HBM Method and Nonlinear Oscillators Under Resosnant

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Harmonic Balance, Melnikov Method andNonlinear Oscillators Under Resonant Perturbation

Michele Bonnin1

1Politecnico di Torino, Torino, Italy

SUMMARY

The Subharmonic Melnikovs method is a classical tool for the analysis of subharmonic orbits

in weakly perturbed nonlinear oscillators, but its application requires the availability of an an-

alytical expression for the periodic trajectories of the unperturbed system. On the other hand,

spectral techniques, like the Harmonic Balance, have been widely applied to the analysis and de-

sign of nonlinear oscillators. In this manuscript we show that bifurcations of subharmonic orbits

in perturbed systems can be easily detected computing the Melnikovs integral over the Harmonic

Balance approximation of the unperturbed orbits. The proposed method significantly extend the

applicability of the Melnikovs method since the orbits of any nonlinear oscillator can be easily

detected by the Harmonic Balance technique, and the integrability of the unperturbed equations

is not required anymore. As examples, several case studies are presented, the results obtained are

confirmed by extensive numerical experiments.

1. I

Nonlinear oscillators subject to periodic perturbations and coupled nonlinear oscillators

are enduring problems in the classical theory of synchronization. While the full range of

dynamical behavior, including chaos, is exhibited by these types of dynamical systems,

periodic orbits are perhaps of primary importance from the point of view of applications.

If the single unperturbed oscillator is not structurally stable, a weak perturbation can

drastically change the phase portrait. In the case of aggregates of structurally stable oscil-lators, an appropriate perturbation can introduce mutual entrainment or synchronization

in the network, and this phenomenon is believed to play a major role in self organization

in nature. The importance of synchronization in the self organization lies in the fact that

what looks like a single process on a macroscopic level often turns out to be a collective

oscillation resulting from the mutual synchronization among the tremendous number of

the constituent oscillators [1].

It is thus of primary importance to understand how synchronization is achieved under

the effect of perturbations or due to the presence of couplings. In these cases, control

Correspondence to: Michele Bonnin, Politecnico di Torino, Corso Duca degli Abruzzi, 24, I-10129

Torino, Italy, Tel. +390115644199, Fax. +390115644099e-mail: [email protected]


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parameters can be the amplitude and frequency of the external forcing or the coupling

strength. The problem is simplified if the amplitude of the external input or the strength

of the connections are assumed to be small. This allow to employ perturbative techniques

as the classical Melnikovs method [2], based on the idea to make use of the computable

solutions of the unperturbed system to determine the solutions of the perturbed one.

The critical point is the required availability of an analytic expression for the unper-

turbed solutions. Unfortunately, the periodic trajectories of almost all non trivial oscil-

lators can only be determined through numerical integrations. Consequently, Melnikovs

method has been mainly applied to integrable systems [36].

On the other hand, spectral techniques, like the Harmonic Balance, have been widely

used to study periodic solutions of nonlinear systems [912]. The idea is to expand the

periodic solution and the vector field in Fourier series and to equate the coe fficients of the

same order harmonics. A system of nonlinear ordinary differential equations is then trans-

formed in a systems of nonlinear algebraic equations whose unknowns are the amplitude

and the period of the harmonics.In this manuscript nonlinear oscillators under the effect of weak perturbation are in-

vestigated. As a main contribution, we show that the Melnikovs integrals can be accu-

rately evaluated over the analytical expression of the periodic orbit obtained through the

Harmonic Balance technique. The proposed approach, permits to investigate both limit

cycles bifurcating from continuous family of periodic orbits in non hyperbolic oscillators

subject to external forcing, and frequency entrainment in hyperbolic oscillators driven by

resonant excitations.

The paper is organized as follows. In Section 2 the Harmonic Balance approach is

outlined, under the assumption that the dynamical system is described by a system of

differential equations. In Section 3 the Melnikovs method for subharmonic orbits isillustrated in details. Non hyperbolic oscillators driven by an autonomous perturbation

are analyzed in subsection 3.1, and the stability analysis of the bifurcating limit cycle is

studied in subsection 3.2. The case of a non hyperbolic oscillator under the effect of a

non autonomous periodic perturbation is analyzed in subsection 3.3. Subsection 3.4 is

devoted to the analysis of dynamical systems whose free oscillation is a limit cycle. In

Section 4 some key examples are presented. In section 5 conclusions are drawn.

2. HBT

We consider a system of nonlinear ordinary differential equations (ODEs), of the form

x(t) = fx(t),

, (1)

where x Rn, f : Rn Rn is a nonlinear vector field, and Rm is a vector of real

parameters. We assume that there exists a set of parameters values for which system (1)

admits aTperiodic orbit(t) =(t+ T) described by a regular curve(t) Rn. Therefore

(t) can be developed in Fourier series, giving

(t) =

+

k

=

kei k t (2)

where kis the vector of the complex valued Fourier coefficients and =2/T.


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For computational purposes, the Fourier series has to be truncated to a suitable number

of harmonics, high enough to accurately represent the solution (t), thereby obtaining

(t) =

Nk=N

kei k t

. (3)

The coefficients kand the angular frequency identify the T periodic function (t),

which is not, in general, a solution of (1). However, the higher the numberNof harmonics

in (3), the smaller the error made approximating the exact solution(t) with the truncated

Fourier series (t). In other words (t) approaches(t) asNtends to infinity.

It is well known that any function of a Tperiodic argument is itselfTperiodic, hence

also the vector field can be expanded in Fourier series

f(t), =N

k=N

Fkei k t, (4)

where the vector of the Fourier coefficients can be determined as

Fk = 1

2

f

N

m=N

mei m t,

ei k td( t). (5)Such integrals can in all cases be evaluated numerically, but in many cases they can be

expressed in a closed analytical form [12].

By substituting equation (4) and the derivative of (3) in (1), we obtain the following

equationN

k=N

i k kei k t

=

Nk=N

Fkei k t. (6)

Due to the orthogonality of the base functions, equation (6) implies that the coe fficients

of the same order harmonics are equal, that is

i k k Fk =0. (7)

System (7) will be referred to as the Harmonic Balance system. Two main situation might

be encountered. If the nonlinear system (1) is autonomous (i.e f(x(t)) does not depend

explicitly on time) the period of the solution is unknown. Therefore system (7) has 2N+ 1

equations and 2N + 2 unknowns, the 2N + 1 spectral coefficients k and . Take intoaccount that ke

i kt= ke

i( kt+k) where k = kei k. Since we are interested in steady

state behaviors, one of the initial phases k is irrelevant and can be set to any desired

value. As a consequence one of the associated coefficients, for example Im{1}, can be

imposed to be null and the equation Im{1} =0 is added to system (7).

