Resonance of isochronous oscillators [ Back ]

16:00 to 17:00
UAB - Dept. Matemàtiques (C1/-128)
David Rojas
Universidad de Granada


Consider an oscillator with equation \[ \ddot x + V'(x) = 0,\ x\in\mathbb{R} \] 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 $T=2\pi$. We are interested in the phenomenon of resonance for periodic perturbations. More precisely, we ask for the class of $2\pi$-periodic function $p(t)$ such that all the solutions of the non-autonomous equation \[ \ddot x + V'(x) = \epsilon p(t),\ x\in\mathbb{R} \] are unbounded. Here $\epsilon\neq 0$ 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 \[ I_n(p)\!:=\int_0^{2\pi}p(t)e^{int}dt \] 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 $I_n(p)\neq 0$. 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.