Instability of rapidly-oscillating periodic solutions for discontinuous differential delay equations
Consider the scalar delay differential equation \[ x'(t)=-h(x(t-1))+f(x(t)), \tag{1} \] with a continuous initial condition \(x_0:[-1,0]\to \mathbb{R}\) having finitely many zeros, the function \(h\) is odd, takes the value \(a\) for \(0<x<c\), and \(b\) for \(x\geq c\), where \(a,b,c\) are positive constants, \(a>b\). The function \(f\) is odd, Lebesgue integrable, with essential supremum in \(\mathbb{R}\) less than \(b\). A solution \(x:[-1,\infty)\to \mathbb{R}\) to (1) is said a rapidly oscillating symmetric 2-phase periodic solution (ROS2PP solution) if \(x\) is \(T\)-periodic, \(x(t+T/2)=-x(t)\) for all \(t\), there exists \(t_1\geq -1\) such that \(x\) changes signs exactly once between \(t_1\) and \(t_1+T\), and \(x\) has two different zeros in some interval of length less than one. In the main result of the paper, the authors characterize the ROS2PP solutions to (1) with \(\|x\|_{\infty}>c\), and they show that all are unstable.
scientific article; zbMATH DE number 1836163