The second important case is the one in which the vector field is non-autonomous and

contains a periodic forcing term (f(x(t), t) = f(x(t), t+ T)). In this situation the periodic

solutions of the forced nonlinear system (1) are expected to have either the same period

of the forcing term (harmonic solutions, T = T) or be resonant with the perturbations

(mT = n T with m, n Z). A m : 1 resonance with the external forcing is called

subharmonic of order m and the periodic solutions are called subharmonics. A 1 : nresonance is calledultraharmonicand am : n resonance is called ultrasubharmonic. The


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angular frequency is now determined to be = mn

= m

n2T

and the unknowns in system

(7) are simply the 2N + 1 spectral coefficients.

In both cases the system of differential equations (1) is reduced to a system of non-

linear algebraic equation (7) involving the aforementioned unknowns and which can be

efficiently solved exploiting standard numerical techniques.

Once the harmonic balance system is solved, equation (3) gives the analytical, even

though approximated, expression of the periodic trajectory. Any point (0 (t)) belong-

ing to the orbit can be written as

0 =

Nk=N

kei k t0 (8)

and the associated flow leaving from0is

(t, 0) =

Nk=N

kei k (t0+t) =

Nk=N

0ke

i k t. (9)

3. MM SO

Melnikovs method represents one of the few cases in which global information on spe-

cific systems can be obtained analytically. The original Melnikovs method [2], is applied

to homoclinic orbits passing through a hyperbolic saddle point. It defines an integral

function which measure the first variation of the separation between the perturbed sta-

ble and unstable manifolds of the hyperbolic saddle point [4, 68,13]. The subharmonic

Melnikovs method, on the contrary, introduces an integral function which measure the

distance between two consecutive intersections of a perturbed orbit and a suitable cross

section. If there exist a cross section and a perturbed orbit such that the distance is zero,

the two intersections coincide and the perturbed orbit is periodic. The prospecting of a pe-

riodic orbit is therefore reduced to the research of a zero of the integral function [5,14,15].

The integral function, usually called Melnikovs integral, yields a formal way for eval-

uating the distance, provided that an explicit expression of the unperturbed periodic tra-

jectory is known.

Recently some significant improvements have been developed for the classical sub-harmonic Melnikovs method [16, 17]. These enhancements allow to investigate either

limit cycles which bifurcate from continuous family of periodic orbits or weakly per-

turbed limit cycle oscillators. However, in both cases the integrability of the unperturbed

system is still required.

In the remaining of this section, we recall classical results and recent developments

concerning the subharmonic Melnikovs method. Before proceed to main results, we

introduce some fundamental concepts on the integration of both homogeneous and in-

homogeneous variational equations of planar autonomous differential equations along a

given trajectory. We consider a smooth plane vector field f : R2 R2, f =(f1, f2) with

flow(t,x) (i.e. (t,x) = f((t,x))), and the orthogonal vector field f =(f2, f1). Thesolutions can be expressed in terms of geometric quantities which involve the divergence


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f, the curl f = f2

x

f1y

and the curvature

K =def f(x) f(x)

f(x)3 . (10)

Theorem 1 (Dilibertos Theorem [16,18]) Suppose(t,x)is the flow of the differential

equation

x(t) = fx(t)

(11)

and consider a point x0 R2. I f f (x0) 0, then the fundamental matrix solution (t)

satisfying det((0)) =1, of the variational equation

y(t) = D f((t,x0)) y(t) (12)

with D f is the jacobian matrix of f , is such that

(t) f(x0) = f((t,x0)) (13)

(t) f(x0) = (t) f((t,x0)) +(t) f ((t,x0)) (14)

where

(t) =def

t0

2K f

(s,x0)

f

(s,x0)

(s) ds (15)

(t) =def f(x0)

2

f((t,x0)) 2 e

t0

f((s,x0))ds . (16)

For the proof of the theorem the reader is referred to [16, 18]

Dilibertos theorem contains all the information about the solutions of the linear varia-tional equation along the trajectories of the plane vector field f. The next lemma gives an

explicit formula for the solution of the inhomogeneous linear variational equation along

a trajectory of f.

Lemma 2 (Variational Lemma [16]) Let f : R2 R2 and g : R2 R2 denote smooth

vector fields. Consider x0 R2 and let(t,x)denotes the flow of f. If f(x0) 0 then the

solution of the initial value problem

z(t) = D f((t,x0)) z(t) + g ((t,x0))

z(0) = 0,

(17)

is

z(t) =[N(t,x0) + (t) M(t,x0)] f((t,x0)) +(t) M(t,x0)

f ((t,x0)) . (18)

where

M(t,x0) =def

t0

f((s,x0)) g ((s,x0))

f((s,x0)) 2 (s) ds, (19)

N(t,x0) =def

t0

1

f((s,x0))2

f

(s,x0)

, g

(s,x0)

(s)

(s) f

(s,x0)

g

(s,x0)

ds,

(20)while and are defined as in the statement of Dilibertos theorem.


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Once again the interested reader can found a proof of the lemma in [16].

Having these results at disposal it is possible to investigate the existence of periodic

orbits in the perturbed system, which bifurcate from a continuous family of periodic orbits

of the unperturbed one. We make one basic assumption about the unperturbed system, in

particular we assume that it possesses a period annulus, i.e. a one-parameter family of

periodic orbits(t), (1, 2) with periodT((t))> 0.

It is always possible to find a cross section transverse to the flow of f in the period

annulus (see figure 1). Then there is some 0 > 0 such that the flow of the perturbed

system is also transverse to .

(t)

Figure 1: The period annulus of the unperturbed system and the cross section .

For the sake of simplicity we fix the reference frame so that y =0 on the cross section

. Let us consider a point x0 belonging to the period annulus. For the perturbed

system we define the trajectory (t,x0, ) leaving from x0, the parameterized Poincare

mapP(x0, ) :

R

, asP(x0, ) =

(T(x0, ),x0, ) on the Poincare section

andtheperiod function T(x0, ) : R R

+, which assign to x0 the time required for the

return on . Settingto zero, the corresponding quantities for the unperturbed system are

obtained.

