# Differential Geometry Applied to Dynamical Systems

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Differential Geometry Applied to Dynamical Systems
JEAN-MARC GINOUX BRUNO ROSSETTO Laboratoire PROTEE, I.U.T. de Toulon Université du Sud, B.P , 83957, LA GARDE Cedex, France

OUTLINE A. Modeling & Dynamical Systems B. Flow Curvature Method
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
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 Systems
velocity Hairer's 60th birthday

MANIFOLD DEFINTION A manifold is defined as a set of points in
satisfying a system of m scalar equations : where for with The manifold M is 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 In dimension 3 curve surface Hairer's 60th birthday

The Lie derivative is defined as follows:
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 over flowing 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 SYSTEMS Dynamical Systems: Fixed Points Local Bifurcations
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

« Classical » analytic methods
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

velocity  acceleration  over-acceleration  etc. …
FLOW CURVATURE METHOD Geometric Method Flow Curvature Method (Ginoux & Rossetto, 2005  2009) velocity velocity  acceleration  over-acceleration  etc. … position  Hairer's 60th birthday

n-Euclidean space curve
FLOW CURVATURE METHOD “trajectory curve” n-Euclidean space curve plane or space curve curvatures Hairer's 60th birthday

Flow curvature manifold:
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 METHOD Flow Curvature Manifold: In dimension 2:
curvature or 1st curvature In dimension 3: torsion ou 2nd curvature Hairer's 60th birthday

Flow Curvature Manifold:
FLOW CURVATURE METHOD Flow Curvature Manifold: In dimension 4: 3rd curvature In dimension 5: 4th curvature Hairer's 60th birthday

Fixed points of the flow curvature manifold
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
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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) http://ginoux.univ-tln.fr
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VAN DER POL SYSTEM (1926) slow part slow part Singular approximation
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VAN DER POL SYSTEM (1926) slow part slow part
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VAN DER POL SYSTEM (1926) Slow manifold Lie derivative
Singular approximation Slow manifold Lie derivative Hairer's 60th birthday

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 (1926) Flow Curvature Method vs
Singular Perturbation Method (up to order ) Singular perturbation Flow Curvature Hairer's 60th birthday

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

Slow manifold attractive domain
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

CHUA's piecewise linear model:
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

CHUA's piecewise linear model:
APPLICATIONS 3D CHUA's piecewise linear model: Invariant Planes Invariant Planes Hairer's 60th birthday

APPLICATIONS 3D CHUA's cubic model: with and http://ginoux.univ-tln.fr
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APPLICATIONS 3D CHUA's cubic model: Slow manifold Slow manifold
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Edward Lorenz model (1963):
APPLICATIONS 3D Edward Lorenz model (1963): Hairer's 60th birthday

Slow invariant analytic manifold (Theorem 4)
APPLICATIONS 3D Edward Lorenz model: Slow invariant analytic manifold (Theorem 4) Hairer's 60th birthday

APPLICATIONS 3D Hairer's 60th birthday

APPLICATIONS 3D Autocatalator Neuronal Bursting Model
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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]
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APPLICATIONS 5D Edgar Knobloch model: http://ginoux.univ-tln.fr
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APPLICATIONS 5D MagnetoConvection http://ginoux.univ-tln.fr
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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

Fixed Points & Stability: Center, Slow & Linear
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

Flow Curvature Method: n-dimensional dynamical systems
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

Differential Geometry Applied to Dynamical Systems
Publications Book Differential Geometry Applied to Dynamical Systems World Scientific Series on Nonlinear Science, series A, 2009 Hairer's 60th birthday