimensions p and q of the invariant manifolds W s

p and W uq remain the same for all

c close enough to the initial value� hence the type of the equilibrium Op�q remains thesame� Otherwise the point in space of parameters cb will be bifurcational one�

The two simplest and the most common bifurcational situations for equilibrium so�lutions are

� there is one eigenvalue equal to zero�� there is a complex pair on the imaginary axis�The surfaces No and N� of codimension � in the space of parameters M correspond

to these situations� The equation for No is

Det

��F

�x

��x�xe

��

The surface N� is de�ned by the equation


Det

��F

�x

��x�xe

� i�E

��

Consider the characteristic equation of the linearized system in the vicinity of theequilibrium

�n � a��n�� a��n�� an�� an � ��

Using coe!cients ai � ai �c� � i � �� n� one can express the equations for the

bifurcational surfaces No �� and N� �� as follows

an �

and

#n��

a� � � � �a� a� a� � � � �a� a a� a� � � ��

�

where #n�� is the last but one Raus determinant for the characteristic equation ��When the point in the space of parameters crosses the bifurcational boundary No�

then the following bifurcations may occur with the initially stable equilibrium point

� stable and saddle�type unstable equilibrium solutions merge and vanish� An abruptloss of stability takes place� Bifurcation scheme is

On�o �On��

� stable point becomes saddle�type unstable� at the same time two more stableequilibrium points appear� The �soft� loss of stability occurs� Bifurcation is

On�o � On�� On�o

� the merge of two saddle�type and one stable equilibrium solutions� The loss ofstability is abrupt�

On�o � �On�� On��

When the bifurcational boundary N� is crossed� the Andronov�Hopf bifurcationoccurs� There are two forms of the Andronov�Hopf bifurcation with initially stableequilibrium point � �g� ��

� the equilibrium becomes oscillatory unstable� At that moment stable periodicsolution is detached from the equilibrium� Soft�type loss of stability takes place�

On�o � On�� n��


Figure �� Bifurcations of the equilibrium points

�� BIFURCATIONAL ANALYSIS �

� the stable equilibrium solution merges with saddle�type unstable periodic trajec�tory and becomes oscillatory unstable� The abrupt loss of stability occurs�

On�o � �n�� On��

The type of the loss of stability mentioned above takes place when the point in thespace of parameters slowly enough passes through the bifurcational boundary�

Soft�type loss of stability means that the trajectory in the state space �started inthe vicinity of the equilibrium� slowly goes away from the equilibrium point while thevalues of parameters are close to the critical ones� If the parameters return to the initialvalues� then the trajectory return to the neighbourhood of the equilibrium point�

When the loss of stability is abrupt� then the trajectory in the state space quicklyleaves the equilibrium point for some other attractor even if the deviations of the pa�rameters from critical values are very small� If the parameters then return to the initialvalues� the trajectory usually would not return to the initial equilibrium point�

The bifurcation points where two or more branches of the equilibrium solutionsintersect ��g�� has the codimension greater or equal to �� So at least two conditionsmust be satis�ed� Hence� in general case� such points are structurally unstable� Possiblechanges in the extended state space �the parameter axis is added� in the vicinity of thebranching point are shown in �g��

�� Bifurcations of the periodic trajectories

The stability type of the �xed point O of the mapping P and of the periodic trajectory� may change when the characteristic multipliers cross the unit circle on the complexplane� That�s why only those values of parameters cb are bifurcational when there existsa multiplier belonging to the unit circle�

Consider the bifurcations of the stable closed orbits� Three simplest cases are

� multiplier crosses the unit circle at the point �� multiplier crosses the unit circle at the point �� a pair of multipliers crosses the unit circle at the point e�i��

In these cases the simplest bifurcational surfaces N��N��N� of codimension � in thespace of parameters are generated �see �g� ��

Bifurcational surfaces N�� and N� for the �xed point are equivalent to the bifurca�tional surfacesNo and N� for the equilibrium point� the surfaceN�� represents somethingnew�

There are three more situations when periodic trajectory vanishes�

� the periodic trajectory � shrinks into a point�� an equilibrium point emerges on the closed orbit ��

� some point belonging to � goes to in�nity� thus the curve is no more closed�


Figure ��


Figure �� Bifurcations of the closed orbits


When the boundary N�� is crossed� then two periodic trajectories either merge andvanish or emerge simultaneously �in the latter case the abrupt loss of stability occurs��

� �n�� n�� n�� n��

When the boundary N� is crossed then stable closed orbit �n�� becomes unstable�n�� At the same time the stable ��dimensional toroidal invariant manifold T n��

emerges� In the second possible pattern unstable periodic solution �n�� appears whenunstable ��dimensional toroidal invariant manifold T n�� and stable closed orbit �n��

merge� The bifurcations may be written as follows

� �n�� T n�� n�� soft loss of stability�

� �n�� T n�� n�� abrupt loss of stability

If the boundary N�� is crossed� there may be bifurcations when branching of theperiodic solutions takes place�

� �n�� n�� n�� soft loss of stability

� �n�� n�� n�� abrupt loss of stability

When the bifurcational boundary N�� is crossed� then the changes in comparisonwith equilibrium bifurcations are somewhat unusual� Stable periodic trajectory withthe period �n�� becomes unstable one �n�� and at the same time stable periodictrajectory with double period �n�� detaches from the initial closed orbit�

� �n�� n�� n��

Similarly� the merge of the saddle�type unstable double period closed orbit �n��

with stable periodic solution �n�� is possible

� �n�� n�� n��

�� How to determine the phase portrait of the dy�

namical system

Local methods of the qualitative analysis �the investigation of the state space structurein the vicinity of the equilibrium points and closed trajectories and the methods forinvestigation of the bifurcational changes in the small neighbourhood of these singulartrajectories� were discussed in detail in the previous section�

One of the most important problems to be solved for the reconstruction of the globalportrait of the state space is the determination of the whole set of the isolated equi�librium points and periodical trajectories existing in the dynamical system under givenconditions �i�e� given values of parameters�� Locally stable singular trajectories are the

�� HOW TO DETERMINE THE PHASE PORTRAIT �

attracting sets �attractors�� thus they correspond to observable modes of motion of thedynamical system�

The problem of determining the stability regions �domains of attraction� of the locallystable singular trajectories is also very important for numerous applications includingaircraft dynamics� These data su!ciently improve the understanding of the global phaseportrait of the system in question�

The numerous methods proposed in the literature for estimating the stability regioncan be roughly divided into two classes �� those using Lyapunov functions� and thoseusing qualitative analysis technique� The estimated �using Lyapunov functions� stabilityregion usually is only a subset of the true stability region� The qualitative methodsprovide the true stability region by determining its boundaries ��

For multi�dimensional dynamical system� there can exist more complex attractors�called strange attractors�� which bring about rather complex behaviour of the system�� Such a behaviour might be irregular and chaotic� The theory of strangeattractors is closely coupled with some problems in hydrodynamics �turbulence� forexample� ��

�� Stability regions

Suppose that xs is a stable equilibrium point of �� A set of points of the state space�for which phase trajectories started from these points approach the equilibrium point xsas t�� is called stability region or domain of attraction S�xs�� Then

S�xs� � fx � Rn � �t�x�� xs as t�gS�xs� is an open and invariant set� The boundary �S�xs� of the stability region S�xs�

is an invariant closed set� the dimension of the boundary is less then the dimension ofthe state space�

If the dynamical system has at least two equilibrium points� then the codimensionof the stability boundary of each of then is �� and the boundary is nonempty� The sameis true for stable closed orbits�

It is worth noting� that the boundary �S is the union of the stable manifolds W s�O�and W s�� of the unstable singular trajectories existing on the boundary �S�

The unstable equilibrium point xu belongs to �S�xs� if �see �g� ��

W u�xu��S�xs� ��

There are the conditions� that determine the number and the properties of the criticalelements �equilibrium points and closed orbits� belonging to �S �� In particular� underfairly general conditions the number of equilibrium points on �S is even� For the non�linear autonomous dynamical system containing two or more stable equilibrium points�the stability boundary �S must contain at least one saddle�type unstable equilibriumpoint On�� If� furthermore� the stability region is bounded� then �S must contain atleast one saddle point and one �source� equilibrium point Oo�n �see �g� ��

Thus� the stability boundary �S is the union of stable manifolds of the equilibriumpoints and closed orbits on the stability boundary �Oi � �S��i � �S��


Figure �� Stability region S

Figure ��

�� HOW TO DETERMINE THE PHASE PORTRAIT ��

�S ��i

W s�Oi��j

W s��i� ��

Using �� the following algorithm to determine the stability region through re�construction of its boundary is proposed�

�� Find all the equilibrium points and closed orbits�

�� Identify those critical elements whose unstable manifolds contain trajectories ap�proaching the stable equilibrium point �or stable closed orbit��

�� The stability boundary is the union of stable manifolds of the equilibrium pointsand closed orbits identi�ed in Step ��

Each of the algorithm�s steps in general can be e!ciently performed only usingnumerical methods�

Finding the whole set of the isolated critical elements is a nontrivial problem� Variousnumerical procedures may be used� with di�erent chance of success�

To accomplish Step � of the algorithm mentioned above� one can linearize the systemin the neighbourhood of the critical element and �nd the Jacobian and its eigenvectors�then select proper initial points and after that integrate vector �eld from each of thesepoints�

The stable manifolds W s�Oi� and W s��i� can be reconstructed by determining of anumber of their two�dimensional cross�sections�

�� Continuation method

The succession of the qualitative changes of the equilibrium point or a closed orbit due tothe variations of the values of parameters may include local bifurcations� the successivebifurcations sometimes form certain �chain of events�� Restoration of such a �chaine�greatly deepens the insight of the system�s global structure� For example� the so�called�catastrophic jumps� from one stable attractors to the others may be predicted�

The method of the �continuation� of some previously found steady�state solutionwhile varying the value�s� of parameter�s� is an extremely e!cient for such investigations�Since present�day versions of the method perform local analysis of the solution duringcontinuation and are able to handle such situations as branching and turning points�sometimes it is possible to �nd some �extra� solutions under initial conditions�

The examples of the bifurcation diagrams �typical to the simplest nonlinear systems�are shown in �g� �� They correspond to the so�called supercritical and subcriticalbifurcations� Solid lines indicate stable solutions� dashed lines correspond to saddle�type unstable solutions On�� or �n��

Under certain values of the parameter � �section � and �� or even �section � and�� di�erent solution may coexist�

Since saddle�type unstable points On�� belong to stability boundaries �S of thestable eqilibrium points� the distance between stable and saddle points �see �g� �� canbe used as a measure of the size of the stability region� For example� when the parameter


Figure �� Supercritical and subcritical bifurcations

�� HOW TO DETERMINE THE PHASE PORTRAIT ��

approaches the point B on �g� �from left to the right�� the domain of attraction of thepoint x � vanishes� At the same time abrupt loss of stability and �catastrophic jump�take place into one of the asymmetrical solutions x �� The return to the symmetricalsolution may be only near point A� Thus we have hysteresis�like behaviour of the system�

If the value of the parameter is somewhere between points A and B� then the transferfrom the state with x � to asymmetrical states with x �� is possible only if thestability boundary is passed� This may take place due to �nite perturbations�

When two parameters are varied� the continuation method supplies the two�dimen�sional equilibrium surface� Such surfaces may have various singularities known in thecatastrophe theory �� such as �fold��cusp�� butter�y� and others� A set of points�where the map of the equilibrium surface onto the plane of parameters becomes singular�is called bifurcational� The projection of the bifurcational set on the plane of parame�ters provides bifurcational portrait of the dynamical system� the regions with the samenumber of critical elements are identi�ed�

Fig� �� illustrates the surface with �cusp��type singularity� either one or threerobust equilibrium solutions may exist under given conditions� If c� � � the hysteresis�type behaviour is possible while varying c�� The transfer from the �upper� point on thesurface to the �lower� �at the conditions as in the point A in the plane of parameters�may proceed continuously �route I� or with a jump �route II��


Figure �� Cusp��type catastrophe

Chapter �

Basic numerical methods

This chapter is devoted to basic numerical algorithms and methods that are applied forcomputer�aided qualitative analysis of dynamical systems�

These algorithms are aimed at �nding sets of equilibrium and periodic solutions�computing eigenvalues and characteristic multipliers� sections of the boundary of thestability regions and bifurcational portraits in the state of parameters� Poincar�e pointmapping technique is used for investigation of periodic solutions and phase portrait ofdynamical system�

