Post on 14-Dec-2015
JEAN-MARC GINOUX JEAN-MARC GINOUX BRUNO ROSSETTO BRUNO ROSSETTO ginoux@univ-tln.frginoux@univ-tln.fr rossetto@univ-tln.frrossetto@univ-tln.fr
http://ginoux.univ-tln.frhttp://ginoux.univ-tln.fr http://rossetto.univ-tln.frhttp://rossetto.univ-tln.fr
Laboratoire PROTEE, I.U.T. de Toulon Laboratoire PROTEE, I.U.T. de Toulon Université du Sud, Université du Sud,
B.P. 20132, 83957, LA GARDE Cedex, FranceB.P. 20132, 83957, LA GARDE Cedex, France
Differential Geometry Differential Geometry Applied to Applied to
Dynamical SystemsDynamical Systems
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A. Modeling & Dynamical SystemsA. Modeling & Dynamical Systems 1. Definition & Features1. Definition & Features 2. Classical analytical approaches2. Classical analytical approaches
B. Flow Curvature MethodB. Flow Curvature Method 1. Presentation1. Presentation 2. Results2. Results
C. ApplicationsC. Applications 1. n-dimensional dynamical systems1. n-dimensional dynamical systems 2. Non-autonomous dynamical systems2. Non-autonomous dynamical systems
OUTLINEOUTLINE
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MODELING DYNAMICAL SYSTEMSMODELING DYNAMICAL SYSTEMS
ModelingModeling::
Defining states variables of a system (predator, prey)Defining states variables of a system (predator, prey) Describing their evolution with differential equations (O.D.E.)Describing their evolution with differential equations (O.D.E.)
Dynamical SystemDynamical System::
Representation of a differential equation in phase spaceRepresentation of a differential equation in phase space
expresses variation of each state variableexpresses variation of each state variable
Determining variables from their variation (velocity)Determining variables from their variation (velocity)
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dXX
dt
��������������
1 2, ,...,t
nnX f X f X f X E
��������������
1 2
t nnX x x x E
n-dimensional Dynamical Systn-dimensional Dynamical Systeemsms
V X���������������������������� velocity
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MANIFOLD DEFINTIONMANIFOLD DEFINTION
A manifold isA manifold is defined as a set of points indefined as a set of points in satisfying a system of satisfying a system of mm scalar equations : scalar equations :
where for with where for with The manifold The manifold MM is is differentiabledifferentiable if if is differentiable and if theis differentiable and if therank of the jacobian matrix is equal to rank of the jacobian matrix is equal to in each point . in each point .
Thus, in each pointThus, in each point of of the différentiable manifold , the différentiable manifold , a tangent a tangent space ofspace of dimension dimension is defined.is defined.
In dimension 2 In dimension 3In dimension 2 In dimension 3 curve surfacecurve surface
nnM
0X
: n m m n 1 2, ,...,t n
nX x x x E
XD m
n m
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Let a function defined in a compact E included in and Let a function defined in a compact E included in and
the integral of the dynamical system defined by (1). the integral of the dynamical system defined by (1).
The Lie derivative is defined as follows:The Lie derivative is defined as follows:
If If then is first integral of the dynamical system (1). then is first integral of the dynamical system (1).
So, is constant along each So, is constant along each trajectory curvetrajectory curve and the first and the first
integrals are drawn on the integrals are drawn on the hypersurfaceshypersurfaces of level set of level set
( is a constant) which are over( is a constant) which are over flowing invariant.flowing invariant.
