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JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.fr rossetto@univ-tln.fr

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

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JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.fr rossetto@univ-tln.fr

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  1. Differential Geometry Applied to Dynamical Systems JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.frrossetto@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

  2. 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

  3. 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

  4. n-dimensional Dynamical Systems velocity Hairer's 60th birthday

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

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

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

  13. 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

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

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

  16. 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

  17. 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

  18. FIXED POINTS STABILITY Unforced Duffing oscillator and Thus is a saddle point or a node Hairer's 60th birthday

  19. 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

  20. CENTER MANIFOLD Guckenheimer et al. (1983) Local Bifurcations Hairer's 60th birthday

  21. 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

  22. VAN DER POL SYSTEM (1926) Hairer's 60th birthday

  23. VAN DER POL SYSTEM (1926) slow part slow part Singular approximation Hairer's 60th birthday

  24. VAN DER POL SYSTEM (1926) slow part slow part Hairer's 60th birthday

  25. VAN DER POL SYSTEM (1926) Slow manifoldLie derivative Singular approximation Hairer's 60th birthday

  26. 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

  27. VAN DER POL SYSTEM (1926) Flow Curvature Method vs Singular Perturbation Method (up to order ) Flow Curvature Singular perturbation Hairer's 60th birthday

  28. 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

  29. VAN DER POL SYSTEM (1926) Slow manifold attractive domain Hairer's 60th birthday

  30. 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

  31. APPLICATIONS 3D CHUA's piecewise linear model: Hairer's 60th birthday

  32. APPLICATIONS 3D CHUA's piecewise linear model: Slow invariant manifold analytical equation Hyperplanes Hairer's 60th birthday

  33. APPLICATIONS 3D CHUA's piecewise linear model: Invariant Hyperplanes (Darboux) Hairer's 60th birthday

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

  35. APPLICATIONS 3D CHUA's cubic model: with and Hairer's 60th birthday

  36. APPLICATIONS 3D CHUA's cubic model: Slow manifold Slow manifold Hairer's 60th birthday

  37. APPLICATIONS 3D Edward Lorenz model (1963): Hairer's 60th birthday

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

  39. APPLICATIONS 3D Hairer's 60th birthday

  40. APPLICATIONS 3D Neuronal Bursting Model Autocatalator Hairer's 60th birthday

  41. APPLICATIONS 4D Chua cubic 4D [Thamilmaran et al., 2004, Liu et al., 2007] Hairer's 60th birthday

  42. APPLICATIONS 5D Chua cubic 5D [Hao et al., 2005] Hairer's 60th birthday

  43. APPLICATIONS 5D Edgar Knobloch model: Hairer's 60th birthday

  44. APPLICATIONS 5D MagnetoConvection Hairer's 60th birthday

  45. NON-AUTONOMOUS DYNAMICAL SYSTEMS Forced Van der Pol Guckenheimer et al., 2003 Hairer's 60th birthday

  46. NON-AUTONOMOUS DYNAMICAL SYSTEMS Forced Van der Pol Guckenheimer et al., 2003 Hairer's 60th birthday

  47. NON-AUTONOMOUS DYNAMICAL SYSTEMS Forced Van der Pol Hairer's 60th birthday

  48. 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

  49. 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

  50. 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

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