Many well�known algorithms are used� It is necessary to mention Runge�Kutta andAdams numerical methods for integration of systems of di�erential equations� SOLV Eand DECOMP �� algorithms for solving linear systems� QR�algorithm is a uniqueand extremely e!cient tool for �nding eigenvalues and eigenvectors of a matrix�

Sometimes stable limit sets of equilibrium and periodic solutions may be found usingmathematical simulation� Such a method is very time�consuming if the stability of thesolutions is weak� It is unable to �nd saddle�type critical elements� which form theboundaries of stability regions�

Newton�like iterative algorithms are more e�ective for computing equilibrium andperiodic solutions� For periodic and complex solutions they are used in conjunctionwith point mapping technique for �nding �xed points of maps� These algorithms arediscussed in detail�

Continuation method for computing solutions of nonlinear system depending on aparameter�s� is used in a number of new numerical methods� They include the algo�rithms for calculation of manifolds of equilibrium solutions and ��dimensional sections ofstability region boundaries� Continuation method is also used for tracking bifurcationalsituations� The original continuation method is described in this chapter�

�� Mathematical simulation

Mathematical simulation will always remain one of the most widely used tools for study�ing nonlinear dynamical systems� But it is very e!cient to use it in conjunction withdi�erent iterative numerical methods of qualitative analysis� thus speeding up the re�search program�

�

�� CHAPTER �� BASIC NUMERICAL METHODS

We prefer �xed�step integration methods for mathematical simulation� Runge�Kutta��th order method proved itself to be reliable and robust� Adams methods �order up to�� also show good performance� For rigid systems specialized methods can be applied�

Simulation is used for determining the trajectories in the state space� thus it allowsto collect data about phase portrait of the dynamical system� Mathematical simulationis a vital tool for �nding stable steady�state motions �equilibrium points� closed tra�jectories and even more complex attractors�� The expenses �integration time� dependon the level of stability� At the same time� reconstruction of the phase portrait of thedynamical system only by means of �blind� simulation �with numerous starting points�is an enormous problem even for the most powerful modern computers�

Simulation is used by various complex methods of the qualitative analysis �searchfor closed orbits� reconstruction of the stability boundaries etc�� Thus we may say� thatmathematical simulation is a necessary addition to the family of methods of the �direct�search of the critical elements of the state space�

�� Investigation of equilibrium points

For a dynamical system of the form

dx

dt� F�x� c� �x � Rn� c � Rm ��

equilibrium values of the state vector x and vector of parameters c satisfy to

F�x� c� � �x � Rn� c � Rm ��

�� Newton�like and gradient methods

It is possible to solve a nonlinear system of equations �� by means of Newton�Rafsonmethod� The method rapidly converges in the neighbourhood of the solution�

xi�� xi � r�F

�x

��

�xi� c� � F �xi� c�

where r � � �

Under certain conditions gradient method can also be helpful�

xi�� xi � k�F

�x

�

�xi� c�F �xi� c�

where coe!cient k is chosen using normalization and continuity �of the state vector�

conditions�Newton method has an excellent rate of convergence� but it is very sensitive to the

choice of initial approximation to solution� Gradient method may be used to expand theregion of convergence� So it is reasonable to combine these methods during numericalsearch for a solution thus greatly improving overall performance�

�� INVESTIGATION OF EQUILIBRIUM POINTS ��

�� The continuation method for �nding the dependence

of the solution of nonlinear system on a parameter

The method is intended for solving numerically the system of nonlinear equations

F� �x�� xn� c� � ��

Fn �x�� xn� c� �

or� using vector notation�F�x� c� � ��

where vector function F � Rn� x is a vector in n�dimensional state space� c is a scalarparameter�

Functions Fi� i � �� n are continuous and have continuous partial derivativeswith respect to all variables xi and the parameter c� In general� F is some smoothmapping �i�e� as often di�erentiable as needed� F � Rn�� Rn� Poincar�e point mappingfor example�

Figure �� The solution x�c� in extended state space

If the parameter c is changed continuously� then the solution x�c� set implicitly byequation �� will also change continuously �see �g� �� To establish the dependencex�c�� the following equation can be considered�

�F

�xdx�

�F

�cdc � ��


If Jacoby matrix �F�x is nonsingular then the solution curve x �c� may be obtained

by integration of system of di�erential equations

dx

dc� �

�dF

dx

��dF

dc��

The initial conditions for �� are in the form x �co� � xo� where F �xo� c� � ��F�x �

��Fi

�xi

�i� j � �� n is a Jacoby matrix� �F�c �

��F��c �

�F��c � � � � �

�Fn

�c

�

Matrix �F�x

becomes singular in turning points and branching points of the curve

xvc �see �g� �� In this case equation �� cannot be used�The solution curve xc in extended state space z � �x� c� may be parameterized using

length along curve as a parameter� thus avoiding singularities in the turning points�When following the solution curve �x� c�� the increments of the state vector and the

parameter will be de�ned by the components of the unit vector s tangent to the curve�

This unit vector is normal to the vector�rows of the matrix��F�x

� �F�c

� �F

�z� which are

gradients of the functions Fi�x� c�� i � �� n�

If there are no linear dependent vectors among �Fi

�z � i � �� n � i�e� rank of the

matrix �F�z equals n� then these vectors along with unit tangent vector s will de�ne a

basis in extended state space z� In this case matrix �

� �

BBBBBB�

�F��x�

�F��x�

� � � �F��xn

�F��c

� � � � � � ��Fn

�x��Fn

�x�� Fn

�xn�Fn

�cs� s� � � � sn sn��

�CCCCCCA�

� �F

�z

s�

�A ��

will be regular�

The elements of the unit vector s can be obtained by solving the system of equations�F�z s � �

Since vector s is normal to the rows of the matrix �F�z � the elements of s may be

de�ned from equations

si �An��i

det��

where An��i � algebraic complements to the elements of the matrix� last row�

det� �

vuutn��Xj��

A�n��j

Similar to eqs� �� the solution curve �x� c� in extended state space may be get indi�erential form� This form de�nes motion along curve with unit velocity

dz

dt� k�s

�� INVESTIGATION OF EQUILIBRIUM POINTS �

or ��

dxidt

� k�si� i � �� n

dcdt

� k�sn�� k�det

��F�x

det�

��

Initial conditions are z � zo� F�zo� � � t is a scalar parameter along the curve� Inte�grating numerically equations �� one can obtain a continuous branch of the solutioncurve �� in the extended state space z� As seen from the last equation in �� de�

terminant of the matrix �F�x

changes its sign in turning points� These points correspond

to the bifurcation points of dynamical system $x � F�x� c��When integrating the di�erential equations �� numerically� one inevitably makes

some error and deviate from the genuine solution curve�To build a continuation method stabilizing the motion along the solution curve ��

for a system of nonlinear equations �� append equations �� with linear equation

dF

dt� k�F � � ��

The solution of this auxiliary equation is in the form F � Foe�k�t� where positive

constant k� sets the speed of decreasing the error

kFk �vuutn��X

j�i

F �i

while increasing t�Making substitutions in the left side of �� and using z variables� we obtain

�F

�x

dx

dt��F

�c

dc

dt�

�F

�z

dz

dt� �k�F

Scalar product of this equation with s vector may be appended to eqs�� resultingin joint system of equations

�dz

dt�

� �k�Fk�

��

At the turning points on the curve when matrix �F�x

becomes singular� matrix �

remains regular� and system of equations �� also remains regular� Thus we canrewrite �� in the following form�

dz

dt� ��

� �k�Fk�

��

When integrating equations �� with k� � �� not only �following the curve�� butalso an asymptotic convergence to the solution curve starting from rather wide regionin z space is assured� If k� � then only convergence to the solution curve z�t� along

� CHAPTER �� BASIC NUMERICAL METHODS

subspace perpendicular to a �family� of curves F�x� c� � Fo is done� Fo being a constantvector�

Equations �� enable the computation of the desired solution with automaticelimination of the error kFk �� due to numerical integration procedure with �xed step t� and the errors in the initial conditions�

Assuming coe!cient k� �� t� where t is a parameter increment �i�e� step of

numerical integration along the curve�� the correction vector z can be obtained fromthe system of linear equations

�F�z z � �F

z�s � ��

Correction vector z is calculated using a condition that it is normal to unit tangentvector s�

To calculate the solution curve x�c� one can use a numerical method which includesthe following steps�

�� Start� Calculation of the initial point xo� co � F�xo� co� � either using knownmethods� with �xed c � co or using eqs� �� Iterations continue until errorjjFjj becomes less than some prede�ned value�In many cases convergence is much better if the parameter c is allowed to vary�

�� A unit tangent vector s is calculated using ��

�� Finite step along trajectory t is obtained from a condition� that increments ofall variables zi do not exceed some prede�ned values z�i � t � min

k� z�i sk��

�� The next point on a curve is calculated using equations �� After that a correc�tion is made �see �� until required accuracy is reached�

� Go to ��

�� Algorithm of systematic search for a set of solutions

SSNE algorithm �for Systematic Search for a set of solutions of Nonliner system ofEquations� is intended for �nding as many solutions as possible for a given N �th ordersystem� There is no guarantees of total success� but the method is a regular one thatproved itself to be e!cient for a number of nonlinear problems� We are sure that anyregular method able to �nd even only one previously unknown solution can be helpfulfor specialists� and this method provides a chance to �nd all the solutions%

This is a two�stage method� On the �rst �preliminary� stage there is a search forsome solution of nonlinear system using known methods �Newton method� for example��This stage will not be discussed any more�

We shall discuss in detail the second stage when algorithm tries to �nd other solutions�


Outline of the method

Suppose that the system of nonlinear equations is in the form

f �x� � where f � RN � x � RN ��

and let xo be a known solution of ��

f �xo� �

Consider the auxiliary system of equations Gk� �

Gk� �y� u� � where Gk� � RN�� y � RN�� u � R� ��

and function g k� �y� u�� that satisfy to

��

yi �

�xi � i kxi�� i � k

� i � � � � � N � �u � xk ��

��G

k�i �y� u� �

�fi �x� i kfi�� x� i � k

i � � � � � N � �g k� �y� u� � fk �x�

��

Thus system �� includes N � � equations depending on N � � variables yi i �� N � � and one parameter u�

From �� it follows� that if xo is a solution of initial system �� then �yo� uo��where

yoi �

�xoi � i kxoi�� i � k

� i � � � � � N � �uo � xok

��

is a solution of �� with initial conditions ��

Gk� �yo� uo� � g k� �yo� uo� �

��

Now a continuation method is applied to eqs�� with initial conditions ��

�� As a result one�dimensional spatial curve Lk�� expressing the dependence of solution

of auxiliary system on the parameter is computed� The curve is implicitly de�ned asfollows

Gk� �y� u� �

For every point of the curve Lk�� the value of the function g k� �y� u� is determined�

If there is a point ��y� �u� � Lk�� on the curve where function g k� �y� u� changes its

sign� then in the neighbourhood of it the point �y�� u�� Lk�� can be found that

g k� �y�� u��


If �see ��

x�i �

��

y�i i ku� i � ky�i�� i � k

� i � � � � � N�

then f �x�� If x� �� xo� hence a new solution of the initial nonlinear system ��is found�

Computing the points on the curve Lk�� and selecting those where function g k�

changes sign� one can �nd a number of new solutions of ��

Discussion of the algorithms behaviour

When the algorithm is able to �nd all the solutions& The following theorem is valid�

Theorem� If there exists an auxiliary system Gk�� such that it de�nes one�branch

curve Lk�� then the following is true�

�� The method will �nd all the solutions of the initial system

�� It is enough to consider only one auxiliary system Gk�

Proof� If Lk�� is a one�branch curve� it is easy to obtain it using continuation

method� Since all solutions of the initial system belong to Lk�� they all will be found�

This completes the proof�

Example

Consider the system of equations

�fi��k � ai��x� � � � �� ai�NxNfk � � �x�

where A � is a two�dimensional �N��N� matrix of rank N�� and � is an arbitrarynonlinear function�

The subsystem of linear equations Gk� � Ax de�ne a straight line in state space�hence the conditions of the theorem are satis�ed and SSNE method will �nd all thesolutions�

Both theoretical analysis and experience of application of SSNE show that since thecurve Lk�

