Interactive Computational Search Strategy of Periodic Solutions in Essentially Nonlinear Dynamics

Abstract

We consider essentially nonlinear autonomous and nonautonomous dynamic systems described by ordinary differential equations. In such systems, for the same parameters of the system and forcing, different stable and unstable periodic solutions of different periods can exist. In addition, along with the ordered movements, the existence of a strange attractor is known. In such circumstances, the search for periodic solutions and their stability analysis is not a trivial problem. In order to find periodic solutions of the dynamical systems, we offer an interactive computer algorithm based on finding the initial conditions corresponding to the periodic solutions with the possibility of interactive intervention and operational control of the computing process. We demonstrate the algorithm and various numerical examples of finding new and complex stable and unstable periodic solutions in strongly nonlinear dynamical systems with one and two degrees of freedom. We also consider the mutual influence of oscillations in multidimensional nonlinear dynamic systems.