Search results for "Dynamic"
showing 10 items of 12329 documents
Darboux systems with a cusp point and pseudo-abelian integrals
2018
International audience; We study pseudo-abelian integrals associated with polynomial deformations of Darboux systems having a cuspidal singularity. Under some genericity hypothesis we provide locally uniform boundedness of on the number of their zeros.
Global dynamical behaviors in a physical shallow water system
2016
International audience; The theory of bifurcations of dynamical systems is used to investigate the behavior of travelling wave solutions in an entire family of shallow water wave equations. This family is obtained by a perturbative asymptotic expansion for unidirectional shallow water waves. According to the parameters of the system, this family can lead to different sets of known equations such as Camassa-Holm, Korteweg-de Vries, Degasperis and Procesi and several other dispersive equations of the third order. Looking for possible travelling wave solutions, we show that different phase orbits in some regions of parametric planes are similar to those obtained with the model of the pressure …
Building Anosov flows on $3$–manifolds
2014
We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different Anosov vector fields; 3. we build transitive attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive attractors; 4. we build a transitive Anosov vector field admitting infinitely many pairwise non-isotopic trans- verse tori.
Bifurcations in the elementary Desboves family
2017
International audience; We give an example of a family of endomorphisms of $\mathbb{P}^2(\mathbb{C})$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non-empty interior.
Rotation Forms and Local Hamiltonian Monodromy
2017
International audience; The monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article, we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non-degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach …
INSTABILITY OF HAMILTONIAN SYSTEMS IN THE SENSE OF CHIRIKOV AND BIFURCATION IN A NON LINEAR EVOLUTION PROBLEM EMANATING FROM PHYSICS
2004
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 de…
Laminations and tilings of the Hyperbolic upper half plane
2005
This thesis is devoted to the study of dynamical systems associated with tilings of theEuclidean plane or of the Hyperbolic half-plane. A such tiling codes an action of a group ofisometries (namely the group of translations of the plane or the group of affine maps) on a compactmetric space $\Omega$ such that the properties of this action are related with the combinatoricproperties of the tiling. The behaviors of the actions obtained by this way are really various. Insome cases, like for example for the Penrose's tiling, this action is free and minimal. This givesto the set $\Omega$ a structure of a specific lamination called {\it solenoid}. This space islocally the product of a Cantor set w…
The tennis racket effect in a three-dimensional rigid body
2017
We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a three-dimensional rigid body. This effect is characterized by a flip ($\pi$- rotation) of the head of the racket when a full ($2\pi$) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Differen…
Dynamic instability in absence of dominated splittings.
2006
We want to understand the dynamics in absence of dominated splittings. A dominated splitting is a weak form of hyperbolicity where the tangent bundle splits into invariant subbundles, each of them is more contracted or less expanded by the dynamics than the next one. We first answer an old question from Hirsch, Pugh and Shub, and show the existence of adapted metrics for dominated splittings.Mañé found on surfaces a $C^1$-generic dichotomy between hyperbolicity and Newhouse phenomenons (infinitely many sinks/sources). For that purpose, he showed that without a strong enough dominated splitting along one periodic orbit, a $C^1$-perturbation creates a sink or a source. We generalise that last…
Control of the One Dimensional Map Dynamics of the Cardiac Action Potential Duration
2015
International audience; The aim of this work is to investigate the control of the chaos in the one dimensional map which modelizes the duration of the current cardiac action potential (APD) as a function of the previous one. Using OGY control method, we obtain very satisfactory numerical results to stabilize the irregular heart rhythm into the normal rhythm.