The idea of the bifurcation theory is to find the periodic trajectories of the perturbed

system as fixed points of the parameterized Poincare mapP, that is, to find initial condi-

tions x0 lying on the Poincare section such that (T(x0, ),x0, ) = x0. With this goal

in mind, the displacement function (x0, ) = (T(x0, ),x0, ) x0 and the normalized

displacement function

(x0, ) =def

f(x0)

(x0, ) (21)

are defined. It is evident that (x0, 0) = 0 for all x0 , and that (x0, ) = 0 if and onlyif the trajectory(t,x0, ) passing through x0 is periodic. With the choice made about the

reference frame, it follows that P(x0, ) =(P(x0, ), 0) and(x0, ) =(d(x0, ), 0)

.

Before to analyze the influence of small perturbations on the periodic orbits of the

unperturbed system, one final lemma is presented, which provides useful information

about the functions(t) and(t) defined in Dilibertos theorem (equations (15) and (16)).

Lemma 3 Let f : R2 R2 be a plane vector field with flow (t,x), and x0 . If x0 is

contained in a period annulus, the functions (t)and(t)defined in Dilibertos theorem

are such that

(T(x0)) =T(x0) (T(x0)) =1, (22)

where T(x)is the period function.


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Proof By hypothesis (t,x) is the flow of f = (f1, f2), let (t,x) be the flow of the

orthogonal field f =(f2, f1), thus

(t,x) = f((t,x)) (23)

(t,x) = f ((t,x)) . (24)

Consider a point x(s) = (s,x0) on the trajectory of the orthogonal vector field, and still

belonging to the period annulus. Abbreviate T(s) = T((s,x0)) and consider that, since

(s,x0) lies in the period annulus, the following equation holds

(T(s), (s,x0)) (s,x0) =0.

A differentiation with respect to syields

(T(s), (s,x0)) T(s) + D (T(s), (s,x0)) (s,x0) (s,x0) =0 (25)

and taking into account equations (23) and (24) we obtain

f (T(s), (s,x0))

T(s) + D (T(s), (s,x0)) f

((s,x0)) f ((s,x0)) =0.

An evaluation ats =0, remembering that (0,x0) = x0(which implies T(0) =T((0,x0)) =

T(x0)) gives

f( (T(x0),x0)) T(x0) + D (T(x0),x0)) f

(x0) f(x0) =0. (26)

Applying Dilibertos theorem the required formulas can be obtained. We begin by ob-

serving that both D(t,x0) f(x0) and (t) f((t,x0)) +(t) f

((t,x0))are solutions of

the initial value problem (12) with initial condition y(0) = f(x0), thus

D(t,x0) f(x0) =(t) f((t,x0)) +(t) f ((t,x0)) . (27)

Substituting equation (27) evaluated at t = T(x0) into (26) we derive the following equa-

tion

f((T(x0),x0)) T(x0) + (T(x0)) f((T(x0),x0)) +(T(x0)) f

(x0) f(x0) =0. (28)

By hypothesis x0 belongs to the period annulus, therefore (T(x0),x0) = x0, and consid-

ering that f and f are orthogonal, equation (28) holds if and only if

(T(x0)) =T(x0) and (T(x0)) =1.

which prove the thesis of the lemma.

3.1. Autonomous Perturbations

We now consider the case of a nonlinear dynamical systems with an autonomous pertur-

bation

x = f(x) + g(x), (29)

and denote by(t,x, ) the associated flow

(t,x, ) = f((t,x, )) + g ((t,x, )) . (30)

The following theorem [16,17] provides a mean to identify periodic trajectories in a periodannulus of the unperturbed system from which limit cycles emerge.


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Theorem 4 (Andronov-Poincare) Suppose for = 0 system (29) has a period annulus.

If the integral function

M(x0) =

T(x0,0)

0 e

t

0f((s,x0)) ds

f(t,x0)

g

(t,x0)

dt (31)

has a simple zero at x0, that is M(x0) = 0 andM(x0)/x 0, then, for small value of

, there is a limit cycle (t,x0, ) of the perturbed system (29) passing through at x0,

emerging from the periodic orbit(t,x0) =(t,x0, 0)of the unperturbed system.

ProofBy differentiating both the normalized displacement function and the displacement

function with respect to we have

(x0, )

=

f(x0)

(x0, 0)

, (32)

(x0, )

= (T(x0, ),x0, ) T(x0, )

+

(T(x0, ),x0, ).

The latter, evaluated at =0, gives

(x0, 0)

= (T(x0, 0),x0, 0)

T(x0, 0)

+

(T(x0, 0),x0, 0). (33)

We now consider that(t,x0, 0) is a periodic trajectory of the unperturbed system, hence

the following equations hold

(t,x0, 0) = f((t,x0, 0)) (34)

(T(x0, 0),x0, 0) = x0. (35)

By introducing equations (34) and (35) in equation (33) we obtain

(x0, 0)

= f(x0)

T(x0, 0)

+

(T(x0, 0),x0, 0), (36)

and consequently, equation (32) becomes

(x0, 0)

=

f(x0)

(T(x0, 0),x0, 0). (37)

To find the proper expression of/ we consider that (t,x0, ) is a trajectory of theperturbed system leaving from x0

(t,x0, ) = f((t,x0, )) + g(t,x0, )

(0,x0, ) = x0.

A differentiation with respect to and an evaluation at = 0 yield the variational initial

value problem

d

dt

(t,x0, 0)

= D f((t,x0, 0))

(t,x0, 0)

+ g ((t,x0, 0))

(0,x0, 0)

= 0,

(38)


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which is of the same kind of (17), hence the solution is given by the Variational Lemma

(equation (18)). Inserting such solution in equation (37) we have

(x0, 0)

=(T(x

0))M(T,x

0)f((T,x

0))2.

Keeping into account that (t,x0, 0) =(t,x0) and the second one of (22) it can be inferred

that(x0, 0)

=

T(x0,0)0

et

0f((s,x0))ds f

(t,x0)

g

(t,x0)

dt. (39)

Since x0 belongs to the period annulus (x0, 0) = 0. If there exist an > 0 and a con-

tinuous functionh : (,) such that (h(), ) = 0, then for each (,) there

is a periodic trajectory(t, h(), ) of the perturbed vector field passing through the point

h().

