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Differential geometry and mechanics: applications to chaotic dynamical systems. (English) Zbl 1111.37021

Summary: The aim of this article is to highlight the interest to apply differential geometry and mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk).The attractivity of the slow manifold is characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration makes it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold is introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra-Gause are proposed.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C10 Dynamics induced by flows and semiflows
53A04 Curves in Euclidean and related spaces
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics

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References:

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