Mel’nikov Methods and Homoclinic Orbits in Discontinuous Systems

Abstract

We consider a discontinuous system exhibiting a, possibly non-smooth, homoclinic trajectory. We assume that the critical point lies on the discontinuity level. We study the persistence of such a trajectory when the system is subject to a smooth non-autonomous perturbation. We use a Mel'nikov type approach and we introduce conditions which enable us to reformulate the problem in the setting of smooth systems so that we can follow the outline of the classical theory.

On the Exponents of Exponential Dichotomies

“…However, the exponents of the dichotomy determine the rate of convergence to zero of bounded solution either at ∞ (when the dichotomy is in R + ) or at −∞ (when the dichotomy is in R − ). Sometimes it becomes important to determine this rate of convergence, and hence the exponents of the dichotomy, for example when studying chaotic behaviour of discontinuous systems [8] or developing Melnikov theory for implicit nonlinear differential equations [9]. As a matter of fact in [8] the following result has been proved.…”

“…Sometimes it becomes important to determine this rate of convergence, and hence the exponents of the dichotomy, for example when studying chaotic behaviour of discontinuous systems [8] or developing Melnikov theory for implicit nonlinear differential equations [9]. As a matter of fact in [8] the following result has been proved. x has an exponential dichotomy on [T, ∞) (and hence also on R + ) with the same exponents α, β.…”

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

“…So, let Y (t) be the fundamental matrix of (A.1), where A lu (t) is replaced by A ls (t). Then, for any τ ∈ R there is a constant K = K(τ ) > 1, and a projection P + such that see again [12,Section 4], and [8,Appendix]. Denote by P + (τ ) := Y (τ )P + Y (τ ) −1 , and by s (τ ) the 1-dimensional range of P + (τ ).…”

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

“…However, the temporal discontinuities, i.e., impulses, addressed in this article are special in that time is a privileged independent variable in the augmented phase space, whose evolution is always given byṫ = 1. Several recent spatially discontinuous studies [37,38,39,41,42] do have connections to this article in that they share the goal of determining conditions on heteroclinic connections, while also being in the spirit of Melnikov theory [43,1,2,44].…”

“…x u ε (p, t) ≈ − sech 2 p tanh p sech 2 p + εM u (p, t) 4 sech 4 p tanh 2 p+[cosh (2p)−2] 2 sech 8 p 2 sech 2 p tanh p [cosh (2p)−2] sech 4 p(38)whereM u (p, t) = I (0,∞) (t)Λ(p − t) for p < 0 , in which Λ(ξ) := 2 sech 2 ξ tanh ξ − sech 2 ξ tanh ξ −x 1 + [cosh (2ξ) − 2] sech 4 ξ sech 2 ξ −xThe pseudo-separatrix formed by the stable pseudo-manifold [green], the unstable pseudo-manifold [red] and the gate [magenta] for the explosion centred at (0.5, 0.8) and ε = 0.02 using (38) at times t = 0.1 [left] and t = 0.6 [right].The restriction p < 0 ensures that this pseudo-manifold is only drawn from the point (0, 0) until it intersects the gate drawn at (0, 1). The stable pseudo-manifold would be given by the expression(38) with the superscript u replaced by s, and where M s (p, t) = −I (−∞,0) (t)Λ(p − t) for p > 0 .…”