We take the Taylor expansion of (x0, ) in the neighborhood of =0

(x0, ) = (x0, 0) + (x0, 0)

+ O(2).

and observe that, as a consequence of (39), the hypothesis of the theorem can be recast as

(x0, 0)

=0 and

2(x0, 0)

x0,

therefore the implicit function theorem ensures the existence of the required functionh().

Integral (31) is often referred to as the Melnikovs integral and is the basic measure

of the distance between x0 and the first iteration of the Poincare map for the perturbedsystem. For every initial condition x0the associated periodT(x0) is uniquely determined.

Since the integral is evaluated over the period T(x0), the dependency on time in (31) is

not written explicitly.

Remark If the unperturbed vector field is hamiltonian, f = 0 and the Melnikovs

integral simply reduces to

M(x0) =

T(x0,0)0

f(t,x0)

g

(t,x0)

dt.

Remark The proposed approach is not unique. Here a cross section is fixed, and the

Melnikovs integral is evaluated as the initial condition x0 is varied. This approachis preferred here in view of the joint application of the Melnikovs method and the Har-

monic Balance technique, since any initial condition can be written as x0 =N

k=N 0kand

the associated flow is given by (t,x0) =N

k=N 0k

ei k t. Another approach is to keep

fixed x0 and evaluate the Melnikovs integral on a cross section which moves along the

unperturbed trajectory, however the two approaches are completely equivalent.

3.2. Stability of Subharmonic Orbits

The natural context to study stability and bifurcations of periodic orbits is the Poincare

map. Thus it is not surprising that Melnikovs method, with its close relation to Poincaremap permits a simple approach to such problem.


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If the perturbative vector fields is time dependent and periodic, to investigate the sta-

bility properties of the emerging limit cycles a symplectic transformation to action angle

coordinates has to be performed, but such transformation requires the unperturbed vector

field to be hamiltonian.

Conversely, if the forcing vector field is autonomous, the stability analysis is much

simpler. The following theorem provides information about the stability of the limit cycle

emerging from a period annulus:

Theorem 5 Suppose system (29) has, for = 0, a period annulus containing a point

x0 from which a limit cycle of the perturbed system emerges, and consider the function

(x0, 0) =

f(x0)

T(x0, 0)

+

T(x0,0)0

g

(t,x0)

+

d

d f

(t,x0)

dt

.

(40)

If(x

0,

0)

0then the

limit cycle is asymptotically unstable.

Proof Since(x, 0) = 0 for all xin the period annulus, the scalar displacement d(x, 0) is

also null. The Taylor expansion of the scalar displacement d(x, ) in the neighborhood of

(x, 0) has the form

d(x, ) =d(x, 0)

+ O(2), (41)

and a differentiation with respect to x gives

d(x, )

x

=2d(x, 0)

x

+ O(2). (42)

By hypothesis the limit cycle of the perturbed system (29) passes through x0, hence

d(x0, ) = 0. Equation (42) implies that, for sufficiently small values of,d(x, ) crosses

the value(x0, ) =0 asx passes throughx0, with positive or negative slope depending on

the signs of and of the mixed partial derivative. We consider the case of negative slope,

first

2d(x, 0)

x 0.


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80 60 40 20 0 20 40 60 8040

30

20

10

0

10

20

30

40

x

x0

P(x)

Figure 2: A stable limit cycle. The sequence of intersections of the trajectory with the Poincare

section converges to x0.

In order to complete the proof of the theorem it is necessary to devise the proper

expression of(x0, 0). It is convenient to rewrite system (29) in the form

x =h(x, ) (43)

whereh(x, ) = f(x) + g(x), hence the flow(t,x, ) is a solution of

(t,x, ) =h ((t,x, ), ) . (44)

To simplify the notation(T(x, ),x, ) = xhereafter. Considering the directional deriva-

tives of the scalar Poincare map along the cross section , we obtain

P(x, )

x( x) = ( x)

T(x, )

x+ D( x) (x), (45)

where (x) is the versor tangent to the cross section in x. If the cross section is not

a simple straight line ( x) and (x) may differ. It is always possible to project in the

directions parallel toh andh

(x) =a h(x, ) + b h(x, ) ( x) =c h( x, ) + d h( x, ) (46)

where

a = (x), h(x, )

h(x, )2 , b =

(x), h(x, )

h(x, )2 =

h(x, ) (x)

h(x, )2 ,

c = ( x), h( x, )

h( x, )2 , d =

( x), h( x, )

h( x, )2 =

h( x, ) ( x)

h( x, )2 . (47)

Substituting equations (44) and (46) in (45) we obtain

P(x, )

x

c h( x, ) + d h( x, )

=h( x, )

T(x, )

x+D( x)

a h(x, ) + b h(x, )

. (48)

The application of Dilibertos theorem to the last term of (48) yields, after some algebraic

manipulations

cP(x, )

x T(x, )

x a b(T(x, ))

h( x, )+

dP(x, )

x b(T(x, ))

h( x, ) =0.


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Taking into account equation (16) and the orthogonality of the vector fields we have

P(x, )

x=

h(x, ) (x)

h( x, ) ( x) exp

T(x,)0

h((t,x, ), ) dt

. (49)

To evaluate the derivative of (49) with respect to at = 0, we consider that x =(T(x, 0),x, 0) = x. Therefore the first factor is equal to one and its derivative

h(x, ) (x)

h( x, ) ( x) =

h(x, )

(x)

(h( x, ) ( x)) (h(x, ) (x))

h( x, )

( x)

(h( x, ) ( x))2

(50)

is null. For the second factor we take into account that

T(x,)0

h ((t,x, ), ) dt = h ((T(x, ),x, ), ) T(x, )

+

T(x,)

0

d

d h ((t,x, ), ) dt, (51)

and sinceh(x, ) = f(x) + g(x), the following equation holds

d

d h ((t,x, ), ) =

d

d f((t,x, )) + g((t,x, )) +

d

d g((t,x, )). (52)

Taking into account equations (51), (52), and the considerations made about (50), a dif-

ferentiation of equation (49) with respect to and an evaluation at (x, ) =(x0, 0) yields

2P(x0, 0)

x = f(x0)

T(x0, 0)

+

T(x0,0)0

g

(t,x0)

dt+

T(x0,0)0

d

d f

(t,x0)

dt.

where(t,x0) =(t,x0, 0). Keeping in mind that

2P(x0, 0)

x=

2d(x0, 0)

x

the thesis of the theorem follows.

