Modeling and Simulation of Periodic Solutions in Nonlinear Oscillatory Systems

Abstract

Nonlinear oscillatory systems are key to many physical, engineering, and biological systems, in which periodic solutions are a manifestation of stable long-term behavior. Nonlinearities in the system and the inability to express them analytically make such solutions difficult to extract, however. In this paper, a single computational framework has been developed to study the periodic solutions of nonlinear oscillatory systems, including their stability and modeling and simulation. Representative models, such as the Van der Pol and Duffing oscillators, are thought to represent both self-excitatory and forced behaviour. It uses a numerical method of determining periodic orbits, which consists of RungeKutta integration, transient elimination and analysis of Poincare sections. Stability is also characterized with the Floquet theory to find out the strength of periodic solutions with changes in parameters. The simulation outcomes show the development of stable limit cycles and the changes to the more intricate dynamics when the nonlinearity and the forcing amplitude increase. The suggested framework offers a systematic and scalable methodology of examining nonlinear vibratory systems with an application in engineering and applied sciences.