Unbounded solutions to a system of coupled asymmetric oscillators at resonance

TL;DR

This paper constructs unbounded solutions for a coupled asymmetric oscillator system at resonance by analyzing a higher-dimensional resonance function and the associated Poincaré map dynamics.

Contribution

It introduces a higher-dimensional resonance function and demonstrates how to find unbounded solutions when this function has zeros with specific properties.

Findings

Unbounded solutions exist at resonance under certain conditions.

The resonance function's zeros are key to constructing solutions.

Analysis of the Poincaré map reveals the system's unbounded behavior.

Abstract

We deal with the following system of coupled asymmetric oscillators \[ \begin{cases} \ddot{x}_1+a_1x_1^+-b_1x^-_1+\phi_1(x_2)=p_1(t) \\ \ddot{x}_2+a_2\,x_2^+-b_2\,x^-_2+\phi_2(x_1)=p_2(t) \end{cases} \] where $\phi_i: \mathbb{R} \to \mathbb{R}$ is locally Lipschitz continuous and bounded, $p_i: \mathbb{R} \to \mathbb{R}$ is continuous and $2\pi$-periodic and the positive real numbers $a_i, b_i$ satisfy $$ \dfrac{1}{\sqrt{a_i}}+\dfrac{1}{\sqrt{b_i}}=\dfrac{2}{n}, \quad \mbox{ for some } n \in \mathbb{N}. $$ We define a suitable function $L: \mathbb{T}^2 \to \mathbb{R}^2$, appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever $L$ has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincar\'e map, in action-angle coordinates.