X t
1
n
iVi i
dL V x
x dt
������������������������������������������
0VL ��������������
LIE DERIVATIVELIE DERIVATIVE
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Darboux Theorem for Invariant ManifoldsDarboux Theorem for Invariant Manifolds::
An invariant manifold (An invariant manifold (curvecurve or or surfacesurface) is a manifold) is a manifold
defined by where is a defined by where is a
function in the open set U and such that there exists a function in the open set U and such that there exists a
function in U denoted and called cofactor such that:function in U denoted and called cofactor such that:
for all for all
This notion is due to Gaston Darboux (1878)This notion is due to Gaston Darboux (1878)
k X
: nU 1C1C
VL X k X X ��������������
0X
X U
INVARIANT MANIFOLDSINVARIANT MANIFOLDS
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Manifold implicit equation: Manifold implicit equation:
Instantaneous velocity vector:Instantaneous velocity vector:
Normal vector:Normal vector:
attractive manifoldattractive manifold tangent manifoldtangent manifold repulsive manifoldrepulsive manifold
This notion is due to Henri Poincaré (1881)This notion is due to Henri Poincaré (1881)
0X
0 VL ��������������
0 VL ��������������
0 VL ��������������
������������� � V t
��������������
M
V t��������������
C
ATTRACTIVE MANIFOLDSATTRACTIVE MANIFOLDS
Poincaré’s criterionPoincaré’s criterion : :
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Fixed PointsFixed Points
Local BifurcationsLocal Bifurcations
Invariant manifolds Invariant manifolds center manifoldscenter manifolds slow manifolds (local integrals)slow manifolds (local integrals) linear manifolds (global integrals)linear manifolds (global integrals)
Normal FormsNormal Forms
DYNAMICAL SYSTDYNAMICAL SYSTEEMSMS
Dynamical SystemsDynamical Systems::Integrables or non-integrables analyticallyIntegrables or non-integrables analytically
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Courbes définies par une équation différentielleCourbes définies par une équation différentielle (Poincaré, 1881(Poincaré, 1881 1886) 1886)
…………………….…..….
Singular Perturbation MethodsSingular Perturbation Methods (Poincaré, 1892, Andronov 1937, Cole 1968, Fenichel 1971, O'Malley 1974)(Poincaré, 1892, Andronov 1937, Cole 1968, Fenichel 1971, O'Malley 1974)
Tangent Linear System ApproximationTangent Linear System Approximation
(Rossetto, 1998 & Ramdani, 1999)(Rossetto, 1998 & Ramdani, 1999)
« Classical » analytic methods« Classical » analytic methods
DYNAMICAL SYSTDYNAMICAL SYSTEEMSMS
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Flow Curvature MethodFlow Curvature Method(Ginoux & Rossetto, 2005 (Ginoux & Rossetto, 2005 2009) 2009)
velocityvelocity
velocity velocity acceleration acceleration over-acceleration over-acceleration etc. … etc. …
Geometric MethodGeometric Method
FLOW CURVATURE METHODFLOW CURVATURE METHOD
position position
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plane or space curve
““trajectory curve” trajectory curve”
curvatures
FLOW CURVATURE METHODFLOW CURVATURE METHOD
n-Euclidean space curve
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Flow curvature manifoldFlow curvature manifold::
The The flow curvature manifoldflow curvature manifold is defined as the location is defined as the location
of the points whereof the points where the curvature of the flow the curvature of the flow, i.e., the, i.e., the
curvaturecurvature of of trajectory curvetrajectory curve integral of the dynamical integral of the dynamical
system vanishes.system vanishes.
where represents the where represents the nn-th derivative-th derivative
, , , , 0n n
X X X X X det X X X X
nX
X
FLOW CURVATURE METHODFLOW CURVATURE METHOD
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Flow Curvature ManifoldFlow Curvature Manifold::
In dimension 2In dimension 2: :
curvaturecurvature or 1 or 1stst curvature curvature
In dimension 3In dimension 3: :
torsiontorsion ou 2 ou 2ndnd curvature curvature
, , 0X det X X X
, 0X det X X
FLOW CURVATURE METHODFLOW CURVATURE METHOD
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Flow Curvature ManifoldFlow Curvature Manifold::
In dimension 4In dimension 4: :
33rdrd curvature curvature
In dimension 5In dimension 5: :
44thth curvature curvature
5
, , , , 0X det X X X X X
4
, , , 0X det X X X X
FLOW CURVATURE METHODFLOW CURVATURE METHOD
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Theorem 1Theorem 1 ( (GinouxGinoux, 2009), 2009)
Fixed points of any n-dimensional dynamical system Fixed points of any n-dimensional dynamical system
are singular solutionare singular solution of the flow curvature manifold of the flow curvature manifold
Corollary 1Corollary 1
Fixed points of the Fixed points of the flow curvature manifoldflow curvature manifold
are defined byare defined by
*
*
0
0
X
X
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FIXED POINTSFIXED POINTS
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Theorem 2Theorem 2::(Poincaré 1881(Poincaré 1881 GinouxGinoux, 2009), 2009)
Hessian of Hessian of flow curvature manifoldflow curvature manifold
associated to dynamical system enables differenting associated to dynamical system enables differenting foci from saddles (resp. nodes).foci from saddles (resp. nodes).
