6533b857fe1ef96bd12b4146

RESEARCH PRODUCT

INSTABILITY OF HAMILTONIAN SYSTEMS IN THE SENSE OF CHIRIKOV AND BIFURCATION IN A NON LINEAR EVOLUTION PROBLEM EMANATING FROM PHYSICS

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We prove the existence of a minimal geometrico-dynamical condition to create hyperbolicity in section in the vicinity of a transversal homoclinic partially hyperbolic torus in a near integrable Hamiltonian system with three degrees of freedom. We deduce in this context a generalization of the Easton's theorem of symbolic dynamics. Then we give the optimal estimation of the Arnold diffusion time along a transition chain in the initially hyperbolic Hamiltonian systems with three degrees of freedom with a surrounding chain of hyperbolic periodic orbits .In a second part, we describe geometrically a mechanism of diffusion studied by Chirikov in a near integrable Hamiltonian system with three degrees of freedom and depending of two parameters, involving a layer of nearby parallel resonant planes and a resonant plane crossing this layer in a same given energy manifold. Thus, we prove that under some threshold about the dominating parameter, we can construct a transition orbit drifting through this modulational layer. One of the sketches proposed, the mechanism of modulational diffusion, based on the existence of heteroclinic connections between partially hyperbolic tori of two nearby resonant planes of the layer, is valid when an overlapping condition is satisfied.We finally study the bi-dimensional model describing a laminar flow with mixed convection between two parallel plane plates and inside a vertical tube. With reduced boundary conditions, we prove via the central manifold theorem that the system has a primary pitchfork bifurcation for a critical value of the Rayleigh number.