FEIGENBAUM UNIVERSALITY, TIME SERIES ANALYSIS AND LYAPUNOV EXPONENTS IN NONLINEAR TWO DIMENSIONAL CHAOTIC MODELS

Abstract

In this paper, we consider a two dimensional nonlinear chaotic model as

 where , ,  are adjustable parameters. M.J. Feigenbaum showed around 1980 how a route can be established from a regular system to a chaotic system in many nonlinear systems. Here we establish the universal route with the above mentioned model by determining the sequence of bifurcation points with the help of numerical methods and computer software. Time series analysis is carried out with different graphs in order to reveal how stability and instability of the periodic points appear in different ranges of the parameters. We evaluate Lyapunov exponents along with their graphs in order to confirm the regular and chaotic regions of the system. Different techniques are applied how to control the chaos i.e., how to go from the chaotic region to the regular one.  Many other relevant results are discussed, and a few open problems are posed.