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Research Papers

# New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria

[+] Author and Article Information
Haijun Wang

College of Mathematical Science,
Yangzhou University,
Yangzhou 225002, China
e-mail: mathchaos@126.com

Xianyi Li

College of Mathematical Science,
Yangzhou University,
Yangzhou 225002, China
e-mail: mathxyli@yzu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 11, 2014; final manuscript received January 23, 2015; published online June 9, 2015. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 10(6), 061021 (Nov 01, 2015) (14 pages) Paper No: CND-14-1207; doi: 10.1115/1.4030028 History: Received September 11, 2014; Revised January 23, 2015; Online June 09, 2015

## Abstract

In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system $x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy$ is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system $x·1=-ax1- fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22$ by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.

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## Figures

Fig. 1

The figure of the first Lyapunov exponent l1. It implies that the values of l1(b, c) can be positive, zero, and negative when (a,b,c) ∈ S212.

Fig. 2

Two orbits of system (7) for time integration: [0, 250], the initial conditions: (a) (x0,y0,z0)=(0,±3.82 × 10-5,±6.18 × 10-5), (b) (x0,y0,z0)=(1,±3.82 × 10-5,±6.18 × 10-5) and the parameters: a = 0, b = 1 and c = 2, which imply the existence of two singularly degenerate heteroclinic cycles of system (7)

Fig. 3

Degenerate heteroclinic cycles of system (7) with (a) (x0,y0,z0)=(-2,±3.82 × 10-5,±6.18 × 10-5), (b) (x0,y0,z0)=(0,±3.82 × 10-5,±6.18 × 10-5), and (c) (x0,y0,z0)=(2,±3.82 × 10-5,±6.18 × 10-5) when (a,b,c)=(0,2,5). This figure suggests that system (7) has infinitely many degenerate heteroclinic cycles.

Fig. 4

Phase portraits of system (7) for the parameters b = 2, c = 3 and (a) a = 0, (b) a = 0.07, time of integration: [0, 250], and the initial conditions: (x0,y0,z0)=(0,±3.82 × 10-5,±6.18 × 10-5), implying the existence of chaotic attractors bifurcated from the singularly degenerate heteroclinic cycles of system (7)

Fig. 5

Two homoclinic orbits of system (7) to E0 as t→±∞ for a = 3.27, b = 2, c = 3 and different initial values: (a) (x0,y0,z0)=(0,3.82 × 10-5,6.18 × 10-5) and (b) (x0,y0,z0)=(0,-3.82 × 10-5,-6.18 × 10-5)

Fig. 6

Two orbits of system (7) for time integration: [0, 2000], the initial conditions: (x0,y0,z0)=(0,±3.82 × 10-5,±6.18 × 10-5) and the parameters: (b,c)=(1,7) and a satisfying a > 0((a,b,c) ∈ S214): (a) a = 0.01, (b) a = 0.07, (c) a = 0.7, and (d) a = 3, implying the existence of two heteroclinic orbits of system (7) joining E0 and E±, respectively

Fig. 7

Phase portraits of system (7) when (x0,y0,z0)=(0,±3.82×10-5,±6.18×10-5) and (a) (a,b,c)=(3.9,2,3), (b) (a,b,c)=(500,2,3), which illustrate that system (7) has two heteroclinic orbits to E0 and E± when (a,b,c)∈S213

Fig. 8

Phase portrait of system (7) at infinity

Fig. 11

Phase portrait of the first integral H1=(z12+z2)/(1+z22)+arctanz2 for different H1, where black orbits mean H1 > 0, red orbit refers to H1=0, and green orbits point toward H1 < 0

Fig. 12

Phase portrait of the first integral H2=(z12+z2)/(1+z22)-arctanz2 for different H2, where black orbits mean H2 > 0, red orbit refers to H2=0, and green orbits point toward H2 < 0

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