Qualitative analysis for a class of plane systems.
Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé
Henri Poincaré avait déjà remarqué que les variétés stable et instable du pendule perturbé, défini par l’hamiltonienne coïncident pas lorsque que le paramètre n’est pas nul, mais qu’on peut leur associer un même développement formel divergent en puissance de . Cette divergence est ici analysée au moyen de la récente théorie de la résurgence, et du calcul étranger qui permet de trouver un équivalent asymptotique de l’écart des deux variétés pour tendant vers zéro - du moins cela est-il montré...
Retracts, fixed point index and differential equations.
Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions.
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.
Saddle connections in planar systems
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.
Singular problems on the half-line
The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form
Soluzioni di tipo «multibump» e dinamiche caotiche in una classe di equazioni differenziali periodiche
Some remarks on the Melnikov function.
Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom.
Tangencies for real and complex Hénon maps: An analytic method.
The existence of homoclinic solutions for hyperbolic equations.
The period function near a polycycle with two semi-hyperbolic vertices
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.
Traveling wave fronts in spatially discrete reaction-diffusion equations on higher-dimensional lattices.
Travelling wave solutions for the Painlevé-integrable coupled KdV equations.
Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium
Viscous profiles for traveling waves of scalar balance laws: The uniformly hyperbolic case.
Wave fronts to some modifications of Korteweg-de Vries and Burgers-Korteweg-de Vries equations
The existence of a traveling wave with special properties to modified KdV and BKdV equations is proved. Nonlinear terms in the equations are defined by means of a function f of an unknown u satisfying some conditions.