Roots of unity and unbounded motions of an asymmetric oscillator
The piecewise linear equation of the form \[ \ddot x+ ax^+- bx^-= f(t)\tag{1} \] is considered, where \(a\) and \(b\) are positive constants with \(a\neq b\), \(x^+= \max\{x, 0\}\), \(x^-= \min\{x, 0\}\) and \(f(t)\) is a \(2\pi\)-periodic function. A resonance phenomenon for the asymmetric oscillator can be described by equation (1). In a previous paper, the authors [J. Math. Anal. Appl. 196, No. 3, 965-986 (1995; Zbl 0844.34047)] used a technique based on Lyapunov functions to study the existence of unbounded solutions to differential equations at resonance. The general questions that the authors have in mind are: 1) under what conditions on \(a\), \(b\) and \(f(t)\) one can say that equation (1) has unbounded solutions; 2) are there equations of the form (1), where unbounded and periodic solutions can coexist? To answer the questions, the authors use a class of Poincaré mappings in polar coordinates (the action-angle variables on the plane).
scientific article; zbMATH DE number 1130540