On Bounded Trajectories for Some Non-Autonomous Systems

Abstract

We study conditions for the existence of heteroclinics connecting ±1 for a nonautonomous equation of the form

$$ \ddot u = a\left( t \right)f\left( u \right) $$

(1)

where a(t) is a bounded positive function and f(±1) = 0. In addition, we consider the existence of a solution to the boundary value problem in the half line

$$ \left\{ {\begin{array}{*{20}c} {\ddot x + c\ddot x = a\left( t \right)V\prime \left( x \right)} \\ {x\left( 0 \right) = 0,{\mathbf{ }}x\left( { + \infty } \right) = 1.} \\ \end{array} } \right. $$

(2)

where c ≥ 0 and V is a C ¹, non-negative function, such that V (0) = V (1) = 0. If c = 0 and a and V are even, it turns out that these solutions yield heteroclinics for a special class of symmetric systems which connect the two non-consecutive equilibria ±1 at the same minimum level of the potential V. Therefore, the existence of such a solution in the case c = 0 means that the system (0.2) behaves in significantly different way from its autonomous counterpart.

A. Gavioli is supported by CNR, Italy and L. Sanchez is supported by GRICES and Fundação para a Ciência e Tecnologia, program POCI (Portugal/FEDER-EU).