Differential Geometry
Download
1 / 52

JEAN-MARC GINOUX BRUNO ROSSETTO [email protected] [email protected] - PowerPoint PPT Presentation


  • 52 Views
  • Uploaded on

Differential Geometry Applied to Dynamical Systems. JEAN-MARC GINOUX BRUNO ROSSETTO [email protected] [email protected] http://ginoux.univ-tln.fr http://rossetto.univ-tln.fr Laboratoire PROTEE, I.U.T. de Toulon Université du Sud,

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' JEAN-MARC GINOUX BRUNO ROSSETTO [email protected] [email protected]' - patricia-mills


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Differential Geometry Applied to Dynamical Systems

JEAN-MARC GINOUX BRUNO ROSSETTO

[email protected]@univ-tln.fr

http://ginoux.univ-tln.frhttp://rossetto.univ-tln.fr

Laboratoire PROTEE, I.U.T. de Toulon

Université du Sud,

B.P. 20132, 83957, LA GARDE Cedex, France


OUTLINE

A. Modeling & Dynamical Systems

1. Definition & Features

2. Classical analytical approaches

B. Flow Curvature Method

1. Presentation

2. Results

C. Applications

1. n-dimensional dynamical systems

2. Non-autonomous dynamical systems

Hairer's 60th birthday


Modeling dynamical systems
MODELING DYNAMICAL SYSTEMS

Modeling:

  • Defining states variables of a system (predator, prey)

  • Describing their evolution with differential equations (O.D.E.)

Dynamical System:

Representation of a differential equation in phase space

expresses variation of each state variable

 Determining variables from their variation (velocity)

Hairer's 60th birthday


N dimensional dynamical syst e ms
n-dimensional Dynamical Systems

velocity

Hairer's 60th birthday


MANIFOLD DEFINTION

A manifold isdefined as a set of points in

satisfying a system of m scalar equations :

where for with

The manifold Mis differentiable if is differentiable and if the

rank of the jacobian matrix is equal to in each point .

Thus, in each point of the différentiable manifold ,

a tangent space of dimension is defined.

In dimension 2 In dimension 3

curve surface

Hairer's 60th birthday


LIE DERIVATIVE

Let a function defined in a compact E included in and

the integral of the dynamical system defined by (1).

The Lie derivative is defined as follows:

If then is first integral of the dynamical system (1).

So, is constant along each trajectory curve and the first

integrals are drawn on the hypersurfaces of level set

( is a constant) which are overflowing invariant.

Hairer's 60th birthday


INVARIANT MANIFOLDS

Darboux Theorem for Invariant Manifolds:

An invariant manifold (curve or surface) is a manifold

defined by where is a

function in the open set U and such that there exists a

function in U denoted and called cofactor such that:

for all

This notion is due to Gaston Darboux (1878)

Hairer's 60th birthday


ATTRACTIVE MANIFOLDS

Poincaré’s criterion :

Manifold implicit equation:

Instantaneous velocity vector:

Normal vector:

attractive manifold

tangent manifold

repulsive manifold

This notion is due to Henri Poincaré (1881)

Hairer's 60th birthday


Dynamical syst e ms
DYNAMICAL SYSTEMS

Dynamical Systems:

Integrables or non-integrables analytically

  • Fixed Points

  • Local Bifurcations

  • Invariant manifolds

  •  center manifolds

  •  slow manifolds (local integrals)

  •  linear manifolds (global integrals)

  • Normal Forms

Hairer's 60th birthday


Dynamical syst e ms1
DYNAMICAL SYSTEMS

« Classical » analytic methods

  • Courbes définies par une équation différentielle

    (Poincaré, 1881 1886)

    ………….….

  • Singular Perturbation Methods

    (Poincaré, 1892, Andronov 1937, Cole 1968, Fenichel 1971, O'Malley 1974)

  • Tangent Linear System Approximation

    (Rossetto, 1998 & Ramdani, 1999)

Hairer's 60th birthday


Flow curvature method
FLOW CURVATURE METHOD

Geometric Method

Flow Curvature Method

(Ginoux & Rossetto, 2005  2009)

velocity

velocity  acceleration  over-acceleration  etc. …

position 

Hairer's 60th birthday


Flow curvature method1
FLOW CURVATURE METHOD

“trajectory curve”

n-Euclidean space curve

plane or space curve

curvatures

Hairer's 60th birthday


Flow curvature method2
FLOW CURVATURE METHOD

Flow curvature manifold:

The flow curvature manifold is defined as the location

of the points where the curvature of the flow, i.e., the

curvature of trajectory curve integral of the dynamical

system vanishes.

where represents the n-th derivative

Hairer's 60th birthday


Flow curvature method3
FLOW CURVATURE METHOD

Flow Curvature Manifold:

In dimension 2:

curvature or 1st curvature

In dimension 3:

torsion ou 2nd curvature

Hairer's 60th birthday


Flow curvature method4
FLOW CURVATURE METHOD

Flow Curvature Manifold:

In dimension 4:

3rd curvature

In dimension 5:

4th curvature

Hairer's 60th birthday


FIXED POINTS

Theorem 1 (Ginoux, 2009)

Fixed points of any n-dimensional dynamical system

are singular solution of the flow curvature manifold

Corollary 1

Fixed points of the flow curvature manifold

are defined by

Hairer's 60th birthday


FIXED POINTS STABILITY

Theorem 2:

(Poincaré 1881 Ginoux, 2009)

Hessian of flow curvature manifold

associated to dynamical system enables differenting foci from saddles (resp. nodes).

