On generalized p-adic logistic dynamical system

Abstract

Applications of p-adic numbers in p-adic mathematical physics, quantum mechanics stimulated increasing interest in the study of p-adic dynamical system. One of the interesting p-adic dynamical system is p-adic logistic map. It is known such a mapping is chaotic. In the present thesis, we consider its cubic generalization, namely we study a dynamical system of the formf(x)=ax(1-x^2 ).Thethesis is devoted to the investigation of trajectory of the given system. We investigate the generalized logistic dynamical system with respect to parameter a. For the value of parameter, we consider three cases:|a|_p<1,|a|_p>1and|a|_p=1. For each case, we study the existence of the fixed points. Moreover, 2-periodic points are also studied for the case |a|_p<1.Not only that, their behavior also being investigated whether such fixed points are attracting, repelling or neutral. Moreover, we describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs. For the case |a|_p>1, we establish that the dynamical system is conjugate to the shift of symbolic dynamics.