JEAN-MARC GINOUX BRUNO ROSSETTO JEAN-MARC GINOUX BRUNO ROSSETTO [email protected]...

JEAN-MARC GINOUX JEAN-MARC GINOUX BRUNO ROSSETTO BRUNO ROSSETTO [email protected]@univ-tln.fr [email protected]@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

http://ginoux.univ-tln.frhttp://ginoux.univ-tln.fr Hairer's 60th birthdayHairer's 60th birthday 22

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


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)


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


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


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


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


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 : :


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


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


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


plane or space curve

““trajectory curve” trajectory curve”

curvatures


n-Euclidean space curve


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 ManifoldFlow Curvature Manifold::

In dimension 2In dimension 2: :

curvaturecurvature or 1 or 1stst curvature curvature


torsiontorsion ou 2 ou 2ndnd curvature curvature

, , 0X det X X X

, 0X det X X



Flow Curvature ManifoldFlow Curvature Manifold::


33rdrd curvature curvature


44thth curvature curvature

5

, , , , 0X det X X X X X

4

, , , 0X det X X X X



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

��

FIXED POINTSFIXED POINTS


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


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


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


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


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


VAN DER POL SYSTEM (1926)VAN DER POL SYSTEM (1926)

31

3

xx yx

Vy

x

��

1 1, , x y

0.05


-3 -2 -1 1 2 3X

-3

-2

-1

1

2

3

Y


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


-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


, 0X X


Slow manifoldSlow manifold Lie derivativeLie derivative

-2 -1 0 1 2

-2

-1

0

1

2

X

Y

Singular approximationSingular approximation



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


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



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


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…



Slow manifold attractive domainSlow manifold attractive domain

0 VL ��


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


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



Slow invariant manifold analytical equation Slow invariant manifold analytical equation

HyperplanesHyperplanes

1,2 2.8759 3.9421 2.8139 0X x y z




Invariant Hyperplanes (Darboux)Invariant Hyperplanes (Darboux)

1,2 1 1,2VL X X ��


1,2 0X X X X X Q X




Invariant PlanesInvariant Planes Invariant PlanesInvariant Planes


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::




Slow manifold Slow manifold

Slow manifoldSlow manifold

CHUA's cubic modelCHUA's cubic model::


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

��


810 ; 28 ;

3r


Edward Lorenz modelEdward Lorenz model::

Slow invariant analytic manifold (Theorem 4)Slow invariant analytic manifold (Theorem 4)


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


Autocatalator Autocatalator Neuronal Bursting ModelNeuronal Bursting Model



Chua cubic 4DChua cubic 4D [Thamilmaran [Thamilmaran et al.et al., 2004, , 2004, Liu Liu et al.et al., 2007, 2007]]


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


Chua cubic 5DChua cubic 5D[Hao [Hao et al.et al., 2005] , 2005]


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


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

��


0.09683 ; 14.47 ; 0.1081 ; 5 ; 1r q



MagnetoConvection MagnetoConvection


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

��




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


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


• 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::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


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


Thanks for your attention.Thanks for your attention.

To be continued…To be continued…

JEAN-MARC GINOUX BRUNO ROSSETTO JEAN-MARC GINOUX BRUNO ROSSETTO [email protected]...

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