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

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60th birthday2 A. Modeling & Dynamical Systems 1. Definition & Features 1. Definition & Features 2. Classical analytical approaches 2. Classical analytical approaches B. Flow Curvature Method 1. Presentation 1. Presentation 2. Results 2. Results C. Applications 1. n-dimensional dynamical systems 1. n-dimensional dynamical systems 2. Non-autonomous dynamical systems 2. Non-autonomous dynamical systems OUTLINE

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60th birthday3 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)

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60th birthday4 n-dimensional Dynamical Systems velocity

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60th birthday5 MANIFOLD DEFINTION A manifold is defined as a set of points in A manifold is defined as a set of points in satisfying a system of m scalar equations : 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 2 In dimension 3 curve surface curve surface

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60th birthday6 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: 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. LIE DERIVATIVE

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60th birthday7 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) INVARIANT MANIFOLDS

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60th birthday8 Manifold implicit equation: Instantaneous velocity vector: Normal vector: attractive manifold attractive manifold tangent manifold tangent manifold repulsive manifold repulsive manifold This notion is due to Henri Poincaré (1881) ATTRACTIVE MANIFOLDS Poincaré’s criterion :

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60th birthday9 Fixed Points Local Bifurcations Invariant manifolds center manifolds center manifolds slow manifolds (local integrals) slow manifolds (local integrals) linear manifolds (global integrals) linear manifolds (global integrals) Normal Forms DYNAMICAL SYSTEMS Dynamical Systems : Dynamical Systems : Integrables or non-integrables analytically

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60th birthday10 Courbes définies par une équation différentielle (Poincaré, 1881 1886) (Poincaré, 1881 1886)………….…. Singular 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 Approximation (Rossetto, 1998 & Ramdani, 1999) « Classical » analytic methods DYNAMICAL SYSTEMS

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60th birthday11 Flow Curvature Method (Ginoux & Rossetto, 2005 2009) velocity velocity acceleration over-acceleration etc. … Geometric Method FLOW CURVATURE METHOD position

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60th birthday12 plane or space curve “trajectory curve” curvatures FLOW CURVATURE METHOD n-Euclidean space curve

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60th birthday13 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 where represents the n-th derivative FLOW CURVATURE METHOD

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60th birthday14 Flow Curvature Manifold: In dimension 2: curvature or 1 st curvature In dimension 3: torsion ou 2 nd curvature FLOW CURVATURE METHOD

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60th birthday15 Flow Curvature Manifold: In dimension 4: 3 rd curvature In dimension 5: 4 th curvature FLOW CURVATURE METHOD

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60th birthday16 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 FIXED POINTS

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60th birthday17 Theorem 2: (Poincaré 1881 Ginoux, 2009) Hessian of flow curvature manifold associated to dynamical system enables differenting foci from saddles (resp. nodes). FIXED POINTS STABILITY

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60th birthday18 Unforced Duffing oscillator and and Thus is a saddle point or a node FIXED POINTS STABILITY

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60th birthday19 Theorem 3 (Ginoux, 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 manifold may be directly deduced from flow curvature manifold with with CENTER MANIFOLD

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60th birthday20 Guckenheimer et al. (1983) Local Bifurcations CENTER MANIFOLD

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60th birthday21 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. SLOW INVARIANT MANIFOLD

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60th birthday22 VAN DER POL SYSTEM (1926)

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60th birthday23 VAN DER POL SYSTEM (1926) slow part slow part slow part slow part Singular approximation

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60th birthday24 slow part slow part slow part slow part VAN DER POL SYSTEM (1926)

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60th birthday25 Slow manifold Lie derivative Singular approximation VAN DER POL SYSTEM (1926)

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60th birthday26 Slow Manifold Analytical Equation Flow Curvature Method vs Singular Perturbation Method (Fenichel, 1979 vs Ginoux 2009) VAN DER POL SYSTEM (1926)

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60th birthday27 VAN DER POL SYSTEM (1926) Flow Curvature Method vs Singular Perturbation Method (up to order ) Singular perturbation Flow Curvature

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60th birthday28 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 Pr. 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 by high 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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60th birthday29 VAN DER POL SYSTEM (1926) Slow manifold attractive domain

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60th birthday30 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. LINEAR INVARIANT MANIFOLD

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60th birthday31 CHUA's piecewise linear model: APPLICATIONS 3D

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60th birthday32 CHUA's piecewise linear model: Slow invariant manifold analytical equation Hyperplanes APPLICATIONS 3D

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60th birthday33 CHUA's piecewise linear model: Invariant Hyperplanes (Darboux) APPLICATIONS 3D

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60th birthday34 CHUA's piecewise linear model: APPLICATIONS 3D Invariant Planes

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60th birthday35 with and CHUA's cubic model: APPLICATIONS 3D

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60th birthday36 APPLICATIONS 3D Slow manifold CHUA's cubic model:

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60th birthday37 Edward Lorenz model (1963): APPLICATIONS 3D

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60th birthday38 Edward Lorenz model: Slow invariant analytic manifold (Theorem 4) APPLICATIONS 3D

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60th birthday39 APPLICATIONS 3D

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60th birthday40 Autocatalator Neuronal Bursting Model APPLICATIONS 3D

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60th birthday41 Chua cubic 4D [Thamilmaran et al., 2004, Liu et al., 2007] APPLICATIONS 4D

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60th birthday42 Chua cubic 5D [Hao et al., 2005] APPLICATIONS 5D

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60th birthday43 Edgar Knobloch model: APPLICATIONS 5D

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60th birthday44 APPLICATIONS 5D MagnetoConvection

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60th birthday45 NON-AUTONOMOUS DYNAMICAL SYSTEMS Forced Van der Pol Guckenheimer et al., 2003

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60th birthday46 NON-AUTONOMOUS DYNAMICAL SYSTEMS Forced Van der Pol Guckenheimer et al., 2003

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60th birthday47 NON-AUTONOMOUS DYNAMICAL SYSTEMS Forced Van der Pol

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60th birthday48 Theorem 6 : (Poincaré 1879 Ginoux, 2009) Normal form associated to any n-dimensional Normal form associated to any n-dimensional dynamical system may be deduced from flow dynamical system may be deduced from flow curvature manifold curvature manifold Normal Form

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60th birthday49 Fixed Points & Stability: Fixed Points & Stability: - Flow Curvature Manifold: Theorems 1 & 2 - Flow Curvature Manifold: Theorems 1 & 2 Center, Slow & Linear Center, Slow & Linear Manifold Analytical Equation: - Theorems 3, 4 & 5 - Theorems 3, 4 & 5 Normal Forms: Normal Forms: - Theorem 6 - Theorem 6 FLOW CURVATURE METHOD

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60th birthday50 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… DISCUSSION

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60th birthday51 Book Differential Geometry Applied to Dynamical Systems World Scientific Series on Nonlinear Science, series A, 2009 Publications

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60th birthday52 Thanks for your attention. To be continued…

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