A global perspective for non-conservative dynamics
Bifurcation of multi-bump homoclinics in systems with normal and slow variables.
Canard cycles and homoclinic bifurcation in a 3 parameter family of vector fields on the plane.
Classification of heteroclinic orbits of semilinear parabolic equations with a polynomial nonlinearity.
Construction of entire solutions for semilinear parabolic equations.
Contracting singular cycles
Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems
We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincaré-Melnikov function.
Entropy-expansiveness and domination for surface diffeomorphisms.
Existence of homoclinic orbit for second-order nonlinear difference equation.
Existence of homoclinic solutions for a class of nonlinear difference equations.
Fixed points of two-sided fractional matrix transformations.
Forcing relations for homoclinic orbits of the Smale horseshoe map.
From transverse heteroclinic cycles to transverse homoclinic orbits
Generic diffeomorphisms on compact surfaces
We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion,...
Generic robustness of spectral decompositions
Heteroclinic orbits, mobility parameters and stability for thin film type equations.
Heterodimensional cycles, partial hyperbolicity and limit dynamics
We study one-parameter families of diffeomorphisms unfolding heterodimensional cycles (i.e. cycles containing periodic points of different indices). We construct an open set of such arcs such that, for a subset of the parameter space with positive relative density at the bifurcation value, the resulting nonwandering set is the disjoint union of two hyperbolic basic sets of different indices and a strong partially hyperbolic set which is robustly transitive. The dynamics of the diffeomorphisms we...
Higher dimensional Melnikov mappings
Higher order branching of periodic orbits from polynomial isochrones.
Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications.