Sequence of Routes to Chaos in a Lorenz-Type System

Fangyan Yang,Qingdu Li,Lijuan Chen,Yongming Cao

doi:10.1155/2020/3162170

Abstract

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

Highlights

Nonlinear phenomena in physics and engineering are o en described by quadratic systems, such as the famous Lorenz system [1], the Nose-Hoover equations [2, 3], the brushless DC motor system [4], the pendulum system [5], etc
The isolated sub-branches are from a period- limit cycle, otherwise the sub-branches are from twin period-푛/2 limit cycles
E detailed bifurcations are shown in Figure 1(b), which all follows the known pattern: When is odd, like the branch-3 shown in Figure 5, the sub-branches first start with a periodlimit cycle introduced by a fold bifurcation, bifurcate into a pair of twin limit cycles with the same period, a er that the twin cycles become a pair of twin chaotic attractors via period doubling bifurcations, at last the twin chaotic attractors merge into one larger chaotic attractor

Summary

Introduction

Nonlinear phenomena in physics and engineering are o en described by quadratic systems, such as the famous Lorenz system [1], the Nose-Hoover equations [2, 3], the brushless DC motor system [4], the pendulum system [5], etc. The period of the initial limit cycle increases with step size 1, the pattern is different from the period-adding phenomena reported in [24,25,26] since there is no parameter overlap between these branches and the period-doubling and period-adding happen in different parameter-changing directions For this phenomenon, the maximal found is 7, and the systems are all nonsymmetrical. The isolated sub-branches are from a period- limit cycle, otherwise the sub-branches are from twin period-푛/2 limit cycles All these branches lead to chaos as a parameter increases, this pattern is completely different (see Figure 1) from the existing literature, according to the best of the authors’ knowledge. E rest of paper is organized as follows: Section 2 introduces the Lorenz-type system, studies its symmetry, equilibria and their stability, and briefly explores the nonlinear dynamics; Section 3 carries out detailed numerical research on the sequence of bifurcation branches; Section 4 draws the conclusion

The Lorenz Type System e Lorenz type system is given by following equations:

Sequence of Routes to Chaos

Connections with Periodic Windows of the Main

Conclusions