2 2
2
2 2
2
X
x x y
y x y
FIXED POINTS STABILITYFIXED POINTS STABILITY
, , , , 0n n
X X X X X det X X X X
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Unforced Duffing oscillatorUnforced Duffing oscillator
andand
Thus is a saddle point or a node Thus is a saddle point or a node
4 0X
FIXED POINTS STABILITYFIXED POINTS STABILITY
3
,
,
f x yx yV
g x yy x x
����������������������������
22 2 2 21 3 0X y x x y
3 5 2
3
2 4 3 3
2 6
x x x xyX
x y
��������������
0,0O
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Theorem 3Theorem 3 ( (GinouxGinoux, 2009), 2009)
Center manifold associated to any n-dimensional Center manifold associated to any n-dimensional dynamical system is a polynomial whose coefficients dynamical system is a polynomial whose coefficients may be directly deduced from flow curvature manifoldmay be directly deduced from flow curvature manifold
withwith
CENTER MANIFOLDCENTER MANIFOLD
2 30 20 30
0
. . .n
pp
p
y h x a x a x a x h o t
10
1,00
1
1 !
n
n nx
aa lim
n x
10
y h x
xa
y
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Guckenheimer Guckenheimer et al. (1983)et al. (1983)
Local BifurcationsLocal Bifurcations
CENTER MANIFOLDCENTER MANIFOLD
2
,
,
f x yx xyV
g x yy y x
����������������������������
3 2 2 2 2 4,x y x y x y x y x
2 3 420 30y h x a x a x O x
10 2020
0
5 31
2! 2x
a aa lim
x
20a
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Theorem 4Theorem 4
((Ginoux & RossettoGinoux & Rossetto, 2005 , 2005 2009) 2009)
Flow curvature manifold of any n-dimensional slow-fast Flow curvature manifold of any n-dimensional slow-fast
dynamical system directly provides its slow manifold dynamical system directly provides its slow manifold
analytical equation and represents a local first integral analytical equation and represents a local first integral
of this system.of this system.
SLOW INVARIANT MANIFOLDSLOW INVARIANT MANIFOLD
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VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
31
3
xx yx
Vy
x
��������������
1 1, , x y
0.05
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-3 -2 -1 1 2 3X
-3
-2
-1
1
2
3
Y
VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
3
3
xy x
slow part slow partslow part slow part
-3 -2 -1 1 2 3X
-3
-2
-1
1
2
3
Y
0.05
Singular approximationSingular approximation
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-3 -2 -1 1 2 3X
-3
-2
-1
1
2
3
Y
slow part slow partslow part slow part
-3 -2 -1 1 2 3X
-3
-2
-1
1
2
3
Y
VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
, 0X X
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Slow manifoldSlow manifold Lie derivativeLie derivative
-2 -1 0 1 2
-2
-1
0
1
2
X
Y
Singular approximationSingular approximation
VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
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Slow Manifold Analytical Equation Slow Manifold Analytical Equation
Flow Curvature Method vs Singular Perturbation MethodFlow Curvature Method vs Singular Perturbation Method
(Fenichel, 1979 vs Ginoux 2009) (Fenichel, 1979 vs Ginoux 2009)
23
2 342 2
1
3 1 1
x xx xy x O
x x
VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
2 3 4 6 2, , 9 9 3 6 2 9 0x y y x x y x x x
2 30 1 2y Y x Y x Y x O
32 3
32 23 1 1
x x xy x O
x x
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VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
Flow Curvature Method vs Flow Curvature Method vs
Singular Perturbation Method (up to order )Singular Perturbation Method (up to order )2
Singular perturbation Flow Curvature
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VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)Slow Manifold Analytical Equation given bySlow Manifold Analytical Equation given by
Flow Curvature Method & Singular Perturbation MethodFlow Curvature Method & Singular Perturbation Method
are identical up to order one in are identical up to order one in
Pr. Eric BenoîtPr. Eric Benoît
High order approximations are simply given by High order approximations are simply given by
high order derivatives, e. g., order 2 in is given byhigh order derivatives, e. g., order 2 in is given by
the Lie derivative of the flow curvature manifold, etc…the Lie derivative of the flow curvature manifold, etc…
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VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)
Slow manifold attractive domainSlow manifold attractive domain
0 VL ��������������
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Theorem 5Theorem 5
(Darboux, 1878 (Darboux, 1878 Ginoux,Ginoux, 2009) 2009)
Every linear manifold (line, plane, hyperplane) invariant Every linear manifold (line, plane, hyperplane) invariant
with respect to the flow of any n-dimensional dynamical with respect to the flow of any n-dimensional dynamical
system is a factor in the flow curvature manifold.system is a factor in the flow curvature manifold.