Remark In general, stability of periodic orbits is determined computing Floquets mul-

tipliers. Recently some authors [19] have proved that Floquets multipliers can be accu-

rately computed by exploiting numerical algorithms described in [20] evaluated over the

Harmonic Balance solutions. However that approach is not reliable for weakly perturbed

non hyperbolic oscillators. In fact, in this case at least two Floquets multipliers lie on the

unit circle, and since their moduli depend continuously on perturbative terms, under the

effect of a weak perturbation their value remains very close to the unity. Therefore, it is

difficult to discriminate between the variation due to the perturbations and the numerical

inaccuracies.

RemarkIf the unperturbed system is hamiltonian f =0, the function(x0, 0) reduces

to

(x0, 0) =

T(x0,0)0

g(t,x0)

dt. (53)

It is worth noting that the stability of the bifurcating limit cycles depends not only on the

perturbationg(x) and its strength , but also on the unperturbed system, since the forcingvector field is evaluated over the free oscillation.


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NOURP 13

3.3. Non Autonomous Perturbations

This section is devoted to study the system

x = f(x) + g(x, t) (54)

where both f(x) andg(x, t) are smooth functions, g is aTperiodic function and for =0

system (54) has a period annulus. Since the perturbed system is expected to exhibit peri-

odic trajectories resonant with the perturbation, it is natural to search for the persistence

of periodic orbits of the unperturbed system passing through x0, whose period T(x0) is

commensurable with the periodTof the forcing term, that is

m T = n T(x0) m, n Z.

It is possible to find periodic trajectories of the perturbed system finding conditions on

the functions f andgsuch that, for small values of 0, fixed points of the parametrizedPoincare map remain. We consider the time derivative of the displacement function

D(x, ) x = (mT,x, ) x = f((mT,x, )) f(x)

Since(mT,x, 0) = x, an evaluation at =0 gives,

D(x, 0) f(x) =0, (55)

thereforeD(x, 0) is singular and the implicit function theorem cannot be applied directly.

However, different kinds of reduction can be made depending on the degeneracy of the

period annulus. The most degenerate condition is the case of an isochronous annulus, forwhich the period function is constant, i.e. T(x) =0, for all xbelonging to the annulus.

Theorem 6 Suppose system (54) has, for = 0an isochronous period annulus. If there

are two positive integers m and n such that the unperturbed system has a mT/n periodic

trajectory passing through x0and the functions

M(x0) =

mTn

0

et

0f((s,x0)) ds f

(t,x0)

g

(t,x0)

dt (56)

and

N(x0) =

mTn

0

1

f(t,x0)

2

f

(t,x0)

, g

(t,x0), t

(t)

(t) f(t,x0)

g

(t,x0), t

dt,

(57)

have both a simple zero at x0 , then the periodic orbit of the unperturbed system

persists for small values of.

Proof For the sake of simplicity we focus on a subharmonic orbit of the unperturbed

system passing through x which is inm : 1 resonance with the external forcing. We

make use of the Taylor expansion of the displacement function in a neighborhood of =0

(x, ) =(x, 0)

+ O(2) =

(mT,x, 0) + O(2). (58)


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It was shown in section 3.1 that / is a solution of the inhomogeneous linear varia-

tional equation (17), hence, according to (18), it can be written as

(mT,x, 0) =[N(x) + (mT) M(x)] f((mT,x, 0)) + (mT) M(x) f ((mT,x, 0))

(59)

As a consequence of (22), (mT) = m T(x) =0 since the period annulus is isochronous,

and(mT) =1. Thus

(mT,x, 0) =N(x) f(x) + M(x) f(x)

Introducing such expression in (58), the following equation is obtained

(x, ) =N(x) f(x) + M(x) f(x) + O()

.

Hence the implicit function theorem can be applied to determine when there is an implicit

solution of the equation(x, ) =0 at some point (x0, 0).

Before to consider the case of a regular period annulus, we introduce some preliminary

considerations. It is possible to split the displacement function into its tangent and radial

projections

(x, ) =(x, ) f(x) +(x, ) f(x) (60)

where

(x, ) = (x, ), f(x)

f(x)2 (x, ) =

(x, ), f(x)

f(x)2 .

In what follows, we need the directional derivatives of(x, ) and(x, ), thus we recall

that

D(x, 0) = D(T,x, 0) x

= D(T,x, 0) I, (61)

and applying Dilibertos theorem we obtain

D(x, 0) f(x) = (T) f(x) +(T) f(x) f(x) (62)

D(x, 0) f(x) = 0. (63)

Now we can compute the derivative of(x, ) in the direction f(x) and f(x) at (x, ) =

(x, 0), we obtain

D (x, 0)f(x) = D (x, 0) f(x), f(x)

f(x)

2 +

(x, 0), D f(x) f(x)

f(x)

2 =(T) (64)

D (x, 0)f(x) = D (x, 0) f(x), f(x)

f(x)2 +

(x, 0), D f(x) f(x)

f(x)2 =0, (65)

where (62) and (63) were used, and since (x, 0) = 0. In a similar way it is possible to

prove that

D(x, 0)f(x) = (T) 1 (66)

D(x, 0)f(x) = 0. (67)

We can now introdue the theorem dealing with a regular period annulus, that is when

T(x) 0. Even in this situation the implicit function theorem cannot be applied directly,since equation (55) still holds, then a Lyapunov-Schmidt reduction is applied.


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Theorem 7 Suppose system (54) has, for =0 a regular period annulus. If there are two

positive integers m and n such that the unperturbed system has a mT/n periodic trajectory

passing through x0and the function

M(x0) =

mTn

0

et

0f

(s,x0)

ds f

(t,x0)

g

(t,x0), t

dt (68)

has a simple zero at x0 , i.e. M(x0) = 0andM(x0)/x 0, then the periodic orbit

of the unperturbed system persists for small values of.

