On ap-adic Cubic Generalized Logistic Dynamical System

Abstract

Abstract. Applications of p-adic numbers mathematical physics, quantum mechanics stimulated increasing interest in the study of p-adic dynamical system. One of the interesting investigations is p-adic logistics map. In this paper, we consider a new generalization, namely we study a dynamical system of the form fa(x) = ax(1 − x 2 ). The paper is devoted to the investigation of a trajectory of the given system. We investigate the generalized logistic dynamical system with respect to parameter a and we restrict ourselves for the investigation of the case |a|p < 1. We study the existence of the fixed points and their behavior. Moreover, we describe their size of attractors and Siegel discs since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs. IntroductionOver the last century, p-adic numbers and p-adic analysis have come to play major role in the number theory. P -adic numbers were first introduced by K. Hensel. During a century after their discovery they were considered mainly objects of pure mathematics. Starting from 1980's various models described in the language of p-adic analysis have been actively studied. P -adic numbers have been widely used in many applications mostly in p-adic mathematical physics, quantum mechanics and others which roses the interest in the study of p-adic dynamical systems ( see for example, [6,18,42,44,45]).On the other hand, the study of p-adic dynamical systems arises in Diophantine geometry in the constructions of canonical heights, used for counting rational points on algebraic vertices over a number field, as in [12]. In [24] the p-adic field have arisen in physics in the theory of superstrings, promoting questions about their dynamics. Also some applications of p-adic dynamical systems to some biological, physical systems were proposed in [8,3,4,15,25]. In [9],[27] dynamical systems (not only monomial) over finite field extensions of the p-adic numbers were considered. Other studies of non-Archimedean dynamics in the neighborhood of a periodic and of the counting of periodic points over global fields using local fields appeared in [26,20,29,28,36]. Certain rational p-adic dynamical systems were investigated in [22], [31],[33], which appear from problems of p-adic Gibbs measures [23,32,34,35]. Note that in [37,10,11] a general theory of p-adic rational dynamical systems over complex p-adic filed C p has been developed.The most studied discrete p-adic dynamical systems (iterations of maps) are so-called monomial systems. In [5], [25] the behavior of a p-adic dynamical system f (x) = x n in the fields of p-adic numbers Q p and C p was investigated. In [25] perturbated monomial dynamical systems defined by functions f q (x) = x n + q(x), where the perturbation q(x) is a polynomial whose coefficients have small p-adic absolute value, have been studied. It was investigated the connection between monomial and perturbated monomial systems. These investigations show that the study of perturbated dynamical systems is important (see [16]). No...

Phase transition for thep-adic Ising–Vannimenus model on the Cayley tree

“…According to theorem 3.1 the solutions x 0 , x 1 and x 2 (If they exist) generate p−adic quasi Gibbs measures μ 0 , μ 1 and μ 2 , respectively. Taking into account the fact that √ −1 exists in Q p if and only if p ≡ 1(mod 4) (see for example [50,54]) together with Proposition 4.4, we conclude that the measures μ 0 , μ 1 , μ 2 exist if and only if the given prime p satisfies p ≡ 1(mod 4). This completes the proof.…”