Hairer's 60th birthday


FIXED POINTS STABILITY

Unforced Duffing oscillator

and

Thus is a saddle point or a node

Hairer's 60th birthday


CENTER MANIFOLD

Theorem 3 (Ginoux, 2009)

Center manifold associated to any n-dimensional

dynamical system is a polynomial whose coefficients

may be directly deduced from flow curvature manifold

with

Hairer's 60th birthday


CENTER MANIFOLD

Guckenheimer et al. (1983)

Local Bifurcations

Hairer's 60th birthday


SLOW INVARIANT MANIFOLD

Theorem 4

(Ginoux & Rossetto, 2005  2009)

Flow curvature manifold of any n-dimensional slow-fast

dynamical system directly provides its slow manifold

analytical equation and represents a local first integral

of this system.

Hairer's 60th birthday


Van der pol system 1926
VAN DER POL SYSTEM (1926)

Hairer's 60th birthday


Van der pol system 19261
VAN DER POL SYSTEM (1926)

slow part slow part

Singular approximation

Hairer's 60th birthday


Van der pol system 19262
VAN DER POL SYSTEM (1926)

slow part slow part

Hairer's 60th birthday


Van der pol system 19263
VAN DER POL SYSTEM (1926)

Slow manifoldLie derivative

Singular approximation

Hairer's 60th birthday


Van der pol system 19264
VAN DER POL SYSTEM (1926)

Slow Manifold Analytical Equation

Flow Curvature Method vs Singular Perturbation Method

(Fenichel, 1979 vs Ginoux 2009)

Hairer's 60th birthday


Van der pol system 19265
VAN DER POL SYSTEM (1926)

Flow Curvature Method vs

Singular Perturbation Method (up to order )

Flow Curvature

Singular perturbation

Hairer's 60th birthday


Van der pol system 19266
VAN DER POL SYSTEM (1926)

Slow Manifold Analytical Equation given by

Flow Curvature Method & Singular Perturbation Method

are identical up to order one in

 Pr. Eric Benoît

High order approximations are simply given by

high order derivatives, e. g., order 2 in is given by

the Lie derivative of the flow curvature manifold, etc…

Hairer's 60th birthday


Van der pol system 19267
VAN DER POL SYSTEM (1926)

Slow manifold attractive domain

Hairer's 60th birthday


LINEAR INVARIANT MANIFOLD

Theorem 5

(Darboux, 1878  Ginoux,2009)

Every linear manifold (line, plane, hyperplane) invariant

with respect to the flow of any n-dimensional dynamical

system is a factor in the flow curvature manifold.

Hairer's 60th birthday


APPLICATIONS 3D

CHUA's piecewise linear model:

Hairer's 60th birthday


APPLICATIONS 3D

CHUA's piecewise linear model:

Slow invariant manifold analytical equation

Hyperplanes

Hairer's 60th birthday


APPLICATIONS 3D

CHUA's piecewise linear model:

Invariant Hyperplanes (Darboux)

Hairer's 60th birthday


APPLICATIONS 3D

CHUA's piecewise linear model:

Invariant Planes

Invariant Planes

Hairer's 60th birthday


APPLICATIONS 3D

CHUA's cubic model:

with and

Hairer's 60th birthday


APPLICATIONS 3D

CHUA's cubic model:

Slow manifold

Slow manifold

Hairer's 60th birthday


APPLICATIONS 3D

Edward Lorenz model (1963):

Hairer's 60th birthday


APPLICATIONS 3D

Edward Lorenz model:

Slow invariant analytic manifold (Theorem 4)

Hairer's 60th birthday


APPLICATIONS 3D

Hairer's 60th birthday


APPLICATIONS 3D

Neuronal Bursting Model

Autocatalator

Hairer's 60th birthday


APPLICATIONS 4D

Chua cubic 4D

[Thamilmaran et al., 2004, Liu et al., 2007]

Hairer's 60th birthday


APPLICATIONS 5D

Chua cubic 5D

[Hao et al., 2005]

Hairer's 60th birthday


APPLICATIONS 5D

Edgar Knobloch model:

Hairer's 60th birthday


APPLICATIONS 5D

MagnetoConvection

Hairer's 60th birthday


NON-AUTONOMOUS DYNAMICAL SYSTEMS

Forced Van der Pol

Guckenheimer et al., 2003

Hairer's 60th birthday


NON-AUTONOMOUS DYNAMICAL SYSTEMS

Forced Van der Pol

Guckenheimer et al., 2003

Hairer's 60th birthday


NON-AUTONOMOUS DYNAMICAL SYSTEMS

Forced Van der Pol

Hairer's 60th birthday


Normal Form

Theorem 6 :

(Poincaré 1879  Ginoux, 2009)

Normal form associated to any n-dimensional

dynamical system may be deduced from flow

curvature manifold

Hairer's 60th birthday


FLOW CURVATURE METHOD

  • Fixed Points & Stability:

    - Flow Curvature Manifold: Theorems 1 & 2

  • Center, Slow & Linear

    Manifold Analytical Equation:

    - Theorems 3, 4 & 5

  • Normal Forms:

    - Theorem 6

Hairer's 60th birthday


DISCUSSION

Flow Curvature Method:

n-dimensional dynamical systems

Autonomous or Non-autonomous

  • Fixed points & stability, local bifurcations, normal forms

  • Center manifolds

  • Slow invariant manifolds

  • Linear invariant manifolds (lines, planes, hyperplanes,…)

    Applications :

  • Electronics, Meteorology, Biology, Chemistry…

Hairer's 60th birthday


Publications

Book

Differential Geometry

Applied to

Dynamical Systems

World Scientific Series on

Nonlinear Science, series A, 2009

Hairer's 60th birthday


Thanks for your attention.

To be continued…

Hairer's 60th birthday


ad