Fractals Parrondo’s Paradox in Alternated Superior Complex System

Yi Zhang,Da Wang

doi:10.3390/fractalfract5020039

Abstract

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

Highlights

The natural process has obvious discrete characteristics; discrete dynamical systems are usually applied for the modeling of actual processes
The connectivity properties of superior Julia set for a complex polynomial of degree 2 and 0 < β ≤ 1 can be identified based on the following cases: (1)
0.35 + 0.59i that its superior Julia set SJ ( Pc1 ) is totally disconnected, the c2 taken from area outside red boundary can lead to totally disconnected SJ ( Pc2 ), the c2 taken from gray area can lead to connected SJ ( Pc1,c2 )

Summary

Introduction

The natural process has obvious discrete characteristics; discrete dynamical systems are usually applied for the modeling of actual processes. In 1999, Parrondo et al [3,4] proposed that two games with loosing gains can paradoxically become a winning game This classical “losing + losing = winning” phenomenon was known as Parrondo’s paradox, which inspired a new research fever in physics and mathematics areas [5,6] about the combination of two systems with negative expected values. Based on the control theory and method, Wang [25,26,27] investigates the Julia sets of a fractional Lotka–Volterra model and Fractal Fract. Motivated by the significant investigations mentioned above, the main motivation of this work is to provide a detailed analysis of the connectivity change law of superior Julia sets in an alternated case.

Preliminaries

Graphical Explorations

Conclusions