KCC theory for a type of nonlinear damped SD oscillators

Abstract

Smooth and discontinuous (SD) oscillators are the nonlinear models that will exhibit SD dynamics due to continuous variation of the smooth parameter. Kosambi–Cartan–Chern (KCC) theory is a differential geometric theory that describes the deviations from the entire trajectory of the variational equation to the nearby trajectory. This paper presents a completely new dynamical analysis of a type of SD oscillators with nonlinear damping based on the KCC theory. First, the paper gives five KCC geometrical invariants of SD oscillators in the smooth and non-smooth cases, respectively. The results show that the geometric quantities in the non-smooth case cannot be derived directly from the geometric quantities in the smooth case. Second, from these geometric quantities, KCC stability of SD oscillators trajectory at any point except [Formula: see text] is analyzed. Lastly, numerical results show that in some regions that deviate from the equilibrium point of system, the system will exhibit complex and variable dynamical behavior due to small changes in parameters. This paper shows that the KCC theory is also a useful tool in the dynamical analysis of non-smooth systems.