Complex time-delay dynamical systems of quadratic polynomials mapping

Yuanyuan Sun,Peng Li,Zhixing Lu

doi:10.1007/s11071-014-1670-0

Abstract

Time delays widely exist in the complex dynamical systems. In this paper, time-delay complex dynamical systems are studied by analyzing the properties of corresponding Julia sets. Experimental results show that the stability of complex dynamical system changes greatly due to the influence of time delay. The variations in time-delay Julia sets are studied. In addition, the conditions holding the dynamical systems stable are found. Time delay affects the stability of complex dynamical systems, and the existence of the fixed and periodic points is also discussed.

Highlights

In the 1970s, Julia sets were obtained by mathematician Gaston Julia who studied the simple iterative map in the complex plane
Definition of time-delay iteration and the numerical experiments is presented in Sect. 3, showing how the time delay affects the stability of complex dynamical systems
The branches of the timedelay Julia sets may disappear in turn, which shows some kind of periodicity

Summary

Introduction

In the 1970s, Julia sets were obtained by mathematician Gaston Julia who studied the simple iterative map in the complex plane. There are many developments in Julia sets, including dynamical systems [1,2,3], complex networks [4,5,6], and the fractal arts [7]. Most research works on time delay were about differential dynamical systems [9,10,11,12] and neglected the problems of the complex dynamical systems. We are concerned with the time-delay complex dynamical systems by drawing corresponding Julia sets. 2 to a discussion of governing equations and the theoretical analysis about complex time-delay dynamical systems. Definition of time-delay iteration and the numerical experiments is presented, showing how the time delay affects the stability of complex dynamical systems Definition of time-delay iteration and the numerical experiments is presented in Sect. 3, showing how the time delay affects the stability of complex dynamical systems

Time-delay equation and theoretical analysis

Time-delay iteration

Numerical experiments and analysis

Conclusions