Using Homotopy Link Function with Lipschitz Threshold in Studying Synchronized Fluctuations in Hierarchical Models

Abstract

In this chapter, we use homotopy and Lipschitz concepts to study synchronization effects in a triangular discrete-time dynamical systems. We have generalized the previous results in this area by defining a new link function with Lipschitz threshold. Using this new link function we continuously deform the orbits of the original model into the coupled model such that we preserve qualitative properties such as stability and periodicity. We analytically obtain this Lipschitz synchronization threshold and we prove that for less than this threshold, two systems are completely synchronized, i.e. they eventually evolve identically in time. We apply this method to a one-dimensional Ricker type population model, whose trajectories as is well known, can be chaotic. We use some qualitative dynamical systems tools such as Poincare section, spectrum and time series to detect the chaotic dynamics and chaotic signals for different synchronization Lipschitz thresholds s and growth rates r. Finally, we numerically find the Lipschitz synchronization threshold for different growth rates using mean phase and amplitude differences.