Limit cycles of non-smooth oscillators
The author considers a nonsmooth oscillator of the form \[ \ddot{u}=f(u), \] where \(f:\mathbb R\to\mathbb R\) is a nonlinear, nonsmooth function with \(f(-u)=-f(u)\). The author develops a method to expand both the solution and the frequency of the periodic solution of the above equation in power series of an artificial parameter. Thereto, an artificial parameter-Linstedt-Poincaré method based on the introduction of a linear stiffness term and an artificial parameter \(p\) is applied before a change of the independent variable enables the author to expand the solution and its frequency into a power series of \(p\). By requiring that the solution at each order in the artificial parameter is free from secular terms, algebraic equations for the approximate frequency of oscillations are obtained. The methods does not require \(f\) to be smooth; if \(f\) is not smooth enough, then the derivatives are interpreted in a generalized sense as generalized functions. The method is applied to three examples, namely \(\ddot{u}=-u| u| \), \(\ddot{u}=-\text{sign}(u)\) and \(\ddot{u}=-u^{1/3}\). The first-order approximation is in full accord with that obtained by other methods such as harmonic balance, linearized harmonic balance and fully iterative methods. Furthermore, this method of this paper also allows to obtain second-order corrections.
scientific article; zbMATH DE number 5286440
scientific article