Heteroclinic solutions for a class of the second order Hamiltonian systems

Abstract

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system q + V q (t, q) = 0, where q ∈ R n and V ∈ C 1 (R × R n , R), V 0. We will assume that V and a certain subset M ⊂ R n satisfy the following conditions. M is a set of isolated points and #M 2. For every sufficiently small ε > 0 there exists δ > 0 such that for all (t, z) ∈ R×R n , if d(z, M) ε then −V (t, z) δ. The integrals ∞ −∞ −V (t, z) dt, z ∈ M, are equi-bounded and −V (t, z) → ∞, as |t| → ∞, uniformly on compact subsets of R n \ M. Our result states that each point in M is joined to another point in M by a solution of our system.

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

“…Chen et al applied the hyperbolic perturbation method to determining the homoclinic and heteroclinic solutions of certain cubic nonlinear systems [13]. However, the results were implicit solutions for the time t. Respectively, Izydorek [14,15], Wang [16], Zhang [17], Tang [18] and their collaborators studied the existence of homoclinic or heteroclinic solutions for Hamiltonian systems.…”

Heteroclinic Solutions for Nonautonomous EFK Equations

“…In this paper we step forward to study heteroclinic solutions of the nonautonomous EFK equation (1). We are inspired by Yeun [9], Rabinowitz [11,12], and Izydorek and Janczewska [13]. However, we are emphasizing that the argument of [9,11] relies on the fact that the equation is autonomous; therefore, the method there cannot be reproduced here to tackle the time-dependent version (1) which is no longer autonomous.…”

“…The nonautonomous case necessitates careful analysis. Another point should be made is that we are working on the ( , ) phase plane; this is essentially different from [11,13]. In our argument, we also benefit from analysis of [3,4] and comments of [5].…”

Heteroclinic orbits for a discrete pendulum equation

“…Izydorek and Janczewska (cf. [9]) proved, without assuming periodicity or almost periodicity in t of the potential, for (3) without forcing term, the existence of heteroclinic solutions joining pairs of equilibria.…”

“…y(ε, ξ) minimizes J| Γε(ξ) .Proof. Let {y m } ∞ m=1 be a minimizing sequence for(9). There exists a positive number M > 0 such that M ≥ J(y m ) ≥ 1 We claim that {y m 0 } ∞ m=1 is a bounded sequence.…”