� �for some k� sometimes don�t connect even solutions nearest to each other inthe state space� then one can greatly improve the e!ciency of the method by computingall the curves L

k�� k � � � � � N using corresponding auxiliary systems Gk� � k � � � � � N �

Is is also desirable to restart SSNE taking as initial every already found solution of�� thus generating new set of curves L

k��

The example shown in �g�� illustrates these recommendations� The system ofequations has four solutions� each solution is connected with two other solutions� Notethat it is impossible to reach more than one new solution moving along one curve L

k��

The example in �g�� shows the situation then the method is unable to �nd secondsolution despite the system of equations being very simple� To overcome this trouble the


x

x

x

x

C

D

B

A

L�� L

��

L��

L��

y

x

f� �x� y� � �x� �� x� ��f� �x� y� � �y � �� y � ��

� solutions�

A � x � � y � �

B � x � � y � ��C � x � �� y � �

D � x � �� y � ��

Figure ��

algorithm must be able to de�ne branching points on the curves Lk�� and to compute

all the branches extending from branching points�

The algorithm cannot cope with the situation shown in �g�� even using improve�ments mentioned above�

Thus we may say� that the SSNE algorithm� though not being a universal tool�may be very useful in applications� and we recommend it for researchers dealing withnonlinear systems�

�� Local stability and phase portrait

In the neighborhood of the equilibrium point xe the local phase portrait of the dynamicalsystem �� is determined by the linear approximation �� Thus the problem of localstability is reduced to the evaluation of the eigenvalues and eigenvectors of the Jacoby

matrix �F�x �xe� c� calculated at the equilibrium point xe�

Eigenvalues and eigenvectors are computed using QR�algorithm�

One can calculate the trajectories belonging to the invariant manifolds W s and W u

by integrating dynamical system� with starting points being displaced from xe alongcorresponding eigenvectors for the small distance �� For the stable manifold W s

p � eitherreverse�time integration is needed or the �reversed� system $x � �F�x� c� is integrated�For the unstable manifold W u

q � the ordinary integration procedure is used� The exam�ple of the selection of the starting points for integration �in the neighborhood of xe�


x

x

B

A

L�� L�

�

L��

L��

y

x

f� �x� y� � �x� �� y � ��f� �x� y� � �x� �� y � ��

� solutions�

A � x � � y � ��B � x � �� y � �

Figure ��

x

x

B

A

L�� L�

�

L��

L��

y

x

f� �x� y� � x� � �f� �x� y� � xy � �

� solutions�

A � x � � y � ��B � x � �� y � �

Figure ��

�� INVESTIGATION OF EQUILIBRIUM POINTS �

according to formulas�

xi � xe � �� cos�i � �� sin �i�� i � � � ��x � xe � ��

is shown in �g��

Figure ��

The value of � is chosen to satisfy to the accuracy condition �the starting point mustbelong to a small neighborhood of xe� and to the integration time limit �if a startingpoint is too close to xe� the integration may become time�consuming��

�� Manifolds of equilibrium solutions

A number of coexisting equilibrium solutions under the same conditions �i�e� values ofparameters� may be di�erent� One can consider regions in the space of the parameters


where under the conditions de�ned by each point of the region there exists the samenumber of equilibrium solutions�

Boundary surfaces of these regions at the parameter space correspond to bifurcationsof dynamical system� In most cases bifurcation results in �birth� or �death� of a pairof equilibrium solutions� one of these solutions always being unstable�

It is possible to reconstruct qualitatively the dynamics of the system when the pa�rameters are varied if manifolds of equilibrium solutions and corresponding bifurcationalsets can be determined�

If a dynamical system is in the form

dx

dt� F�x� c�� x � Rn� c � Rm ��

then a manifold of equilibrium solutions �equilibrium surface� Ee of the system �� is

de�ned as follows

Ee � fx � F�x� c� � � x � Rn� F � Rn� c �M � Rmg

When two parameters are varied� then it is convenient to express the dependence ofthe solution on parameters as a number of two�dimensional surfaces in ��dimensionalspace �two parameters and one of state variables��

A huge amount of information about system�s behaviour can be gathered throughanalysis of the smooth maps of these surfaces onto the plane of parameters�

The most common types of bifurcations are the so�called �fold�� cusp�� butter��y� and �swallow�s tail� bifurcation patterns well�known from catastrophe theory forgradient systems ��

The equilibrium surfaces can be calculated using either continuation method or byscanning across the plane of two selected parameters c� and c��

A set of bifurcation points Be of the equilibrium surface Ee is de�ned by a condition

Be �nx � G�x� c� � � x � Rn� G � Rn�� c �M � Rm

o

where

G �

BB�

F�x� c�

det

��F�x��

�CCA

If a dependence of one state variable on two parameters c� and c� is studied� thenBe is a manifold of dimension �� The projection of Be onto plane of parameters givesboundaries N�� which separate regions with di�erent number of equilibrium solutions�

Boundaries N� are also called bifurcational portrait or diagrams� These diagramsmay have cusp points� The curve Be and bifurcational diagrams N� can be computedusing continuation method�

Continuation method employs the extended Jacoby matrix �G�z

� where z � �x� c��

and thus requires evaluation of gradient of the determinant # � detkFxk of matrix Fx�

�� INVESTIGATION OF THE PERIODIC SOLUTIONS ��

Partial derivatives of the determinant of Jacoby matrix �#�zk

� k � �� n� � may

be expressed using partial derivatives of F�x� c� vector function

�#

�zk�

nXi��

nXj��

��Fi

�zk�xjAij� k � �� n� � ��

where Aij are algebraic complements to the elements of the Jacoby matrix��Fi

�xj

�i��nj��n

�

If there are no analytic expressions for partial derivatives �#�zk� they can be computed

using known multipoint numerical schemes�

�� Investigation of the periodic solutions and their

stability� Point mapping technique

Periodic solutions of the system of nonlinear equations

dx

dt� F�x� c�� x � Rn� c �M � Rm ��

corresponding to closed orbits in the state space� can be found using di�erent iterativenumerical algorithms�

The problem is reduced to a search for �xed point of appropriate mapping� Time�advance mapping �t�x� and mapping of hypersurface of codimension � onto itself maybe used�

�� Timeadvance mapping�

The requirement for a trajectory in state space to be closed results in a search for zerosof nonlinear vector function H

H�x� � � �� x�� x ��

where �� x� is� for example� time�advance mapping of initial point x with time interval

�Thus we need to �nd a period of the solution and some point x belonging to the

closed orbit ��The system �� comprise n equations depending on n�� unknowns� So it de�nes

a set of solutions x � � depending on one parameter�If x � � and H �� one can obtain equations in increments x and in the

vicinity of the closed orbit ��g� ��

��

�x� E

� x� F�� x�� H�x� � ��


Figure �� Time�advance mapping

�� INVESTIGATION OF THE PERIODIC SOLUTIONS �

Matrix ��

�x

��x�

� calculated in some point belonging to the curve � while being

the period� is called the monodromy matrix� Monodromy matrix always has a uniteigenvalue� and the corresponding eigenvector �� coincides with tangent vector to aclosed trajectory at x� �� F�x��

Consider a small shift along the closed trajectory� #x � �F�x�� The result of amapping applied to a new point �� x�#x� so we have

x�#x � �� x�#x� ��

hence

F�x� ��

�x� F�x� ��

Thus the matrix��

�x�E

in the left hand side of the equation �� is singular

in the vicinity of the closed orbit �� i�e� it has zero eigenvalue �with a correspondingeigenvector F�x��

For �nite solutions #x of the equations �� to exist� it is necessary that a vectorin the right hand side of the equations

��

�x�E

�#x � �H�x� �� F�� x��# ��

is normal to vector F�x�� which is eigenvector of ��

�x matrix �� i�e�

F� �x� �H�x� � � F��x��# � � ��

where F� denotes a transposed vector F�

This condition may be used for calculation of the increment of time period #

# � � F��x�H�x� �

F��x�F�� x��

Geometrically the relationship �� means that a �nal point on the trajectory ��x�must belong to a hyperplane "F � hyperplane "F being a set of vectors normal to F�x�vector�

If ��x� � "F � then period doesn�t change during convergence to a closed orbit�Thus hyperplane "F de�nes mapping with a constant period in a vicinity of a closedorbit�

It follows also that all other eigenvectors �i� i � �� n of the monodromy matrix��

�x lie in the hyperplane "F and hence are normal to vector F�x��

General solution #x of the system of equations �� will include a certain shift inthe "F hyperplane and some arbitrary displacement along vector F� In order to minimize

� CHAPTER �� BASIC NUMERICAL METHODS

absolute value of #x� it is reasonable to append equations �� with a condition thatrequires #x to be normal to the tangent vector F�#x � �

��

�x � E F�� x�� #x �H�x� ��

F� #

��

The resulting system of equations �� is regular� Thus one can remain in thehyperplane normal to a closed orbit during convergence to the orbit�

Eigenvalues �� n�� and eigenvectors F�x�� n�� of the matrix

��

�x

��x�

de�ned in a point belonging to a closed orbit provide an information about stability ofthe periodic solution and about the structure of the state space in the vicinity of theclosed orbit�

Monodromy matrix ��

�x is computed by means of numerical integration of equations

�� with initial conditions being varied successively with �xi� i � �� n

��

�x�

��x� �xi�� x� �xi�

��xi

�i��n

��

One can check the accuracy of computation knowing that always there is an eigen�value equal to � with corresponding eigenvector F�x��

�� Poincar�e mapping�

Poincar�e point mapping P� "� " of secant �n��dimensional hyperplane " �transver�sal to phase trajectories in the vicinity of closed orbit� can also be used for �ndingperiodic solutions� Let�s consider the properties of that mapping�

Transversal intersection of the trajectory in the state space with secant hyperplane "results in a sequence of points� One can distinguish these points according to directionof intersection with "� It is possible to consider the subsequence which includes pointswith the same direction of intersection fx��x��x�� g when trajectory leaves half�space"� for half�space "�� Full sequence form an orbit of bidirectional Poincar�e mappingfx��x��x��x�x�� g �see �g� ��

Also K�fold iterated Poincar�e mapping can be considered� In that case point of theeach K�th intersection in the chosen direction is included into the sequence� K�foldmapping Pk �x� is used to study periodical solutions with K�fold period�

" hyperplane is set using two vectors�

" � fx � h��x� xo� � g ��

where vector xo is some point on hyperplane� and vector h is normal to hyperplane "�

�� INVESTIGATION OF THE PERIODIC SOLUTIONS ��

Figure �� Poincar�e mapping

When integrating the trajectory one can determine in what half�space current point xis located�

"� � fx � h��x� xo� � g ��

or

"� � fx � h��x� xo� g ��

The point of intersection x with " can be re�ned using linear interpolation betweentwo successive point on the trajectory x� � "� and x� � "�

x � x�H�

H� �H�� x�

H�H� �H�

H� � h��x� � xo� �

H� � �h��x� � xo� �

��

To satisfy the required accuracy for intersection point

h��x � xo� � � ��

iterative procedure can be used� It comprises a backward step along trajectory �in

reverse time� of the size #t � t� � t� where

t � t�H�

H� �H�� t�

H�

H� �H��


and once more linear interpolation ��

It is possible to look for �xed points of the point mapping xk�� P �xk� or for zerosof vector function H�x� � P �x� � x� x � " using variations �x that keep the vectorx in the hyperplane "� A basis �i of linear independent vectors �i� i � �� n � �for " is to be constructed �rst� the orthogonalization procedure starting with normal tohyperplane vector h may be used�

Matrix �P�x eigenvalues do not depend on the choice of hyperplane " until it remains

transversal to the closed orbit�Let �i� i � �� n � � be eigenvectors of the matrix �P

�x� If �xk � ��

where �� are constants� and �� are eigenvectors forming complex eigenvector�c � �� i�� corresponding to a complex pair of eigenvalues �e

�i�� then the deviationwill be de�ned as follows

�xk�� cos�� sin�� sin�� cos��

Suppose that the other secant hyperplane "h is de�ned� vector h being normal toit� Let the trajectory to intersect with "h in points �x�k and �x�k�� If this trajectoryintersects with "F in �xk and �xk�� then

�x�k � �xk � �x�khF�h

� F�x�k�� xk�� x�k��h

F�h� F

��

The map �� is linear� hence

�x�k � �� cos�� sin�� sin�� cos��

��

where

��i � �i ��ih

F�h� F

It means that Jacoby matrix for the point mapping de�ned on "h hyperplane hasthe eigenvectors ��

�� and eigenvalues as the matrix for mapping de�ned on "F

hyperplane� These eigenvalues are the characteristic multipliers for the periodic solution�Jacoby matrix of the point mapping can also be computed using arbitrary variations

�x of initial point� so that the variations leaving the hyperplane "h are allowed� Theresult of the mapping P �x� �x� belongs to " hyperplane�

