Homoclinic orbit solutions of a one dimensional Wilson-Cowan type model.
Homoclinic tangencies and hyperbolicity for surface diffeomorphisms.
Homoclinics : Poincaré-Melnikov type results via a variational approach
Infinitely many coexisting strange attractors
Infinitely many homoclinic orbits for Hamiltonian systems with group symmetries.
Multidimensional singular -lemma.
Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles
In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C ∞, parametrized families {g t ∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as...
On homoclinic solutions of a semilinear -Laplacian difference equation with periodic coefficients.
On some results of Moser and of Bangert
Orbits connecting singular points in the plane
This paper concerns the global structure of planar systems. It is shown that if a positively bounded system with two singular points has no closed orbits, the set of all bounded solutions is compact and simply connected. Also it is shown that for such a system the existence of connecting orbits is tightly related to the behavior of homoclinic orbits. A necessary and sufficient condition for the existence of connecting orbits is given. The number of connecting orbits is also discussed.
Partial hyperbolicity and homoclinic tangencies
We show that any diffeomorphism of a compact manifold can be approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions
We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of...
Reminiscences on science at IHÉS. A problem on homoclinic theory and a brief review
Rigorous numerics for symmetric homoclinic orbits in reversible dynamical systems
We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and the existence of a symmetric homoclinic orbit is rigorously verified as an example.
Scattered homoclinics to a class of time-recurrent Hamiltonian systems
A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zero whose support is spread out over the real line.
Tangencies for real and complex Hénon maps: An analytic method.
The shadowing chain lemma for singular Hamiltonian systems involving strong forces
We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 {ζ→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits - the shadowing chain lemma - via minimization of action integrals and using simple geometrical arguments.
Transversal homoclinics in nonlinear systems of ordinary differential equations.
Une classe de systèmes dynamiques monotones génériquement Morse-Smale
Dans cet article, nous généralisons les résultats de Fusco et Oliva [8], qui ont montré la transversalité de l’intersection des variétés stable et instable associées à des orbites périodiques hyperboliques, pour un système dynamique de la forme (sur un ouvert de ) où est une matrice de Jacobi cyclique. Dans [8], cette propriété est obtenue en utilisant le nombre de changements de signe de qui est une fonctionnelle monotone le long des orbites. Tout d’abord, nous étendons ce résultat de transversalité...