Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance

Abstract

In this paper, we are concerned with the existence of unbounded orbits of the mapping θ1 = θ + 2π + 1/ρ μ(θ) + o(ρ-1), ρ1 = ρ + c - μ′(θ) + o(1), ρ→∞, where c is a constant and μ(θ) is 2π-periodic. Assume that c≠0, that μ(θ) is non-negative (or non-positive) and that μ(θ) has finitely many degenerate zeros in [0,2π]. We prove that every orbit of the given mapping tends to infinity in the future or in the past for sufficiently large ρ. On the basis of this conclusion, we further prove that the equation x″ + f(x)x′ + V′(x) + φ(x) = p(t) has unbounded solutions provided that V is an isochronous potential at resonance and F(x) (F(x) = ∫0x f(s)ds) and φ(x) satisfy some limit conditions. Meanwhile, we also obtain the existence of 2π-periodic solutions of this equation. © 2008 The Royal Society of Edinburgh.