In this case the matrix of linearized map �P�x can be linked with monodromy matrix

��

�x��

�x��P

�x�Fh�

F�h��

Let F�� n�� be the eigenvectors and �� n�� be corresponding eigen�

values of ��

�xmatrix� One can check that F vector is an eigenvector of ��

�x� but

corresponds in this case to zero eigenvalue�

�� INVESTIGATION OF THE STABILITY REGIONS ��

Indeed��

�xF � F

�P�xF � F� Fh�

F�hF �

��

All other eigenvalues of �P�x � as well as those of��

�x matrix� correspond to charac�teristic multipliers of the periodic solution� One may get the eigenvectors using lineartransformation of this shift along trajectory

�P�x

��i � �i��i

��i � �i � ��ih��iF�h

��

Iterative search for a periodic solution is performed in accordance with �#x � xi��xi� �

�P

�x�E

�#x � �H�x� � �P �x� � x ��

Unlike �� the matrix in the left hand side �� is regular� The value of #x canbe obtained directly solving linear system of equations ��

Continuation algorithm may be e!ciently used to investigate the dependence of theperiodic solutions and their stability on parameters� zeros of the vector function

H�x� c� � P �x� c�� x � ��

can be studied�

�� Investigation of the stability regions

Every stable critical element �equilibrium point� periodic solution� etc�� has its owndomain of attraction or stability region� Domain of attraction is a set of initial pointsof state space� trajectories starting under these initial conditions lead to the criticalelement as t��

Investigation of stability regions is based on reconstruction of their boundaries� Inmany cases boundaries are formed by stable invariant manifolds generated by saddle�type equilibrium and periodic solutions ��

For planar dynamical systems the boundary of stability regions are either �stable�trajectories of the saddle�type equilibrium points �i�e� stable invariant manifolds of thesaddle points� or unstable limit cycles� They de�ne the boundaries of all the domainsof attraction�

If the dimension of the state space n � � then stable manifolds of trajectories W s ofthe codimension � passing through saddle�type equilibrium points of the type On�� orsaddle periodic solutions of the type �n�� are the most widespread�

Qualitative analysis of the critical elements of dynamical system is to precede to theinvestigation of stability regions� In particular� the analysis of the stable and unstable


invariant manifolds of critical elements of type On�� and �n�� must be done� First thosesaddle�type critical elements must be extracted for which unstable manifolds W u havetrajectories approaching to the stable equilibrium point or closed trajectory�

Figure �� The algorithm of �nding the boundary of stability region

Now consider the algorithm of �nding the boundary of stability region or stableinvariant manifolds of trajectories generated by saddle equilibrium point On�� g��

The boundary of stability region W sn�� in the state space can be outlined using a

number of cross sections by ��dimensional planes P�� The intersection between P� andW s

n�� results in curves SP� �

SP� � P� �W s

n��

These curves are the boundaries of the section of the stability region�Trajectories passing through the points of the curve SP

� belong to W sn�� and they

lead to the saddle point On�� as t�� If an initial point of the trajectory is movedaside from the curve SP

� �remaining it in the plane P�� then the trajectory will miss thepoint On��

One can introduce ��dimensional frame of reference on the secant plane P�� Now itis necessary to de�ne a measure of miss� Consider the hyperplane Ln�� passing through

�� INVESTIGATION OF THE STABILITY REGIONS �

On�� and tangent to the W sn�� Ln�� is a linear n � ��dimensional hyperplane with a

basis formed by stable eigenvectors of Jacoby matrix �computed at On��

The hypercylinder CR is de�ned as a set of points being on the same distance R fromthe straight line passing through On�� and orthogonal to the Ln�� hyperplane�

Thus the direction of the axis of the cylinder CR coincides with vector normal toLn�� hyperplane� In general� this direction do not coincide with unstable eigenvector ofJacobian matrix �i�e� with the direction in which trajectory Su

� extends out from On��

There exists some neighbourhood in the vicinity of a point on the curve SP� � P �� the

trajectories started from the points belonging to the neighbourhood cross the surface ofthe cylinder CR� The maximum size if that neighbourhood is connected with the radiusR of the cylinder� A distance from the intersection point �where trajectory crosses thesurface of CR� to the hyperplane Ln�� is used as a measure of miss�

The measure of miss H is

H�x� � �x� xe��ho ��

wherexo � vector of coordinates of On��ho � unit vector normal to Ln�� hyperplane�

Scalar function r�x� evaluates the distance between current point on the trajectoryx and the surface of the cylinder CR�

r�x� � kx� xo �H�x�hok �R ��

where kxk is euclidean norm of the x vector�

If r�x� changes its sign� this means that cylinder CR is crossed by the trajectory�The mapping of a point of a plane P� onto the surface of the cylinder CR is established

by the condition r�x� � � For every point belonging to some neighbourhood of the curveSP� � P� the mapping de�nes a scalar value H�xR�� which we shall denote as HR�� The mapping HR�� is made by numerical integration of dynamical system� the signof r�x� is checked during trajectory integration�

The problem of �nding the curve SP� �i�e� the section of stability region boundary

W sn�� by the plane P�� is reduced to �nding and continuation of solution of the scalar

equation

HR��

The equation depends on two parameters � and � � the coordinates of the initialpoint of the mapping with respect to the frame of reference de�ned on the P� plane�

To use algorithm e!ciently� the radius of the cylinder R is to be set properly� Theradius is to be small enough for the tangent hyperplane Ln�� to be a reasonable approx�imation of the invariant manifolds W s

n�� inside the cylinder CR� At the same time� theradius is to be large enough to minimize calculation time and to establish reasonabledomain of convergence�

Condition �� gives the approximate boundary of stability region� This boundarytends to the exact boundary of the section of invariant manifolds W s

n�� when R� �


The solution of �� is calculated using the continuation method� Continuationmethod does the convergence to a solution of �� and then a continuation of thecurve SP

� �

d�dt �

�kHRH�R �H

R

�H�R�

� � �HR�

�

d�dt �

HR � kHRH

�R

�H�R�

� � �HR�

�

��

The required accuracy is reached by means of proper selection of the coe!cient kand integration step #t in �� The partial derivatives H�

R and HR are evaluated

numerically ��

Chapter �

Application to aircraft �ight

dynamics

�� Problems to be solved

Modern airplane is a highly nonlinear dynamical system� It is clearly seen during spatial�� DOF� maneuvers and the �ights at high angle of attack�

Traditional methodology heavily relies on mathematical simulation of the aircraftdynamics by means of numerical integration of the full set of nonlinear equations ofmotion� and on linear stability analysis using frequency�domain and algebraic methods�It is not able to predict all the nonlinear phenomena exhibited by the airplane� forexample� in stall� spin or in maneuvers with high roll rates�

New methodology based on recent results in mathematics �Bifurcation theory� Qual�itative analysis of the nonlinear dynamical systems� is an e!cient� but delicate tool forstudying aircraft dynamics in these conditions� This methodology is now under devel�opment in the USSR� USA and France� Mathematical simulation is a good complementto it�

New methodology implies calculations of the equilibrium �ight conditions and os�cillatory motions �for example� an aircraft is in equilibrium during spin if the angle ofattack� sideslip and pitch� roll and yaw rates do not oscillate�� The in�uence of thecontrol inputs or �ight regime parameters on the aircraft equilibrium can be easily cal�culated� The data obtained enable the researcher to predict dangerous phenomena suchas stall and spin entries� and to develop a recovery technique�

Krit package was optimized for such calculations� It is especially useful for studyingthe steady�state �ight regimes �with or without oscillations� stable or unstable� andtheir dependence on certain factors� You may even use a set of mathematical models ofthe vehicle during a dialog with the package�

A unique feature of the Krit package is an e�ective set of procedures for an exact

calculation and analysis of the steady�state oscillatory �ight regimes using Poincaremapping technique�

Krit scienti�c package can be applied to the following problems in aircraft �ightdynamics�

��

�� CHAPTER �� APPLICATION TO AIRCRAFT FLIGHT DYNAMICS

� Calculation of the equilibrium conditions for the spatial �� DOF� maneuvers andhigh angle of attack regimes �stall� spin� regimes with rapid rotation�� Local sta�bility analysis of these regimes�

� Calculation of the oscillatory �ight regimes at high angles of attack � wing rock�oscillatory spin�� Local stability analysis of these regimes�

� Numerical simulation of the arbitrary spatial aircraft motion�

� Computation of the critical values of the control inputs and the �ight regimeparameters when an abrupt loss of stability and transfer to the dangerous regimesoccur�

� Calculation of the stability regions or stability to �nite perturbations�

� Determining control for spin recovery�

�� Equations of airplane spatial motion�

Suppose that an airplane is a rigid body with constant mass and inertia characteristics�Usually it has a plane of geometrical and inertia symmetry� The equations are developedusing ��at�earth� assumptions�

The following reference frames will be used�

� Earth inertial reference frame OXgYgZg is motionless with respect to earth� Ori�entation of an airplane and the position of its center of mass are set using thiscoordinate system�

� Body�axis reference frame OXY Z with the origin coinciding with center of massand axes along main inertia axes is used to de�ne angle orientation of the airplanewith respect to the axes of OXgYgZg coordinate system� Orientation may beexpressed using either Euler angles or leading cosines or quaternions�

Here the body X�axis extends forward out the vehicle�s nose� the Z�axis extendsout the right wing� and the Y�axis extends out the top of the vehicle� The �X�Y�plane is usually a plane of geometric symmetry� if the airplane has one�

� Stability�axis reference frame OXaYaZa� with the origin in the center of mass andwith X�axis aligned with the velocity vector and with Y�axis being in the body�axis�X�Y� plane�

The orientation of the stability�axis frame OXaYaZa with respect to body�axis frameOXY Z is de�ned by values of angles of attack � and sideslip �� The components of thevelocity vector with respect to body�axis frame are given by relationships

Vx � V cos � cos�� Vy � �V cos � sin�� Vx � V sin �

�� EQUATIONS OF AIRPLANE SPATIAL MOTION� �

Since arbitrary orientation of the airplane with respect to velocity vector is allowed�the following ranges for the angles of attack and sideslip are chosen

�� o � � �� o� � o � � � o�

Angle of attack is unde�ned if j�j � o�

Kinematic equations�

If Euler angles �i�e� pitch angle �� roll angle � and yaw angle �� are used to de�neorientation of the vehicle with respect to inertial frame of reference� then translationaland angular rates may be expressed as in the following

dR

dt� A ��V ��

andd�

dt� E ��

where

R � �xg� yg� zg�� inertial center�of�mass position vector

OXgYgZg�� Euler angle vector �

pitch� roll� yaw angles�V � �Vx� Vy� Vz�

� � velocity vector in body�axis frame�� x� �y� �z�

� � angular rate vector in body�axis frame�

A ��

��

cos� cos� � sin � cos� cos � sin � cos� sin ��sin � sin� �sin� cos �

sin � cos � cos � � cos� sin �

� cos� sin� cos� sin � cos� cos ��sin � sin� cos � � sin � sin � sin �

��is a matrix of directive cosines between OXY Z and OXgYgZg axes�

E ��

�� sin � cos �� tan � cos � tan � sin �

cos �cos� � sin �cos �

��is a transformation matrix relating rotational velocity components to the rotation rateswith respect to yaw� pitch and roll axes� This matrix is singular at the extreme valuesof the pitch angle � � ��

Since arbitrary spatial orientation of the airplane is allowed� the following ranges forpitch� roll and yaw angles are chosen

� o � � � o� �� o � � � �� o �� o � � � �� o

To eliminate singularities due to Euler angles ' �� directive cosines or quaternionsmay be used�

CHAPTER �� APPLICATION TO AIRCRAFT FLIGHT DYNAMICS

The usage of directive cosines�

The elements of the directive cosines� matrix A are the projections of the inertial frameOXgYgZg unit vectors l�h�s onto the axes of the body�axis frame OXY Z

A �

��

lx ly lz

hx hy hz

sx sy sz

��Since unit vectors l�h�s remain the same in the inertial frame� hence their full deriva�

tive with respect to time equals zero

(dldt

� � � l �

(dhdt � � � h �

(dsdt � � � s �

��

where(ddtis a derivative measured in selected rotating frame of reference�

Equations �� de�ne time derivatives of the nine directive cosines� These equationshave no singularities� The dimension of the system may be less if one of the unit vectoris calculated as a vector product of the other two unit vectors� for example� s � l � h�

The vector equations in �� have the scalar form of representation of the followingform�

dlxdt � lywz � lzwy

dlydt

� lzwx � lxwz

dlzdt

� lxwy � lywx

��

When integrating kinematic equations �� it is necessary to make corrections to thevalues of the directive cosines in order to satisfy to the six conditions of normalizationand orthogonality

�l � l� � � �l � h� � �l � s� � �h � h� � � �s � s� � � �h � s� �

The usage of quaternions�

When using quaternions for de�nition of the vehicle orientation in space� the singularityin kinematic equations is eliminated by introducing one auxiliary parameter� Quaternionis set using either four scalar parameters �q�� q�� q�� q�� or one scalar parameter q� and a��element vector q� Quaternion satis�es normalization condition

q�� q�� q�� q��

�� EQUATIONS OF AIRPLANE SPATIAL MOTION� �

Quaternion components change according to the following di�erential equations

$q� ��xq� � �yq� � �zq��

$q� ��xq� � �zq� � �yq��

$q� ��yq� � �xq� � �zq��

$q� ��zq� � �yq� � �zq��

��

or� using vector notation$q� � ��

�� q�

$q � �� q� q��

where �� q� � is a scalar product of two vectors�� q� � denotes vector product�

To de�ne initial values of quaternion components� one can use the following relation�ships

q� � cos �

�cos �

�cos

�� sin �

�sin �

�sin

�

q� � sin �

� sin�� cos

� � cos�

� cos�� sin

�

q� � sin �

� cos�� cos

� � cos�

� sin�� sin

�

q� � cos �

� sin�� cos

� � sin �

� cos�� sin

�

Current values of the Euler angles are computed using equations

tan� �� q�q� � q�q�� q�� q��

sin � � � �q�q� � q�q��

tan � �� q�q� � q�q�� q�� q��

When integrating numerically kinematic equations �� it is necessary to make cor�rections to the values of the quaternion components in order to satisfy to the normal�ization condition

q�� q�� q�� q��

Dynamical equations�

According to the assumptions made and using vector notation� Newton�s Second Lawmay be written as follows �here Q � mV and K is a momentum of impulse�

mdV

dt� Fa �Pl �Pr �G ��

dK

dt�Ma � �eng � �Pr �Pl� � K � J� �Keng ��

where

� CHAPTER �� APPLICATION TO AIRCRAFT FLIGHT DYNAMICS

Fa�Ma � vectors of aerodynamic force and moment�Pr�l � thrust of the right and left engines ��eng � �xeng� yeng� zeng� � vector of engines� displacement�G � force of gravity�J � inertia matrix�Keng � kinetic moment of engines� rotors�m � mass of the airplane�

Thus equations �� and �� constitute closed system of �� equationsprovided the dependencies of aerodynamic forces and moments and thrust are de�ned�

�� Various scalar forms of the motion equations�

Di�erential equations �� and �� written in scalar notation look di�erently dependingon selected frame of reference� One usually choose the frame taking into account thepeculiarities of the problem in question�

Equations of translational motion in body�axes frame�

Equations of translational motion �� with respect to body�axis frame are

m

�(dV

dt� � �V

�� Fa �P� �G ��

where(ddt is a derivative measured in selected rotating frame of reference�

Using scalar notation� equations �� can be written as

m�dVxdt � �yVz � �zVy

� cx

�V �

� S � P� �mghx

m�dVydt � �zVx � �xVz

� cy

�V �

� S �mghy

m�dVzdt � �xVy � �yVx

� cz

�V �

� S �mghz

��

where hx � sin �� hy � cos � cos�� hz � � sin � cos ��cx� cy� cz � aerodynamic force coe!cients�V � velocity magnitude�� air density�m � mass of the vehicle�S � wing area�g � gravitational acceleration�

�� VARIOUS SCALAR FORMS OF THE MOTION EQUATIONS� �

Equations �� are convenient for numerical computations� The magnitude of ve�locity and the angles of attack and sideslip are given by

V �qV �x � V �

y � V �z

� � arcsin�VzV

� �

��

�arcsin VyqV �x � V �

y

� Vx �

sign �Vy�

�� arcsin jVyjq

V �x � V �

y

�A � Vx

Equations of translational motion in stability�axes frame�

Stability�axis frame of referenceOXaYaZa suits well for structural analysis and analyticalstudies of equations ��

Consider unit vectors along axes of the stability�axis frame OXaYaZa � ia� ja�ka �

The components of these vectors with respect to body�axis frame are

ia �

��cos� cos �� sin � cos �

sin �

�� ja �

��sin �cos�

�� ka �

�� cos� sin �sin� sin �cos �

��

The velocity vector V can be expressed through unit vector ia tangent to the trajec�tory of motion�

V � V ia

hence full derivative of the velocity vector i�e� the acceleration of the center of mass is

dV

dt�

dV

dtia � V

diadt

Taking into account the relationship

diadt

�(diadt� � � ia

and

ia � ja � ka

and transforming double vector product� one obtains

dVdt � dV

dt ia � V d�dt cos �ja � V

d�dt ka � V �� ka�ja � �� ja�ka�

� dVdtia � V

��d�dtcos � � �� ka�

�ja � V

�d�dt� �� ja�

�ka

��


On substituting �� into �� and on making successively scalar productwith unit vectors ia� ja�ka the following scalar form of equations �� can bederived

mdVdt

� �Fa �P� �G�xa

cos� � d�dt � �za ��Fa �P� �G�ya

mVd�dt � �ya �

�Fa �P� �G�zamV

��

where

�xa � �� ia� � �x cos� cos � � �y sin� cos � � �z sin �

�ya � �� ja� � �x sin� � �y cos�

�za � �� ka� � ��x cos� sin � � �y sin� sin � � �z cos �

��

are the components of the rotation rate vector � with respect to stability�axis frame

OXaYaZa� Full vector of stability�axis frame rotation rate �a is

�a �

��

�xa � d�dt sin �

�ya � d�dt

�za � d�dtcos�

��

Equation of angular motion in the body�axes frame�

Momentum equations get the simplest form when written in body�axis frame OXY Z

m

�(dK

dt� � �K

��Ma � �eng � �Pr �Pl�

or� in scalar notation

Jxd�xdt � �Jz � Jy��y�z � mxqSl

Jyd�ydt

� �Jx � Jz��z�x �Keng�z � myqSl� �Pr � Pl� zeng

Jzd�zdt

� �Jy � Jx��x�y �Keng�y � mzqSba � �Pr � Pl� yeng

��

where mx� my� mz � total aerodynamic moment coe!cients �l� ba � wing span and mean aerodynamic chord�

The formulas above were derived assuming that only diagonal elements of inertiamatrix are nonzero and that the vector of kinetic moment of the engines� rotors goesalong OX axis�

�� VARIOUS SCALAR FORMS OF THE MOTION EQUATIONS�

��th order system of equations�

Closed subset of equations governing the parameters V � �� x� �y� �z� �� H��H � Yg� can be separated from full set of equations of spatial motion ��

d�dt� �z � ��ax � �y sin �� sin� � �ay � �x sin �� cos�� cos �

d�dt � az cos� � �ax sin � � �y� cos�� ay sin � � �x� sin�

dVdt � V �ax cos � cos�� ay cos� sin� � az sin ��

d�zdt �

Jx � JyJz

�x�y �� H� V �Sba

�Jzmz � Keng

Jz�y � �Pr � Pl� yeng

Jzd�ydt � Jz � Jx

Jy�x�z �

� �H� V �Sl�Jy

my �Keng

Jy�z �

�Pr � Pl� zengJy

d�xdt

�Jy � JzJx

�y�z �� H� V �Sl

�Jxmx

d�dt� �y sin � � �z cos �

d�dt � �x � tan � ��y cos � � �z sin ��

dHdt

� Vx sin �� Vy cos � cos � � Vy cos� sin � �

� V �cos� cos� sin �� cos � sin� cos � cos � � sin � cos� sin ��

��

where

ax �� H�V S�m ��cx � cp�� g

V sin �

ay �� H�V S�m cy � g

V cos � cos �

az �� H�V S�m cz �

gV cos � sin �

The remaining variables �� xg� zg can be determined integrating solutions of thesystem �� Equations �� weakly interact with equation for H� This interaction isdue to dependence of aerodynamic forces and moments� and thrust on air density � �H��

If one can neglect the variations of air density � during �ight� it is possible to considerthe �rst eight equations in �� separately� If the con�guration of the airplane �i�e�control surfaces� de�ections� remains the same then these eight equations form a closedautonomous system

dX

dt� F �X� ��

where X � �V� �� x� �y� �z� �� R�� s� �a� �s�P�

� � R� are the state vectorand control vector�

Equations �� are used for analysis of spatial motion when mutual in�uence oftrajectory and short period modes of motion is important� As an example one canmention spins and spirals�

Steady�state �ight regimes �when forces and moments are balanced� i�e� equilibri�um points of equations �� correspond to spiral motion regimes �when angular andtranslational motions are synchronized while vehicle spins along vertical axes�� Aircraft


spinning along vertical axes of inertial frame results from stationary conditions for pitch$� � and roll $� � angles� In that case $� � �� If there is no rotation � � � then thetrajectory will be a straight line�

��th order system of equations�

Equations �� can be simpli�ed when spatial manoeuvres with high rate of rotationare considered� When an airplane rotates along the velocity vector� during some initialtime interval the moments are balanced while the balance of forces is not yet reachedand the in�uence of the trajectory curvature on short period motion is negligible� It isassumed here that V � const and the in�uence of the force of gravity is small �one maytake g � o�� Then only �ve equations remain

d�dt

� �z � tan ��y sin �� x cos�� V S��cx � cp� sin ��m cos �

d�dt

� �y cos�� x sin��V S�m cz cos� �

�V S�m ��cx � cp� cos� � cy sin�� sin �

d�zdt

�Jx � JyJz

�x�y �� H�V �Sba

�Jzmz � Keng

Jz�y � �Pr � Pl� yeng

Jzd�ydt � Jz � Jx

Jy�x�z �

� �H� V �Sl�Jy

my �Keng

Jy�z �

�Pr � Pl� zengJy

d�xdt

�Jy � JzJx

�y�z �� H�V �Sl

�Jxmx

��Simpli�ed system of equations �� will be used for analysis of the angular motion

with high roll rates� stall regimes and high angles of attack excursions�Airplane dynamics using equations �� were studied in detail in �� for small

angles of attack and linear aerodynamic model�

Equations for vertical plane motion�

A set of equations used for investigation of motion in vertical plane is worth payingspecial attention� The motion is assumed to be ��at� � �x �� and �x � �y �� This condition can be satis�ed if there is no �or negligible� aerodynamic orgyroscopic coupling with lateral motion� i�e� cz � mx � my � Keng � � On substituting�x � �y � � � � � � equations �� results in

dVdt

��Fx � P � cos� � Fy sin�

m � g sin �

d�dt �

�Fx � P � sin �� Fy cos�mV � g

V cos �

d�dt

� �z

d�zdt

� �V �Sba�Jz

mz

��

where � � �� is a �ight�path angle�

When using equations �� di�erent range for pitch angle will be used� �� o � � �� o�

�� CRITICAL FLIGHT REGIMES� �

It is also possible to separate a set of equations �� into subsets for short periodangular motion and long period trajectory motion if there is a su!cient di�erence intheir speci�c time intervals�

�� Critical �ight regimes�

The maneuvering capabilities of the airplane are limited by some boundary� Outsidethis boundary lies the area of critical �ight regimes i�e� the regimes when uncontrollablemotions �linked with the loss of stability� develop� Pilot�s activities may either provokeor prevent the development of the instability� It depends on the task being performedand on the pilot�s skill in �ying the plane in such regimes�

A desire to enhance maneuverability inevitably results in entering the regions whereaerodynamic characteristics are nonlinear and dynamic cross�coupling between di�erentforms of motion cannot be ignored� As a result� the dynamical problems become highlynonlinear�

All the critical regimes can be divided into two big groups� according to the reasonsof dangerous behavior of the vehicle�

The regimes of �rst group arise when the stability margin is broken and the unstablemodes of motion begin to develop� These regimes are very di�erent� There may be mildor abrupt loss of stability� and the motion may be controllable or uncontrollable� Suchsituations take place during stall or spatial maneuvers with high rotation rate�

The second group comprise stable steady�state �ight regimes with supercritical valuesof parameters �especially angle of attack and rotation rate�� The roll�inertia rotationand spin regimes belong to this group� the abnormal reaction to the control inputs istheir characteristic feature� Determining of the conditions of existence and the methodsof recovery from these regimes is one of the most important questions when the problemof �ight safety is considered�

A variety of critical situations brought about the di!culties in classi�cation� Forexample� a number of di�erent forms of the loss of stability in longitudinal and lateralmotion is called �stall�� One may distinguish among them�

� a sudden rise of the angle of attack ��pitch up�� taking place at moderate and highangles of attack due to pitching moment nonlinearity�

� stable self�oscillating pitching motion at high angles of attack due to the develop�ment of the separated �ow ��bucking��

� �tumbling� in pitch�� divergent increase in the roll angle ��wing drop� or �roll o�� due to asymmetricalaerodynamic roll moment at high angle of attack or aerodynamic autorotation�

� divergent increase in sideslip angle ��nose slice� or �yaw o�� due to aperiodicinstability in yaw or asymmetric aerodynamic moments�

� stable self�oscillating motion in yaw and roll at high angles of attack �the so�called�wing rock��


� the loss of stability due to pilot�s actions� for example� when the attitude stabi�lization or target tracking is performed�

The classi�cation of the steady�state spins also comprise a number of terms�

� erected �normal� spin ny � � and inverted spin ny �

� steep spin � o � � o� and �at spin � � � o�

� stable equilibrium spin without oscillations�

� oscillatory spin� including regimes with �beats� and divergent oscillations�

� �falling leaf� and �deep stall� regimes� etc�

The possibility of the loss of stability an controllability at high roll rate �roll�coupledproblem� is also well�known� The phenomenon is especially dangerous for supersonicaircraft with elongated ellipsoid of inertia and high level of lateral �rolling� aerodynamicstability� which bring about cross�coupling of longitudinal and lateral motions�

Roll�inertia rotation is the uncontrollable rolling of the airplane �the trajectory beingapproximately horizontal� at angle of incidence below stalling� The rotation occursdespite neutral or anti�rotation de�ection of the aileron and rudder�

This regime sometimes is called �aeroinertia rotation�� autorotation�� etc�� one maysuppose that�s due to the desire of the authors to underline the nature of the regime�

Roll�inertia rotation and spin regimes have one vital common feature� In both caseswe have an autorotation regime� the di�erence is in the nature of the aerodynamicmoment that supports the rotation� Flight paths also di�er�

In spin� rapid deceleration is followed by quick transfer to steady descent alongvertical spiral� For a rapid roll regime at �relatively� small incidence� one may assumethe speed of �ight to be constant and to neglect the �ight�path curving due to gravity�for a certain period of time�� The problem then can be investigated using dynamicalequations of the �fth order �in some cases the linear representation of the aerodynamiccharacteristics can be used��

All these types of motion can be linked with features of the equations of spatialmotion and with changes in stability of their singular solutions�

�� Approximate stall criteria�

Aircraft high angle of attack excursions are connected with su!cient deterioration inthe lateral stability� The aerodynamic moments become markedly nonlinear functionsof angle of attack� sideslip and angular rotation rate� Small deviations of parameters mayresult in a loss of lateral�directional stability� aerodynamic autorotation etc� Unsteady

derivatives m��x� m

��y also depend on the angle of attack� They may su!ciently change

their values in the narrow range of the angle of attack� Of course� all that will causechanges in the character of the disturbed motion�

Assume the value of the angle of attack �de�ned from the condition of balance inlongitudinal motion� to be a parameter� The character of the disturbed lateral motion

�� APPROXIMATE STALL CRITERIA�

and its stability towards small perturbations may be evaluated using linearized equationsof motion of the vehicle� To do so� it is necessary to get the dependencies of the lateralaerodynamic derivatives on angle of attack using data from static� forced oscillation�rotary balance wind tunnel tests�

Applying this method of linear analysis� one can determine critical values of theangle of attack when the departures begin to appear due to the loss of stability� Lineartheory is unable to predict further development of the motion� At the same time itprovides simple and reliable stall criteria� that�s why it is widely used for the aircraftdesign ��

Approximate stall prediction criteria are widely used in early design stages� Thesecriteria result from simpli�ed analytic conditions of lateral stability and controllabilitybased on linearized equations of motion� The most important is the fact that these crite�ria use a limited amount of data �from static wind tunnel tests� available at preliminarydesign phase�

Loss of stability criteria�

Analytical formulas for evaluation of roots of characteristic equation of lateral motionwith reasonable accuracy can be obtained using a various known methods� They areused for the analysis of the in�uence of the aerodynamic parameters on the root loci i�e�on the stability and characteristics of transient motion ��

The will to ensure higher level of accuracy brings about complicated algebraic ex�pressions involving maximum possible amount of aerodynamic data� This is not the bestoption since exact values of the roots may easily be calculated numerically�

Consider simpli�ed equations for the quick angular modes of the lateral motion withrespect to stability�axis frame of reference� Assume V � const� � � const� C�

z � �gV � � The equations in the nondimensional form are

d�d � ��ya

d�ya

d � mx

ixsin ��

my

iycos�

d�xa

d� mx

ixcos� � my

iysin�

��

where

� � �m�Sl � relative density of the vehicle

d � dtm m �

m�SV � time scale

�xa ��xal�V � nondimensional roll rate

�ya ��yal�V � nondimensional yaw rate

ix �Ix

m �l �� iy �

Iym �l ��

� nondimensional moments of inertia


Assuming the symmetry of the airplane� the nondimensional lateral aerodynamicmoments mx� my are the odd functions of the parameters of motion

mx�y �� xa� �ya� � �mx�y ��xa��ya�

Retaining the terms of the series up to ��rd order� the moments can be expressed asfollows �for i � x� y�

mi �hm�

i �� k�i�� k�i��xa�

� � k�i��ya��i��

hm

�xai �� ki�

� � k�i��xa�� k�i��ya�

�i�xa �h

m�yai �� k�i�

� � k�i��xa�� k�i��ya�

�i�ya

��

Parameters kj as well as partial derivatives m�i � m

�xai � m

�xai are the functions of the

angle of attack �the cases of static hysteresis are excluded when the series expansion ofthe type �� is invalid��

The stability of the lateral motion in the linear approximation is evaluated using onlypartial derivatives m�

i � m�xai � m�xa

i � Nonlinear terms set by kj coe!cients may in�uencethe lateral stability if sideslip angle or roll rate �xa is nonzero� These terms may alsoa�ect the dynamics of the vehicle motion when oscillation amplitudes in the perturbedmotion are rising �ya ��

On substituting linear terms from �� into �� linearized equations of motionare

d

d

�� ya

�xa

��

a�� a�� a��a�� a�� a��

�� ya

�xa

��

�� a�a�

��

where

a�� m�

xixsin� �

m�y

iycos� a��

m�x

ixcos� � m�

y

iysin �

a�� m�ya

x

ixsin� �

m�yay

iycos� a��

m�yax

ixcos� � m�ya

y

iysin�

a�� m�xa

xix

sin� �m�xa

y

iycos� a��

m�xaxix

cos� � m�xay

iysin �

a� �m�

x

ixsin� �

m�y

iycos� a� �

m�x

ixcos� � m�

y

iysin�

The eigenvalues �roots� of the system �� are de�ned from the equation

�� A�� A�� A� �

�� APPROXIMATE STALL CRITERIA� ��

with coe!cients

A� � � �ix�m�x

x f�o� �m�yy f�o�

IxIy

� ��

ix

A� � ��iy

�m�

y cos��m�x sin�

IyIx

� � �

iy��

A� � � �ixiy

��m�

ym�xax �m�

xm�xay

��

ixiy��

��

where �f�o�� stands for �forced oscillations��

Since �� the terms comprising products of rotational derivatives were neglectedin the expression for A��

Nondimensional parameters � �� and �� in �� are de�ned in the manner similar tothe de�nition of �� in �� These three nondimensional values are called the coe!cientsof dynamic stability of the airplane in the lateral motion�

Coe!cients of dynamic stability are easy to calculate because they are determinedusing partial derivatives of the aerodynamic moment coe!cients� the latter being avail�able from common wind tunnel tests �see table ��

Coe��Aerodynamic partial derivatives tobe used

Wind tunnel testtechnique

� ��

m�xx f�o�

�� m�xx f�o�

�m��x sin �

m�yy f�o�

�� m�yy f�o�

�m��y cos�

Forced oscillations withrespect to OX and OYaxes of the body�axisframe of reference

��

m�x ��

m�y �� Static wind tunnel tests

��

m�xax ��

m�xay �� and also

m�x and m�

y

Rotary balance tests

Table �� Coe!cients of dynamic stability

The conditions of stability of the perturbed motion result from Raus criterion

A� � � A� � � A� � � A�A� �A� � �


They result in the simple stability criteria

� ��

� �� or ��

��

The coe!cients of dynamic stability also depend on the angle of attack� and theymay change their values and even their signs in the narrow range of � at high incidence�

Breaking of some condition in �� indicates the loss of lateral stability� hence itindicates stall� Corresponding angle of attack is called �stall angle� �st�

There are various types of the loss of stability� If only one condition �� is brokenthen a real root of �� becomes positive� Critical point is the branching point wheretwo stable steady�state rotation regimes appear while initial regime becomes saddle�typeunstable�

If a condition � �� is broken then a complex pair of roots becomes unstable�This leads to oscillatory motion provided the frequency of lateral oscillations �� p��does not diminish too rapidly� Such a situation takes place when

��

�� decreases�If simultaneously �� changes its sign and becomes positive� the oscillatory instability

is accompanied with sharp decrease in �� This may result in a breakdown of complexpair into two real roots moving along real axis� After that a branching �� may occur�Taking into account typical dependencies of the aerodynamic derivatives m�

x� m�y � m

�xax

at high incidence� such a behaviour may take place in the narrow range of the angle ofattack � � �o � �o� One may say that this type of instability is similar to the casewhen ��

The criteria �� and � �� govern the dynamic stability in yaw� The breakingof the third criterion �� indicates the loss of stability in roll or the appearance ofthe aerodynamic autorotation�

Yaw departure is typical to the aircraft with variable�sweep wings and highly sweptwings with low aspect ratio�

Instability in roll occurs for aircraft with unswept wings with high aspect ratio� Thereason is aerodynamic autorotation of the wing due to asymmetrical �ow separation athigh angles of attack �when Cy � Cymax��

Controllability criteria�

E�ectiveness of the lateral control is very important for pilot� Conventional controls�ailerons� di�erential stabilizer� etc�� at high angles of attack can hardly be classi�edas been pure lateral or lateral�directional� since aerodynamic moments m�i

x and m�iy are

of the same order� That�s why he modes of the controlled lateral motion are closelycoupled�

It is vital for the pilot to have proportional relationship between aileron de�ectionand angular rotation rate �xa �or angular acceleration $�xa��

Steady�state value of ��xa�e during small de�ections of some control � is evaluatedfrom the condition of the balance of aerodynamic moments� It is possible to neglect

�� APPROXIMATE STALL CRITERIA� ��

inertial moments and nonlinear aerodynamic terms in the equations of the spatial motionand to take away terms with high powers of �xa� This results in simpli�ed relationship

��xa�e � ��

� ��

where �� m�

ym�x �m�

xm�y

�If the perturbed roll motion is stable �� then there will be a �direct� reaction

of the airplane in the response to the control input provided the condition �� issatis�ed� Nondimensional parameter �� is similar to LCDP �Lateral Control DepartureParameter� parameter ��

This criterion is also coupled with stall prevention problem� since reversed reactionto the aileron de�ection ��a changes sign� may bring about instability in controlledlateral motion�

Consider the regime when pilot tries to keep roll angle� Reversed reaction may causeyaw departure ��

Assume that pilot ideally compensates aerodynamic rolling moments arising in theperturbed oscillatory motion� As a result�

$�xa � �xa �

Current control de�ections are de�ned from the equation for $�xa �� Excluding �from the �rst two equations� one obtains the characteristic equation for perturbed yawmotion

)� �A�$� � �A��

where

A� �m�ya

y m�x �m�ya

x m�y

iym�x cos� � ixm

�y sin�

A� ��

iym�x cos� � ixm

�y sin�

� where �� m�

ym�x �m�

xm�y

�

It follows from �� that the parameter �� de�nes the frequency of the yaw per�turbed motion for this regime� Hence the reverse reaction �� results in aperiodicinstability�

The stability and controllability criteria mentioned above can be used to evaluatethe angle of attack when stall becomes possible �see table ��

For modern military aircraft with automatic control switched o�� the critical angleof attack is usually due to the condition of the reversed reaction in roll� Correspondingestimation using wind tunnel data is consistent with �ight tests results ��


