Non-monotone period functions for impact oscillators

Abstract

The existence of non-monotone period functions for differential equations of the form $$ ddot{x}+f(x)+gamma H(x)g(x)=0 $$ is proved for large $gamma$, where H is the Heaviside function and the functions f and g satisfy certain generic conditions. This result is precipitated by an analysis of the system $$ ddot{x}+sin x +gamma H(x) x^{3/2}=0, $$ which models the conservative dimensionless impact pendulum utilizing Hertzian contact during impact with a barrier at the downward vertical position.

Keywords

• Period function
• impact oscillator