ProofUsing equation (64) and the first of (22), we have D(x, 0) f(x) = mT(x) which

is different from zero because the annulus is regular. Applying the implicit function the-

orem to the function (x, 0), it is possible to define a smooth manifold S such that

vanishes on S. In addition, for any x (t,x, 0) equation (65) implies that S is transverse

to the section

and(t,x, 0) S

.Now we focus on the restriction of the radial projection to S. Equations (66) and

(67) imply that D(x, 0) = 0. On the other hand, as a consequence of (58) and (59) we

have(x, 0)

=[N(x) + (mT)] f(x) +(mT)M(x) f(x). (69)

while differentiation of (60) with respect to and an evaluation at =0 yield

(x, 0)

=

(x, 0)

f(x) +

(x, 0)

f(x). (70)

Comparing the last two equations and taking into account the second of (22) we infer

(x, 0)

=M(x).

The hypotheses M(x0) =0 and M(x0)/x 0 imply (x0, 0)/ =0 and 2(x0, 0)/x

0. Therefore there is an > 0 and a smooth function h : (, ) R2 such that

(h(), ) 0, as a consequence

(h(), ) =(h(), ) =0

and(h(), ) =0, which proves the statement of the theorem.

One more result, due to Chow et al. [21], is worth noting. This result is related tosystem possessing a homoclinic orbit to a hyperbolic saddle point as the separatrix of

regions filled with periodic orbits. It implies that a countable sequence of subharmonic

saddle-node bifurcations of periodic orbits converges to a homoclinic bifurcation (see

[14,15] for details).

Theorem 8 Suppose for = 0 system (54) has a homoclinic orbit q0(t) to a hyperbolic

saddle point p0. Assume the interior of q0(t) p0 is filled with a continuous family of

periodic orbits q(t) whose period tends monotonically to infinity as the periodic orbits

approach the homoclinic orbit. Let

Mm(x) =

mT

0

e

t

0f

q(s)

ds f

q(t)

g

q(t), t

dt (71)


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be the Melnikovs integral associated to the mT periodic orbit. Then

limm+

Mm(x) =M(x)

whereM(x) =

+

et

0f

q0(s)

ds f

q0(t)

g

q0(t), t

dt

is the Melnikovs integral whose simple zeroes are associated to homoclinic bifurcations.

3.4. Limit Cycle Oscillators

This section is devoted to the analysis of oscillators whose free oscillation is a limit cycle,

i.e. the unperturbed system is structurally stable. When we consider the forced oscillator

on the manifold R2 S1, the three dimensional system has for =0, an hyperbolic torus

corresponding to the limit cycle. The flow on the torus will be periodic or quasi periodic

depending on wether or not some resonant condition is satisfied. In either case, the orbit

corresponding to the limit cycle will no longer be structurally stable, the stability of the

limit cycle being transferred to the torus. The existence of frequency entrained oscillations

in a limit cycle oscillator driven by a resonant forcing term is described by the following

theorem

Theorem 9 (Limit cycle subharmonic bifurcation theorem [16]) Let us consider the dy-

namical system

x(t) = fx(t)

+ g

x(t), t

and suppose that it admits a limit cycle(t,x0)whose period is resonant with the period

of the external forcing gx(t), t

= g

x(t+ T), t+ T

. If x0is a simple zero of the bifurcation

function

B(x0) =1 (T)

N(x0) + (T)M(x0), (72)

namely, B(x0) = 0 and B(x0)/x 0 then the perturbed system has a limit cycle

(t,x0, )passing through x0.

ProofWith the projections of and defined in equations (64)-(67) we have two possi-

bilities. If(T) 1 we can apply the implicit function theorem to the radial projection.

Conversely, if(T) 0 the implicit function theorem can be applied to the tangent pro-

jection, in both cases we can define a smooth functionh : (, ) R2 such that either

(h(), ) =

0 or(h(), ) =

0. Now we need to identify the bifurcation function. Let usconsider the Taylor expansion of and in the neighborhood of =0

(h(), ) = (h(0), 0) + d(h(0), 0)

d + O(2) (73)

(h(), ) = (h(0), 0) + d(h(0), 0)

d + O(2), (74)

where

d(h(0), 0)

d =

(h(0), 0)

+ D(h(0), 0)

h(0)

(75)

d(h(0), 0)

d =

(h(0), 0)

+ D(h(0), 0)

h(0)

, (76)


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NOURP 17

and comparing (69) and (70) we have

(h(0), 0)

=N(h(0)) + (T)M(h(0))

(h(0), 0)

=(T)M(h(0)). (77)

Sinceh(0) , we can express the vector field h()/as a linear combination of f and

f

h(0)

= a f(h(0)) + b f(h(0))

and taking into account (64)-(67)

D(h(0), 0)h(0)

= b (T) (78)

D(h(0), 0)h(0)

= b (T) 1. (79)

Thus

d(h(0), 0)

d = b (T) + N(h(0)) + (T)M(h(0)) (80)

d(h(0), 0)

d = b

(T) 1

+(T)M(h(0)). (81)

For the case in which(T) 1 we consider equation (81), which implies

b = (T) M((h(0))

1 (T)

which substituted in (80) yields

d(h(0), 0)

d =

1 (T)

N(h(0)) + (T)M(h(0))

1 (T)

The case in which(T) 0 is similar, (80) implies

b =N(h(0)) + (T)M(h(0))

(T)

which substituted in (81) gives

d(h(0), 0)

d =

1 (T)

N(h(0)) + (T)M(h(0))

(T)

as required.

4. A

In the previous section we shoved that the Melnikov technique provides a mean to deter-

mine conditions such that periodic trajectories survive under the effect of an external forc-ing. It was shown how the problem can be reformulated in terms of zeroes of an integral


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function evaluated over the periodic solution of the unperturbed system. Thus, an analytic

expression of the free oscillation must be available. Unfortunately, as stated in the intro-

duction, trajectories of almost all non trivial oscillators can be only determined through

numerical integration. Even for integrable systems, trajectories are often expressed in

terms of special functions, making the integral equations often unsolvable. In some cases,

the problem can be solved recurring to the method of the residues, while in other cases it

is necessary to consider some approximation of the function to be integrated [5, 14, 15].

