1,721,031 research outputs found
Homoclinics and chaotic behaviour for perturbed second order systems
No full textThis paper deals with perturbed Hamiltonian systems. The main assumption is that the unperturbed system has a homoclinic orbit which is non-degenerate in the sense that the linearized equation as a 1-dimensional kernel. If the Poincare Melnikov potential is sufficiently oscillating we construct multi-bump and also infinite-bump solutions. These ideas and results in bifurcation theory have been later used by several author
Homoclinics and chaotic behaviour for perturbed second order systems
No full textThis paper deals with perturbed Hamiltonian systems. The main assumption is that the unperturbed system has a homoclinic orbit which is non-degenerate in the sense that the linearized equation as a 1-dimensional kernel. If the Poincare Melnikov potential is sufficiently oscillating we construct multi-bump and also infinite-bump solutions. These ideas and results in bifurcation theory have been later used by several author
Fast Arnold diffusion in systems with three time scales
No full textWe consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the “splitting determinant” is exponentially small
Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium
No full textWe consider autonomous Lagrangian systems with two degrees of freedom, having a hyperbolic equilibrium of saddle-saddle type (that is, the eingenvalues of the linearized system about the equilibrium are real and distinct. We assume that the system possesses two homoclinic orbits. Under a nondegeneracy assumption on the homoclinics and under suitable conditions on the geometric behaviour of these homoclinics near the equilibrium we prove, by variational methods, that they give rise to an infinite family of multibump homoclinic solutions and that the topological entropy at the zero energy level is positive. A method to deal also with homoclinics satisfying a weaker nondegeneracy condition is developed. An application to a perturbation of an uncoupled system is also given
Fast Arnold diffusion in systems with three time scales
No full textWe consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the “splitting determinant” is exponentially small
Homoclinics and chaotic behaviour for perturbed second order systems
No full textThis paper deals with perturbed dynamical systems of the form:
−u¨+u=∇V(u)+ε∇uW(t,u)
where u(t)∈Rn(n⩾1). By means of a variational approach the existence of multibump homoclinics is proved under general assumptions on the Melnikov function. As a particular case, if (W; u) is T-periodic, the existence of approximate and complete Bernoulli shift structures is proved. An application to partial differential equations is also given
Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium
No full textWe present new existence results about the chaotic dynamics of a Lagrangian systems possising a saddle-saddle hyperbolic equilibrium with two homoclinics orbits. The approach is variational
Bifurcation of free vibrations for completely resonant wave equations
No full textWe prove existence of small amplitude, 2π/ω-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency to belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem
Fast Arnold diffusion in systems with three time scales
No full textWe consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the “splitting determinant” is exponentially small
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