LINEAR INVARIANT MANIFOLDLINEAR INVARIANT MANIFOLD
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CHUA's piecewise linear modelCHUA's piecewise linear model::
1, ,
, ,
, ,
dxy x k xdt f x y z
dyV g x y z x y z
dth x y z y
dz
dt
����������������������������
APPLICATIONS 3DAPPLICATIONS 3D
1/ 9 ; 100 7 ; 8 / 7 ; 5 / 7a b
1
1
1
bx a b x
k x ax x
bx a b x
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CHUA's piecewise linear modelCHUA's piecewise linear model::
Slow invariant manifold analytical equation Slow invariant manifold analytical equation
HyperplanesHyperplanes
1,2 2.8759 3.9421 2.8139 0X x y z
APPLICATIONS 3DAPPLICATIONS 3D
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CHUA's piecewise linear modelCHUA's piecewise linear model::
Invariant Hyperplanes (Darboux)Invariant Hyperplanes (Darboux)
1,2 1 1,2VL X X ��������������
APPLICATIONS 3DAPPLICATIONS 3D
1,2 0X X X X X Q X
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CHUA's piecewise linear modelCHUA's piecewise linear model::
APPLICATIONS 3DAPPLICATIONS 3D
Invariant PlanesInvariant Planes Invariant PlanesInvariant Planes
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with and with and
3 244 41, , 3 2
, ,
, , 0.7 0.24
dxz x x xdt f x y z
dyV g x y z z
dth x y z x y z
dz
dt
����������������������������
2 0.05
CHUA's cubic modelCHUA's cubic model::
APPLICATIONS 3DAPPLICATIONS 3D
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APPLICATIONS 3DAPPLICATIONS 3D
Slow manifold Slow manifold
Slow manifoldSlow manifold
CHUA's cubic modelCHUA's cubic model::
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Edward Lorenz model (1963)Edward Lorenz model (1963)::
, ,
, ,
, ,
dx
dt f x y z y xdy
V g x y z xz rx ydt
h x y z xy zdz
dt
����������������������������
APPLICATIONS 3DAPPLICATIONS 3D
810 ; 28 ;
3r
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Edward Lorenz modelEdward Lorenz model::
Slow invariant analytic manifold (Theorem 4)Slow invariant analytic manifold (Theorem 4)
APPLICATIONS 3DAPPLICATIONS 3D
x, y, z 56448x4 784x6 191352x3y308x5y
367338x2 y2
148x4y2
3
x6 y2 81300xy3670x3y3
3 7200y4
380x2y4
3 100xy5 589120x2z
23968x4z
9 56x6z 189360xyz
599503
x3yz53x5yz 210400y2 z
561409
x2 y2z186803
xy3 z10x3 y3z800y4z
31550864x2 z2
2716x4 z2
x6 z248560027
xyz223003
x3 yz2404800y2z2
273503
x2 y2z2 100xy3 z2
18400x2 z3
934409
xyz3 10x3yz3 800y2 z3
380x2z4
3 0
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APPLICATIONS 3DAPPLICATIONS 3D
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Autocatalator Autocatalator Neuronal Bursting ModelNeuronal Bursting Model
APPLICATIONS 3DAPPLICATIONS 3D
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Chua cubic 4DChua cubic 4D [Thamilmaran [Thamilmaran et al.et al., 2004, , 2004, Liu Liu et al.et al., 2007, 2007]]
APPLICATIONS 4DAPPLICATIONS 4D
1
1 3 12
2 2 3 4
3 1 2 1 3
2 2
4
dx
dtx k xdxx x xdtV
dx x x xdt xdx
dt
��������������
31 1 1 2 1 1 2 1 1 2
ˆ ; 2.1429 ; 0.18 ; 0.0774 ; 0.3937 ; 0.7235k x c x c x c c
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Chua cubic 5DChua cubic 5D[Hao [Hao et al.et al., 2005] , 2005]
APPLICATIONS 5DAPPLICATIONS 5D
1
2 1 2 1 1
2 1 2 3
31 4 2
2 3 54
2 4 1 5
5
dx
dtdx x x k xdt x x xdx
x xVdt
x xdxx xdt
dx
dt
��������������
1 2 1 2 1 2 1 29.934 ; 1 ; 14.47 ; 406.5 ; 0.0152 ; 41000 ; 0.1068 ; 0.3056c c
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Edgar Knobloch modelEdgar Knobloch model::
1
1 2 4 522
2 1 1 33
3 1 2
4 4 1 1 5
5 5 1 4
31
4
4
4
dx
dtx rx qx x
dx
dtx x x x
dxV x x x
dtdx x x x xdtdx x x x
dt
����������������������������
APPLICATIONS 5DAPPLICATIONS 5D
0.09683 ; 14.47 ; 0.1081 ; 5 ; 1r q
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APPLICATIONS 5DAPPLICATIONS 5D
MagnetoConvection MagnetoConvection
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NON-AUTONOMOUS DYNAMICAL SYSTEMSNON-AUTONOMOUS DYNAMICAL SYSTEMS
Forced Van der PolForced Van der Pol
Guckenheimer Guckenheimer et al.et al., 2003 , 2003
31
3, ,
, , 2
, , 1
dx xx ydt f x y z
dyV g x y z x aSin
dth x y z
d
dt
����������������������������
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NON-AUTONOMOUS DYNAMICAL SYSTEMSNON-AUTONOMOUS DYNAMICAL SYSTEMS
Forced Van der PolForced Van der Pol
Guckenheimer Guckenheimer et al.et al., 2003 , 2003
1
31
1 21 1 2 3 42
2 1 2 3 4
1 33 1 2 3 43
44 1 2 3 4
34
1, , ,
3, , ,
, , ,
, , ,
dx
dt xx xf x x x xdx
f x x x xdtV x axf x x x xdx
xdt f x x x x
xdx
dt
����������������������������
2
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NON-AUTONOMOUS DYNAMICAL SYSTEMSNON-AUTONOMOUS DYNAMICAL SYSTEMS
Forced Van der PolForced Van der Pol
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Theorem 6Theorem 6 : :(Poincaré 1879 (Poincaré 1879 GinouxGinoux, 2009), 2009)
Normal form associated to any n-dimensionalNormal form associated to any n-dimensional
dynamical system may be deduced from flow dynamical system may be deduced from flow
curvature manifoldcurvature manifold
Normal FormNormal Form
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• Fixed Points & StabilityFixed Points & Stability: : - Flow Curvature Manifold: Theorems 1 & 2- Flow Curvature Manifold: Theorems 1 & 2
• Center, Slow & LinearCenter, Slow & Linear
Manifold Analytical EquationManifold Analytical Equation: : - Theorems 3, 4 & 5- Theorems 3, 4 & 5
• Normal FormsNormal Forms: : - Theorem 6- Theorem 6
FLOW CURVATURE METHODFLOW CURVATURE METHOD
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Flow Curvature MethodFlow Curvature Method::n-dimensional dynamical systemsn-dimensional dynamical systemsAutonomous or Non-autonomousAutonomous or Non-autonomous
Fixed points & stability, local bifurcations, normal formsFixed points & stability, local bifurcations, normal forms Center manifoldsCenter manifolds Slow invariant manifoldsSlow invariant manifolds Linear invariant manifolds (lines, planes, hyperplanes,…)Linear invariant manifolds (lines, planes, hyperplanes,…)
ApplicationsApplications : : Electronics, Meteorology, Biology, Chemistry…Electronics, Meteorology, Biology, Chemistry…
DISCUSSIONDISCUSSION
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BookBook
Differential Geometry Differential Geometry
Applied to Applied to
Dynamical SystemsDynamical Systems
World Scientific Series on World Scientific Series on
Nonlinear ScienceNonlinear Science,, series A, 2009 series A, 2009
PublicationsPublications
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Thanks for your attention.Thanks for your attention.
To be continued…To be continued…