In this section we show that the Melnikovs integrals can be accurately evaluated over

the Harmonic Balance approximation of the unperturbed system. The proposed approach

presents two main advantages, the Harmonic Balance technique yields the analytical ap-

proximations of the periodic trajectory of any nonlinear oscillator. Hence, the integrability

of the unperturbed system is not required anymore, and the applicability of the Melnikovs

method is considerably extended. Moreover, the unperturbed trajectory is always approx-

imated by a truncated Fourier series, and as a consequence the Melnikovs integrals often

result quite simple to solve making use of the orthogonality of the base functions.

4.1. Hamiltonian system with an autonomous perturbation

As a first application of the proposed technique, we consider the following dynamical

system

x = y

y = x x3 + (y x2y).(82)

For =

0, the system is hamiltonian and has centers at (x,y) =

(1, 0) and a hyperbolicsaddle at (0, 0). There are two homoclinic trajectories passing through the hyperbolic

saddle which separate regions filled with periodic orbits, the phase portrait is presented

in figure 3. The periodic orbits surrounding each center and lying inside the homoclinic

trajectory will be called inner orbits while the larger closed paths surrounding the centers

and the homoclinic trajectories will be called outer orbits. Since the unperturbed system is

integrable, all the trajectories can be determined analytically, however the periodic orbits

are expressed in terms of Jacobi elliptic functions and the resulting Melnikovs integral

can be solved only considering their Fourier series [14,15].

Theorem 4 establishes the conditions under which a limit cycle of the perturbed sys-

tem emerges from periodic solutions of the unperturbed system. We are interested in

determine the values of the bifurcation parameters satisfying such conditions. As a first

step, we have to determine the proper expression of the unperturbed trajectories of period

T =2 /.

Following section 2 we search for solution of the system

x = y

y = x x3,(83)

in the form

x(t)y(t)

=

Nk=N

XkYk

ei k t. (84)


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NOURP 19

2 1.5 1 0.5 0 0.5 1 1.5 21

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Figure 3: The phase portrait of the unperturbed Duffing oscillator obtained through the Harmonic

Balance technique.

The nonlinear term is expressed as follow

x3 =

Nk=N

Fkei k t, (85)

where

Fk = 1

2

N

h=N

Xhei h t

3

ei k t d(t).

Such integral can be expressed in a closed analytical form, in terms of the coe fficientsXkonly.

By substituting (84) and (85) in (83), computing the proper derivatives and equating

the coefficients of the same order harmonics we obtain

i kXk Yk = 0

k2 2 + 1

Xk Fk = 0.(86)

System (86) can be efficiently solved exploiting standard numerical algorithms, giving the

coefficients of the Harmonic Balance approximation of the free oscillation.

According to theorem 4, a limit cycles emerges from a periodic orbit of period T =

2 /, passing through0 =N

k=N kei kt0 , if the Melnikovs integral has a simple zero,

namely if

M(0) =

T(0)0

f(t, 0)

g

(t, 0)

dt =

T(0)0

y(t)y(t) x2(t)y(t)

dt =0.

(87)

Thus, for everyt0 (0, T], the Harmonic Balance coefficients describing the flow(t, 0)

can be computed as 0k

= kei k t0 . By introducing in (87) such coefficients we have

M(0) = I(0) J(0) =0


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20 MB

where

I(0) =

T(0)0

Nk=N

Y0kei k t

2

dt (88)

J(0) =

T(0)0

N

k=N

X0kei k t

2

Nk=N

Y0kei k t

2

dt. (89)

The two integrals can be analytically computed for any number of harmonics, and they

only involve summations of the Harmonic Balance coefficients. Thus a limit cycle arises

when

=

J(0)

I(0) . (90)

In figure 4 the values of the ratio /at which a limit cycle arise, are depicted as a

function of the frequency. For/ < 0.75 the perturbed system has not periodic solution.

At/ =0.75, an outer limit cycle arise from the unperturbed orbit of frequency =0.6

through a saddle node bifurcation, and is the unique periodic solution for 0.75 < / 0.8 there is a limit cycle surrounding each center

and a large limit cycle enclosing all three equilibria.

0 0.5 1 1.50.5

0.75

1.0

1.25

1.5

Figure 4: Bifurcation curves for the hamiltonian system with an autonomous perturbation as a

function of the frequency. (Solid line) Birth of a limit cycle from an outer periodic orbit. (Dashed

line) Birth of a limit cycle from an inner periodic orbit.

In the present example, as 0, i.e. T +, both the inner and the outer periodic

orbits of the unperturbed system approaches the homoclinic trajectory. Consequently, the

values of the ratio / at which limit cycles arise, converge at the value (/ = 0.8 here)

at which a homoclinic tangency occurs, as stated by theorem 8. The convergence of the

bifurcation curves for 0 is evident in figure 4.

According to theorem 5, the stability of the bifurcating limit cycles can be determined

evaluating the sign of equation (53) on the unperturbed periodic solution(t,x0), thus we

consider

T(0)

0 x2(t) dt =

T(0)

0

N

k=N

X0kei k t

2

dt = 2

N

k=N

X0k2 .(91)


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NOURP 21

By substituting in equation (91) the Harmonic Balance coefficients we obtain that the

inner limit cycles are asymptotically unstable, while the outer limit cycle is asymptotically

stable. The values at which limit cycles have birth and their stability are confirmed by

numerical integration of the state equations.

4.2. The Duffing oscillator

As an example of a nonlinear oscillator with a periodic perturbation we consider the

Duffing equation with linear negative stiffness and weak damping and sinusoidal forcingx = y

y = x x3 + ( cos( t) y),(92)

where the forcing amplitude and the damping are assumed as bifurcation parame-

ters. In the unperturbed limit the Duffing oscillator is an integrable system for which

exact results are available. The reader is referred to [14, 15] to compare the accuracy and

simplicity of the proposed approach with respect to classical results.

The unperturbed system is the same of the previous example, but since the forcing

term is periodic, we are now interested in subharmonic orbits of the forced system which

are inm : 1 resonance with the external forcing, therefore we search for harmonic balance

solutions of the unperturbed system in the form

x(t)

y(t)

=

Nk=N

XkYk

ei

km

t (93)

and we express the nonlinear term as

x3 =

Nk=N

Fkei k

m t (94)

where

Fk = 1

2

T0

N

h=N

Xhei h

m t

3

ei k td( t). (95)

Repeating the steps outlined in the previous example we obtain the following harmonic

balance system

i k

mXk

Yk =

0km

2

+ 1

Xk Fk = 0.