CriteriaPossible changesin dynamics

�� m�xm

�xay �m�

ym�xax � Aperiodic loss of stability and

post�stall gyration

R � �� where

�� m�y cos� �m�

x sin�IyIx

� �� m�xx f�o� �m�y

y f�o�

IxIy

Oscillations

��

� �� Aperiodical loss of stability�Oscillations�

�� m�

ym�x �m�

xm�y

��

Reversed reaction to control inputin roll� Pilot�induced aperiodicinstability�

Table �� Stability and controllability criteria

�� STEADY�STATE SPIRAL MOTION �

�� Steady�state spiral motion

Steady�state spiral motion with all forces and moments balanced is an important kind ofspace motion �� There is stationary �steady�state� air �ow conditions since angles ofattack and sideslip� velocity vector and angular velocity vector are constant in body�axisframe of reference OXY Z�

Using body�axis frame and vector notation� the equations of motion are

(dvdt� � � v � Ra

m� gh

(dhdt� � � h �

J d�dt

� � � J� � Ma

��

whereRa � total vector of aerodynamic forces and thrust�Ma � full aerodynamic moment vector�� angular velocity vector�h � local vertical unit vector�J�m � inertia matrix and mass of the vehicle�g � gravitational acceleration�

Steady�state equilibrium regimes of motion of the system �� correspond to thesituation when

(dvdt

�(dhdt

� � d�dt

� �

These are regimes of spiral motion with synchronized rotation

� � ia �c�� h

Fr��

� � h � �

� � J� � m� �

��

where

� � � l�V � vector of nondimensional angular velocity�

� � �m�Sl � normalized density of the vehicle�

Fr �r�V �

gl� Froude number�

J � �Jml�

� normalized inertia matrix�

c � vector of nondimensional coe�� of aerodynamic forces�m � vector of nondimensional coe�� of aerodynamic moments�ia � unit vector going along the velocity vector�

The �rst equation in �� is a condition of the balance of forces� the second re�quires the axis of spiral to be vertical� and the third is the condition of the balance ofaerodynamic and inertial moments�


If there is no rotation � � then the trajectory is a straight line and the followingconditions are satis�ed

c � ��hFr�

�

m �

In the presence of rotation� one can get a vector product of the �rst equation from�� with �� Taking into account second equation� this results in

� �� ia �� c��

In the stability�axis frame vectors c� � and ia are

c � �cxa� cya� cza�� xa� �ya� �za�

�ia � ��

Using scalar notation for equations �� with respect to stability�axis frame� thecomponents of nondimensional angular velocity �ya and �za are expressed as follows�note their dependence on nondimensional rate of rotation along x�axis �i�e�velocityvector� �xa�

�ya �cyacxa � ��cza�xa

cxa� � ��

xa

�xa

�za �czacxa � ��cya�xa

cxa� � ��

xa

�xa

��

Now angle � between local vertical unit vector h and the velocity vector ia can bedetermined

cos � � sign�cxa�

scxa

� � ��xa

cxa� � cya

� � cza� � ��

xa

sin � �

scya

� � cza�

cxa� � cya

� � cza� � ��

xa

�

��

The components of h are

hxa � cos � hya ��ya cos��xa

hza ��za cos ��xa

��

It is convenient to analyze the kinematics of spiral motion using rotating frame ofreference OXsYsZs� with one of the unit vectors identical to h� Other unit vectors r� �rotate along vertical axis so that r points to spiral axis and � is a vector product of the�rst two vectors� Vectors r and � may be expressed through h and ia �

r � h� iasin � � � h� ia � h

sin �

Nondimensional radius of the spiral is

R � �Rl� �V�j�j l �

sin �j�j

�� STEADY�STATE SPIRAL MOTION ��

Using eqs� �� and the relationship �xa � j�j cos� � this leads to

R �sign�cxa�

q�cya

� � cza��cxa

� � ��xa�

�xa�cxa� � cya

� � cza� � ��

xa�

��

The magnitude of the velocity can be obtained from the condition on the balance offorces in the direction of vector ia � Scalar product of the �rst vector equation in ��with ia is

cxa ��hxaFr�

��

It follows from �� that

Fr� � ��jcxaj

scxa

� � ��xa

cxa� � cya

� � cza� � ��

xa

V � � glFr�

�

��

If coe!cients of aerodynamic forces do not depend on velocity �this is true for low�speed subsonic �ight without thrust� then the magnitude of the velocity can easily bederived from ��

The values of ��R and V �� uniquely determine the spiral trajectory�They primarily depend on coe!cients of aerodynamic forces and �xa� i�e� on �� xa

and �i�For a given � � �� xa and �i a condition of the balance of aerodynamic and inertial

moments �� is to be satis�ed� Since � � �h� it follows that

�� h� Jh� �m ��

In body�axis frame equation �� looks as follows

mx � ��xa�Jz � Jy� hyhz

cos� �

my � ��xa�Jx � Jz� hzhxcos� �

mz � ��xa�Jy � Jx� hxhy

cos� �

��

where

hx � cos�

�cos� cos � �

cyacxa � ��cza�xa

cxa� � ��

xa

sin�� czacxa � ��cya�xa

cxa� � ��

xa

cos� sin �

�

hy � cos�

�� sin� cos� � cyacxa � ��cza�xa

cxa� � ��

xa

cos��czacxa � ��cya�xa

cxa� � ��

xa

sin� sin �

�

hz � cos�

�sin � �

czacxa � ��cya�xa

cxa� � ��

xa

cos �

�

are the leading cosines of the vertical unit vector in body�axis frame OXY Z�


The conditions of balance of moments �� in the direction of the vertical unitvector h and the vector of kinetic moment Jh� result in a set of equations independentof inertial moment

mxhx �myhy �mzhz �

Jxmxhx � Jymyhy � Jzmzhz � ��

One can replace the second equation in �� with its linear combination with the�rst equation�

�Jx � Jz�mxhx � �Jy � Jz�myhy � ��

To express thoroughly kinematics of spiral motion it is necessary to de�ne theorientation of the vehicle i�e� transformation from body to rotating axes� One can do itby means of three angles �� Two of them coincide with pitch and roll angles� andthe third is analogous to yaw angle�

To determine � and �� the leading cosines of the vertical unit vector h with respectto body axes are used

hx � sin � hy � cos� cos � hz � �cos� sin �

as well as those with respect to stability axes OXaYaZa� hxa� hya� hza ��

Accounting for �� this leads to

sin � � sign �cxa� ��cxa

� � ��xa� cos� cos � � �cyacxa � ��cza�xa� sin �� czacxa � ��cya�xa� cos� sin �q

�cxa� � ��xa��cxa� � cya

� � cza� � ��

xa�

tan � �

�cxa� � ��

xa� sin � � �czacxa � ��cya�xa� cos �

�cxa� � ��

xa� sin� cos� � �cyacxa � ��cza�xa� cos�� czacxa � ��cya�xa� sin � sin �

��Yaw� angle � is an angle of the turn along vertical axis� between unit vector r of

rotating frame and the projection of the body x�axis unit vector i onto the horizontalplane� Thus

cos� cos� � �i � r� sin� cos� � �i � � �and correspondingly

tan� �i � h� ia � h

i � h� ia��i � ia�� i � h��ia � h�

i � h� ia��

In the stability�axis frame it yields

tan� �cos� cos� � hxa sin �

hza sin �� hya cos� sin ��

Now consider the dependence of the kinematic parameters of the spiral motion onthe rate of rotation along velocity vector �xa�

�� STEADY�STATE SPIRAL MOTION �

It follows from �� that the range of variations of �ya and �za is limited at allvalues of �xa�

The character of these dependencies can be seen from simpli�ed formulas� Assuming�for example� ��xa � � results in

�ya � �cyacxa

�xa �za � czacxa

�xa�

Or� if ��xa � � then

�ya � �cza�� za � cya��

Radius R and angle � become less while the value of �xa rises� It is possible to getapproximate expressions for them in extreme situations ��xa � � and ��xa � ��

Decomposing �� versus small parameter ��xa � � and retaining only seniorterms one receives

R � sign�cxa�qcya

� � cza�

�xa�cxa� � cya

� � cza��

sin � �s

cya� � cza

�

cxa� � cya

� � cza�

��

xa

cxa� � cya

� � cza�

� ��

Similarly� when ��xa � ��

R � sign�cxa�qcya

� � cza�

��xa

sin � �qcya

� � cza�

��xa

��

The relationships �� can be simpli�ed when ��xa � � �it is assumed thatcza � ��

sin � � sign�cxa��cos� cos� � cya

��xacos� sin �

tan � � tan�sin� �

cya��xa cos

� � sin�

tan� � sign�cxa��cos� sin �sin� �

cya � cxa sin��xa sin�

��

Spiral trajectories with small radius �R � �� high angle of attack �� o� and rota�tion rate �xa � � � correspond to spin regimes� That�s why the formulas ��are of value for calculations of steady�state spins� kinematic parameters�

Balance of moments equations �� can be simpli�ed greatly in the case ofsmall�radius spirals� Retaining only senior terms of decomposition� they may be written


in the form

mzbal � ��

xa�Jy � Jx� cos� sin � cos� � �

�mx cos��my sin�� cos � �mzbalsin � � �

� �Jy � Jx� cya�xa sin ��

mx cos��Jy � JzJz � Jxmy sin� �

��

Equations �� are suitable for evaluation of �� xa for equilibrium spin providedthe de�ections of control surfaces are given�

Note that the results obtained may be also used for estimation of spiral motionparameters in the conditions of high�rate rotation along non�vertical axis and ratherhigh speed� so that Fr� � � Here one can neglect the in�uence of the force of gravity�and the parameters of motion can be estimated using ��

The �mean� direction of the velocity vector tends to become vertical due to the forceof gravity� The rate of such process is inversely proportional to the speed of �ight�

�� Aircraft maneuvers with rapid rotation

The analysis of the behavior of the airplane during maneuvers with high rotation rates�inertia or roll�coupled problem� now is the obligatory part in the investigation of thedynamical properties of the vehicle�

If the dependence of the aerodynamic coe!cients of the forces and moments on theparameters of motion is linear and there is no �ight control system� the approximatemethod can be used �� This method allows to get approximate analytical relationshipsfor steady�state parameters of motion and bifurcational values of control parameters�The obtained data may be used for choosing regimes that require extensive mathematicalsimulation�

Modern airplanes are equipped with automatical control systems able to change thedynamical properties radically� It is necessary to take into account the in�uence of the�ight control system and the nonlinearities in the aerodynamic characteristics� but theapproximate method �� is unable to do so�

Consider the equations of the aircraft dynamics in the general form�

dx

dt� F�x� ��

where x � �� x� �y� �z�� h� �e� �rin�� and F is determined by �� Now add

the equations governing the behaviour of the control system�

d�

dt� G�x�

dx

dt� ��R� ��

where R � �x�� x�� x � is a vector of de�ections of the stick and rudder pedal� and the

vector�function G is determined by the control laws� The function G becomes equal tozero at full de�ection of the levers�

Thus� the joint system of equation �� is to be used for determining the steady�state rotation regimes of the airplane

�� AIRCRAFT MANEUVERS WITH RAPID ROTATION ��

F�x� ��

G�x� � ��R� � ��

The elements of the vectors x and � are the unknowns� and vector R de�nes the setof control parameters�

The characteristic features of the equations �� are the limits on the valuesof some state variables �i�e� the de�ections of the control surfaces � � ��hmax �h ��hmin� j�ej � �emax� j�hj � �hmax�� Some additional solutions of the system ��may appear because of these restrictions�

A bulky calculations are needed to investigate the dynamics inside the �ight envelopein the plane H�M � That�s why it is reasonable to perform rather limited qualitativeanalysis for every given �ight regime� and to investigate in detail only the special cases�

The qualitative analysis must be accompanied by extensive mathematical simulationin the critical regions of the �ight envelope� Usually it is enough to simulate �single ordouble� �roll� or Immelmann maneuvers�

In fact� such an approach solves the problem of stability of the motion during �niteperiod of time� The estimates of the bifurcational values of control parameters might bere�ned accounting for the dynamical lag�

Krit scienti�c package incorporates special procedures for the calculation of theparameters of motion at full de�ections of the levers R� and also for the calculation ofthe bifurcational values of control settings and corresponding parameters of motion�

The procedures support the following methodology of the investigation of the regimesin question�

�� The �ight regime is chosen i�e� Mach number� altitude� normal acceleration� andalso lateral control lever to be used�

�� The initial steady�state regime is calculated by solving the equations �� withadditional conditions ny � ntrimy and �x � wy � � The trim de�ections of levers�the values of the parameters of motion and local stability are determined� Thelimitations on normal acceleration� angle of attack and the de�ections of controlsurfaces are also taken into account�

�� The steady�state �trim� values of the parameters of motion and local stability areevaluated at the various de�ections of the selected lateral control lever� The datacorresponding to the critical situations �the appearance of oscillatory instabilityor �turning point� bifurcation� roll�coupled rotation� are stored in permanentmemory� The maximum available roll rate is determined at full lever�

�� Steps �� are repeated for all the regimes inside �ight envelope �with the givenincrements of H� M and normal acceleration��

The dynamical response to extreme control inputs provides information about im�portant aspects of aircraft behavior� the data obtained allow to reveal the �ight regimeswith signi�cant coupling of motions �high levels of lateral acceleration� and degraded


controllability in roll� oscillatory unstable regimes and the regimes with abrupt changesin parameters of rotation�

It is worth noting that dynamic response of the airplane to lateral control input alsodepends on the duration of the control input� That�s why one needs to consider thecontrol impulses of the reasonable duration �say� enough to perform single roll��

�� Computing maneuver limits�

Approximate linear criteria of stability and control departure give approximate maneuverlimits� But it should be noted that these criteria are useful indicators only when � isrelatively small and when unsymmetrical rolling and yawing moments at zero � are alsosmall�

When large lateral�directional asymmetries exist at zero sideslip �as it occurs for longpointed bodies�� more complicated analysis is required � taking into account the strongaerodynamic and kinematic coupling between longitudinal and lateral motions�

Measurements of static stability parameters and control surface e!ciencies are re�quired in the whole control surface de�ection and in the entire �� ranges for trimmingand stability analysis�

Such analysis may be done using the following assumptions� � � const� � � const��x � �y � �z � �

Computer program is scanning the plane �angle�of�attack � � sideslip angle�� For agiven values of these angles�

�� The attempt is made to reduce to zero the derivatives of angle of attack� sideslipangle� roll rates and speed using speed� pitch and roll angles and the de�ectionsof elevator� aileron and rudder as variables� The rotation rates are assumed to bezero�

�� If trim is possible� the stability of the equilibrium regimes using �th order equa�tions is determined and corresponding code is displayed on the graph�

Fig�� shows the examples of calculation of the �stability plot� for aerodynamicallycoupled longitudinal�lateral motions� the coupling is due to asymmetries and nonlinear�ities� with no rotation� The region where trim conditions are possible� is marked usingdi�erent marks�

The maximum available control power and the loss of stability can be the limitingfactors which determine maneuver boundaries of the aircraft� Thus we can evaluate themaneuver limits resulting from aircraft stability conditions and control degradation�

An accurate knowledge of the control characteristics over a large �� range is alsoneeded for the design of departure and spin prevention automatic systems�

�� Analysis of spin characteristics

Numerical methods of determining the parameters of the steady�state regimes and theirstability� as well as mathematical and piloted ground�based simulation play an important

�� ANALYSIS OF SPIN CHARACTERISTICS ��

Y XMIN� �� XMAX� �� DX� ��

��

�� I I I I I ��

�� I I I I I ��

�� I I I I I I

�� I I I I I I

��

�� I I I I

�� I �� I I I

�� I �� I I I

�� I �� I I I

��

�� I �� I I

�� I ��LLL I I

�� I ��LLLLLL� I I

�� I ��LLLLLLL I I

�� LLLLLLL��

�� I ��LLLLLLL I I

�� I ��LLLLLL� I I

�� I ��LLL I I

�� I �� I I

��

�� I ��I I I

�� I ��I I I

�� I �� I I I

� �� I �� I I I

��

�� I I I I I I

�� I I I I I I

�� I I I I I ��

�� I I I I I ��

��

Figure �� Example of the �stability plot� for coupled longitudinal�lateral motions


role in the investigation of spin regimes�These problems can be solved provided consistent mathematical model of the aircraft

motion exists� The model relies on adequate description of the aerodynamic character�istics at high angles of attack and high rotation rates� The experience with modern�ghter airplanes shows that it is possible to make such models by using the results ofthe di�erent wind tunnel tests �methods of forced oscillations� steady�state rotation� inconjunction with �ight tests results�

Equilibrium spin parameters can be e!ciently computed using �nondimensional�equations �� the values of �� xa can be determined� The coe!cients of the aero�dynamic forces cx� cy and moments mx�my�mz are de�ned as smooth functions derivedfrom the data obtained from steady�state rotation tests�

For each set of de�ections of the control surfaces all possible solutions of the nonlinearsystem of equations can be determined� Krit scienti�c package provides the means todo this�

Equations �� can be simpli�ed taking into account small values of the sideslipangle at equilibrium spins� Thus it is possible to linearize the dependences of the rolland yaw moments on sideslip

mx � m�x�� #mx�� xa� �e� �rin�

my � m�y �� #my�� xa� �e� �rin�

and to exclude � from equations �� The two last equations from �� are now

reduced to a single equation

#mx cos�P� �#my sin�P� � P� � ��

where

P� �Jy � JxJy � Jz

�m�y sin� � �xacyamz

�Jy � Jx�baml�

� sin ��

P� �Jy � JxJy � Jz

�m�x cos��mz

bal

P� � �xacyaJy � Jxml�

� sin ��m�

xJz � JxJy � Jz

�m�y sin �

First equation from �� and equation �� form a nonlinear system of the secondorder� which depend on two parameters � and �xa� One can examine this system byplotting zero�value lines for both right�hand side functions� The crossings of these linesgive a full set of solutions of the system in question� For example� the �� xa� planecan be scanned and the signs of the right�hand sides can be checked� Krit packageincorporates suitable command� the example of the calculation is shown in �g� ��

These solutions are good �rst approximations for precise calculation of all possibleequilibrium spin regimes and their stability using ��th order equations ��

Krit package supports such calculations� including the investigation of the changeof the spin regime due to variations of parameters� Figure �� shows the example ofthe calculation of the dependence of eigenvalues �determining local stability� on aileronde�ection�

�� ANALYSIS OF SPIN CHARACTERISTICS �

XMIN� �� XMAX� �� DX� ��

��

�� I I I I I I �� I I I �� I

�� I I I I I I �� I I I �� I

�� I I I I I I �� I I I �� I

�� I I I I I I �� I I I �� I

��

�� I I I I I I �� I I I �� I

�� I I I �� I I �� I I I�� I

�� I I I �� I �� I I �� I

�� I I I ��I �� I �� I I �� I

��

�� I I I �� I �� I �� I I ��I I

�� I I I �� I �� I �� I I �� I I

�� I I I �� I �� I �� I I �� I I

�� I I I �� I �� I �� I �� I I

��

�� I �� I �� I ��I I �� I ��I I I

�� I ��I I �� I �� I �� I I I

�� I �� I I I �� I �� I I I

�� I I I I �� I�� I I I I

��

�� I I I I I �� I �� I I I I

�� I I I I �� I �� I I I I

�� I I I I �� I �� I I I I

�� I I I �� I �� I �� I I I I

�� E � ��

�� I �� I I I�� I �� I I ��

�� I �� I I �� I I �� I �� I I

�� I �� I I �� I �� I �� I I

�� I ��I I �� I �� I ��I I I

��

�� I I �� I �� I ��I I I

�� I I I �� I�� I ��I I I

�� I I I I �� I �� I I I

�� I I I �� I�� I �� I I I

��

�� I I I ��I �� I �� I �� I I

�� I I I �� I �� I �� I �� I�� I I

�� I I I �� I �� I �� I �� I �� I I

�� I I I �� I �� I �� I �� I I

��

�� I I I �� I �� I I �� I �� I

�� I I I ��I �� I I �� I �� I

�� I I I �� I I �� I �� I

�� I I I ��I I I �� I I�� I

��

�� I I I I I I I �� I I �� I

�� I I I I I I I �� I I �� I

�� I I I I I I I �� I I �� I

�� I I I I I I I �� I I �� I

��

Figure ��


Figure ��

Appendix A� Matching of US and USSR notation ��

Appendix A

Matching of US and USSR notation

USA USSR

X X

Y Z

Z �Y

u Vxv Vyw Vz

V V

p �xq �zr ��y

� ��

CX Cx

CD Cx

CY Cz

CZ �Cy

CL Cy

USA USSR

Cl mx

Cm mz

Cn �my

Cl� m�x

Clp m�xx

Clr �m�yx

Cl�a m��x

Cl�r �m�Nx

Cn� �m�y

Cnr m�yy

Cnp �m�xy

Cn�r m�Ny

Cn�a �m��y

CY � C�z

CY �r �C�Nz

CY �a C��z

Cm�e m�Wz

Cm�h m�z

Cmq � �m�zz

S Sb lc bA

USA USSR

p � p�b

�V�x �

�x l�V

q � q�c

�V�z�� z bA

�V

r � r�b�V ��y �

��y l

�V

* *� � �b

�V � � �l�V

�a ��r �N�e �W�h ��S �

JX JX

JY JZ

JZ JY

JXZ �JXY

L Mx

M Mz

N �My

Lift YDrag X

�� Appendix A� Matching of US and USSR notation

Bibliography

�� Poincar�e� Henri �Les methodes nouvelles de la mechaniqes celestre�� vol� I�III� Paris��

�� V�I�Arnold �Additional chapters of the theory of Ordinary Di�erential Equations��Nauka publ�� Moscow� �� in Russian��orV�Arnold Chapitres Suppl�ementaires de la Th�eorie des �Equations Di��erentiellesOrdinaires� �Editions Mir� ��

�� Yu�I�Neymark �Method of point mapping in the theory of nonlinear oscillations��Nauka publ�� Moscow� �� in Russian��

�� Yu�I�Neymark� P�S�Landa �Stochastic and chaotic oscillations�� Nauka publ��Moscow� �� in Russian��

�� M�G�Goman �Di�erential method for continuation of solutions to systems of �nitenonlinear equations depending on a parameter�� Uchenye zapiski TsAGI� vol� XVII�no�� in Russian��

�� M�G�Goman� A�V�Khramtsovsky �Calculation of a boundary of asymptotic stabil�ity region of dynamic system�� Uchenye zapiski TsAGI� vol� XXI� no�� inRussian��

�� Hsiao�Dong Chiang� Morris W� Hirsch� Felix F� Wu �Stability Regions of NonlinearAutonomous Dynamical Systems�� IEEE trans� on Automatic Control� vol�� no��Jan� ��

�� Gilmore R� �Catastrophe Theory for Scientists and Engineers�� John Wiley + Sons�New York Chichester Brisbane Toronto� ��

�� Poston T�� Stewart I�N� �Catastrophe Theory and Its Applications�� London� Pit�man� ��

�� G�E�Forsythe� M�A�Malcolm and C�B�Moler �Computer Methods for MathematicalComputations�� Prentice�Hall Inc�� Englewood Cli�s� N�J��

�� R�Thom �Structural stability and morphogenesis�� Benjamin publ��

�

� BIBLIOGRAPHY

�� Genesio P�� Tartaglia M�� Vicino A� �On the estimation of asymptotic stabilityregions� State of the art and new proposals�� IEEE Trans� on Aut� Contr�� vol�AC�� no� ��

�� Thomas S� Parker� Leon O� Chua �INSITE � A Software Toolkit for the Analysisof Nonlinear Dynamical Systems�� Proc� of the IEEE� vol�� n��

�� G�S�Biushgens� R�V�Studnev �Dynamics of the Longitudinal and Lateral Motion��Moscow� Mashinostroenie publ�� in Russian��

�� G�S�Biushgens� R�V�Studnev �Aircraft Dynamics� Spatial Motion�� Moscow� Ma�shinostroenie publ�� in Russian��

�� Yu�P�Guskov� G�I�Zagainov �Aircraft Flight Control�� Moscow� Mashinostroeniepubl�� in Russian��

�� Manoeuvre Limitations of Combat Aircraft�� AGARD Advisory Report No�AGARD�AR��A�

�� V�K�Svjatodukh �Aircraft Spiral Motion�� Uchenye zapiski TsAGI� vol� VII� no�� in Russian��

�� N�N�Dolzenko �Investigation of Aircraft Spin Characteristics�� MFTI lecture� ��

M.G.Goman and A.V.Khramtsovsky (1993) - Textbook for KRIT Toolbox users

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Transcript of M.G.Goman and A.V.Khramtsovsky (1993) - Textbook for KRIT Toolbox users