Abstract:

Consider an oscillator with equation [ ddot x + V'(x) = 0, xinmathbb{R} ] and assume that it has an isochronous center at the origin. This means that $x=0$ is the only equilibrium of the equation and the remaining solutions are periodic with a fixed period say $T=2pi$. We are interested in the phenomenon of resonance for periodic perturbations. More precisely, we ask for the class of $2pi$-periodic function $p(t)$ such that all the solutions of the non-autonomous equation [ ddot x + V'(x) = epsilon p(t), xinmathbb{R} ] are unbounded. Here $epsilon eq 0$ is a small parameter. The simplest isochronous center is produced by the harmonic oscillator, $V(x)=frac{1}{2}n^2x^2$, $n=1,2,dots$ In this case the previous question has a well-known answer: resonance occurs whenever the integral [ I_n(p)!:=int_0^{2pi}p(t)e^{int}dt ] does not vanish. After this example the study of resonance for general isochronous oscillators seems natural. As far as we know this question was first raised by Roussarie in a meeting held in Lleida in II Symposium on Planar Vector Fields. Concrete examples of functions $p(t)$ producing resonance were presented by Ortega and also by Bonheure. The goal of this work is to identify a general class of forcings leading to resonance. Our main result can be interpreted as a nonlinear version of the condition $I_n(p) eq 0$. The result we present is a sufficient condition for resonance but it is not too far from being also necessary: a partial converse of the resonance result holds.