(96)

Exploiting the Harmonic Balance solution of system (96), it is easy to verify that for each

family of periodic orbits dT/dx 0, and then each family is regular. In particular for

every the system admits one and only one trajectory.

We are interested in determine for which values of the bifurcation parameters and

, am order subharmonic orbit of the unperturbed system survive under the effect of the

external forcing and the presence of damping. We compute the subharmonic Melnikovs

integral for the resonant orbit

Mm(0) =

mT

0

f(t, 0)

g

(t, 0), t

dt =

mT

0

y(t)cos(t) y(t)

dt,


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wherey(t) is determined introducing in (93) the Harmonic Balance coe fficients obtained

solving system (96). Thus

M

m

(0) =

2

mT

0

N

k=N

Y

0

ke

i km

t e

i t+

e

i tdt

mT

0

N

k=N

Y

0

ke

i km

t2

dt

and solving the integral we obtain

Mm(0) = 2

Re{Y0m} N

k=N

Y0k2 . (97)

According to theorem 7, am-order subharmonic orbit with period mT/, appears in the

perturbed system when

=

1

Re{Y0m}

Nk=N

Y0k

2

. (98)

The bifurcation curves corresponding to the birth of a subharmonic of order m in theperturbed system are easily determined by substituting in equation (98) the Harmonic

Balance coefficients of the corresponding unperturbed mT-periodic orbit. Since the ratio

/ is a constant, the bifurcation curves are straight lines. Some of such curves are out-

lined in figure 5, where Sm refers to an inner orbit while Sm refers to an outer orbit. It

is readily seen that, as predicted by theorem 8, the bifurcation curves accumulate on the

curveS0 associate to the homoclinic tangency.

0 0.05 0.10

0.05

0.1

S1S2S3S4S0S5S3

S1

Figure 5: Bifurcation curves for the subharmonics of the Du ffing oscillator. Sm: m-order subhar-

monic orbit inside the homoclinic trajectories; Sm: outer orbit. Note the rapid convergence ofSm

and Sm to the homoclinic tangency S0.

4.3. Coupled Van der Pol Oscillators

As a last application of the proposed technique, we consider two coupled Van der Pol

oscillators running in resonance

u = v

v = u + (1 u2)v

x = yy =

x + (1 x2)y

+ u.

(99)


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NOURP 23

Here the system (x(t),y(t)) is view as the perturbed system subject to a periodic resonant

external input provided by (u(t), v(t)), Z is the resonance factor. The perturbation

admits the Harmonic Balance description

u(t)

v(t)

=

Nk=N

Uk

Vk

ei k t

which leads to the Harmonic Balance system

i k Uk Vk = 0

k22 + i k 1

Uk Gk = 0

where

Gk = 1

2

N

m=N

Umei m t

2

N

n=N

Vnei n t ei k td(t).

The unperturbed system has a stable limit cycle whose Harmonic Balance description

is obtained solving

i kXk Yk = 0

k2 2

+ i k

Xk Fk = 0

where

Fk = 1

2

N

m=N

Xmei m t

2

N

n=N

Ynei n t

ei k td( t).

The Harmonic Balance approximation of the limit cycle is showed in figure (6) where is

compared to the solution obtained through numerical integration of the state equations.

3 2 1 0 1 2 33

2

1

0

1

2

3

3 2 1 0 1 2 33

2

1

0

1

2

3

Figure 6: Accuracy of the Harmonic Balance Technique. The Van der Pol limit cycle obtained

through numerical integration (dashed line) and through the Harmonic balance technique (solid

line) using 5 harmonics (left figure) and 7 harmonics (right figure) respectively. In the right figure

the two curves are almost coincident.

When the Harmonic Balance approximation of the limit cycle is known, it is possibleto determine subharmonic branch points searching zeroes of the bifurcation function (72).


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24 MB

Computing the terms appearing in (72) we obtain

(t) =y2

0+

1 x20

y0 x0

2y2 +

1 x2

y x

2 exp t

0

1 x2 ds

(t) = 2

t0

x2 +y2 + x3y + x y 2y2 1y2 +

1 x2

y x

2 x y + 1

(s) ds

M(0) = 1

y2

0 +

1 x2

0

y0 x0

2 t

0

y u es

0 (1x2) dr ds

N(0) = 1

t0

u

y2 + 1 x2

y x2

1 x2

y x

(s)

(s)y

ds

Unfortunately we are not able to solve the last three integrals by hand, however com-

puting the Harmonic Balance approximations of x(t), y(t) and u(t) for every t0 (0, T]

and introducing such coefficients in the expression of the integrals above, the bifurcation

function can be computed for every0lying on the unperturbed limit cycle.

Figure (7) shows the evolution of the bifurcation function (72) versus the initial con-

dition for = 2. The two zeroes ofB(0) identify two subharmonic branch points. Both

their number and locations are in good agreement with numerical simulations carried out

for small values of.

T

B(x0)

0.25 0.5 0.75

0

Figure 7: Bifurcation function B(0) as a function of the initial condition.

5. C

Considering recent enhancements to the classical subharmonic Melnikovs method, the

effect of periodic perturbations on both non-hyperbolic and hyperbolic oscillators have

been investigated.

As a main contribution we have shown that Melnikovs integrals can be accurately

computed over the Harmonic Balance approximation of the unperturbed trajectories. The

main advantage is that the integrability of the unperturbed system is not required anymore,and the applicability of the Melnikovs method is considerably extended. Moreover, since


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NOURP 25

the unperturbed trajectories are expressed as truncated Fourier series, the Melnikovs in-

tegrals results rather simple to solve.

The joint application of the Harmonic Balance technique and the Melnikovs method,

has allowed to investigate the emergence of periodic orbits from period annulus under

either autonomous or non-autonomous perturbations. We have shown how, in the former

case, the stability of the bifurcating limit cycle can be analytically determined.

In the case of limit cycle oscillators, we have shown the possibility to determine the

number and locations of subharmonic branch points, i.e. intersections between the per-

turbed and the unperturbed orbits.

A

This research was partially supported by the Ministero dellIstruzione, dellUniversita e

della Ricerca, under the FIRB project no. RBAU01LRKJ.

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HBM Method and Nonlinear Oscillators Under Resosnant

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