1. 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

2. 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
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3. 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)
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4. n-dimensional Dynamical Systems
velocity
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5. 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
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6. 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.
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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)
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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)
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9. 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
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10. « 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)
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11. velocity  acceleration  over-acceleration  etc. …
FLOW CURVATURE METHOD
Geometric Method
Flow Curvature Method
(Ginoux & Rossetto, 2005  2009)
velocity
velocity  acceleration  over-acceleration  etc. …
position 
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12. n-Euclidean space curve
FLOW CURVATURE METHOD
“trajectory curve”
n-Euclidean space curve
plane or space curve
curvatures
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13. 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
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14. FLOW CURVATURE METHOD Flow Curvature Manifold: In dimension 2:
curvature or 1st curvature
In dimension 3:
torsion ou 2nd curvature
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15. Flow Curvature Manifold:
FLOW CURVATURE METHOD
Flow Curvature Manifold:
In dimension 4:
3rd curvature
In dimension 5:
4th curvature
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16. 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
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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).
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18. FIXED POINTS STABILITY
Unforced Duffing oscillator
and
Thus is a saddle point or a node
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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
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20. CENTER MANIFOLD Guckenheimer et al. (1983) Local Bifurcations
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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.
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22. VAN DER POL SYSTEM (1926) http://ginoux.univ-tln.fr
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23. VAN DER POL SYSTEM (1926) slow part slow part Singular approximation
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24. VAN DER POL SYSTEM (1926) slow part slow part
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25. VAN DER POL SYSTEM (1926) Slow manifold Lie derivative
Singular approximation
Slow manifold Lie derivative
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26. VAN DER POL SYSTEM (1926) Slow Manifold Analytical Equation
Flow Curvature Method vs Singular Perturbation Method
(Fenichel, 1979 vs Ginoux 2009)
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27. VAN DER POL SYSTEM (1926) Flow Curvature Method vs
Singular Perturbation Method (up to order )
Singular perturbation
Flow Curvature
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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…
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29. Slow manifold attractive domain
VAN DER POL SYSTEM (1926)
Slow manifold attractive domain
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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.
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31. CHUA's piecewise linear model:
APPLICATIONS 3D
CHUA's piecewise linear model:
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32. APPLICATIONS 3D CHUA's piecewise linear model:
Slow invariant manifold analytical equation
Hyperplanes
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33. APPLICATIONS 3D CHUA's piecewise linear model:
Invariant Hyperplanes (Darboux)
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34. CHUA's piecewise linear model:
APPLICATIONS 3D
CHUA's piecewise linear model:
Invariant Planes
Invariant Planes
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35. APPLICATIONS 3D CHUA's cubic model: with and http://ginoux.univ-tln.fr
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36. APPLICATIONS 3D CHUA's cubic model: Slow manifold Slow manifold
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37. Edward Lorenz model (1963):
APPLICATIONS 3D
Edward Lorenz model (1963):
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38. Slow invariant analytic manifold (Theorem 4)
APPLICATIONS 3D
Edward Lorenz model:
Slow invariant analytic manifold (Theorem 4)
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39. APPLICATIONS 3D
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40. APPLICATIONS 3D Autocatalator Neuronal Bursting Model
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41. APPLICATIONS 4D Chua cubic 4D
[Thamilmaran et al., 2004, Liu et al., 2007]
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42. APPLICATIONS 5D Chua cubic 5D [Hao et al., 2005]
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43. APPLICATIONS 5D Edgar Knobloch model: http://ginoux.univ-tln.fr
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44. APPLICATIONS 5D MagnetoConvection http://ginoux.univ-tln.fr
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45. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Forced Van der Pol
Guckenheimer et al., 2003
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46. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Forced Van der Pol
Guckenheimer et al., 2003
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47. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Forced Van der Pol
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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
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49. 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
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50. 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…
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51. Differential Geometry Applied to Dynamical Systems
Publications
Book
Differential Geometry
Applied to
Dynamical Systems
World Scientific Series on
Nonlinear Science, series A, 2009
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52. Thanks for your attention.
To be continued…
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