Consensus of Julia Sets

Weihua Sun,Shutang Liu

doi:10.3390/fractalfract6010043

Abstract

The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods.

Highlights

There are two types of control problems regarding the consensus control: one is with a leader and the other is with no leaders
Two types of consensus of Julia sets are discussed: one is with a leader and the other is with no leaders
Consensus of Julia sets with a leader causes multiple follower Julia sets to change, with one becoming the leader Julia set

Summary

Introduction

Consensus problems with an active leader are discussed and a neighbor-based controller together with a neighbor-based state estimation rule is proposed for each autonomous agent in [9,10]. In [15], the leaderless consensus controller designed for nonlinear multi-agent systems subject to the input saturation nonlinearity by using an event-triggered mechanism is discussed. The fixed-time event-triggered leader-following consensus issue for the nonlinear multi-agent system with uncertain disturbances is discussed in [19]. Based on the applications of Julia sets and the study of the consensus of multi-agent systems, a natural and interesting question is the following: does the consensus of the multiple particles’ behavior depicted by Julia sets happen with a leader or with no leaders?

Preliminaries

The Scheme of Consensus of Julia Sets with a Leader

The Scheme of Consensus of Julia Sets with No Leaders

Discussion

Conclusions