A true three-scroll chaotic attractor coined

Haijun Wang; Hongdan Fan; Jun Pan

doi:10.3934/dcdsb.2021165

Article Contents

2022, Volume 27, Issue 5: 2891-2915. Doi: 10.3934/dcdsb.2021165

This issue Previous Article On the mean field limit for Cucker-Smale models Next Article Analysis of stationary patterns arising from a time-discrete metapopulation model with nonlocal competition

A true three-scroll chaotic attractor coined

1.
School of Electronic and Information Engineering (School of Big Data Science), Taizhou University, Taizhou, 318000, China
2.
Institute of Nonlinear Analysis and Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, China
3.
Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, China

^* Corresponding author: Haijun Wang
^* Corresponding author: Haijun Wang

Received: September 30, 2020

Revised: March 31, 2021

Early access: June 2021

Published: May 2022

The first author is supported by NSF of China (grant: 12001489)

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Abstract

Based on the method of compression and pull forming mechanism (CAP), the authors in a well-known paper proposed and analyzed the Lü-like system: $ \dot{x} = a(y - x) + dxz $, $ \dot{y} = - xz + fy $, $ \dot{z} = -ex^{2} + xy + cz $, which was thought to display an interesting three-scroll chaotic attractors (called as Pan-A attractor) when $ (a, d, f, e, c) = (40, 0.5, 20, 0.65, \frac{5}{6}) $. Unfortunately, by further analysis and Matlab simulation, we show that the Pan-A attractor found is actually a stable torus. Accordingly, we find a new true three-scroll chaotic attractor coexisting with a single saddle-node $ (0, 0, 0) $ for the case with $ (a, d, f, e, c) = (168, 0.4, 100, 0.70, 11) $. Interestingly, the forming mechanism of singularly degenerate heteroclinic cycles of that system is bidirectional, rather than unilateral in the case of most other Lorenz-like systems. This further motivates us to revisit in detail its other complicated dynamical behaviors, i.e., the ultimate bound sets, the globally exponentially attractive sets, Hopf bifurcation, limit cycles coexisting attractors and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate that collapse of infinitely many singularly degenerate heteroclinic cycles and explosions of normally hyperbolic stable nodes or foci generate the aforementioned three-scroll attractor. In particular, four or two unstable limit cycles coexisting one chaotic attractor, the saddle $ E_{0} $ and the stable $ E_{\pm} $ are located in two globally exponentially attractive sets. These results together indicate that this system deserves further exploration in chaos-based applications.

Keywords:

Three-scroll Lü-like system,
three-scroll chaotic attractor,
globally exponentially attractive set,
Lyapunov function,
singularly degenerate heteroclinic cycle.

Mathematics Subject Classification: Primary: 34C45, 34C37; Secondary: 65P20, 65P30, 65P40.

Citation:

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Figure 1. Phase portraits of system (1) with $ (a, d, f, e, c) = (168, 0.4, 100, 0.70, 11) $ and the initial value $ (x_{0}, y_{0}, z_{0}) = (1.618, -1.618, 1.618) $

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Figure 2. Poincaré cross-sections of system (1) with $ (a, d, f, e, c) = (168, 0.4, 100, 0.70, 11) $ and $ (x_{0}, y_{0}, z_{0}) = (1.618, -1.618, 1.618) $

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Figure 3. The period cycle of system (1) located in the globally exponentially attractive set $ \Psi_{1}^{1} $ with $ (a, d, f, e, c) = (-1, 0.5, -0.1, 1.2, -0.2) $ and $ (x_{0}, y_{0}, z_{0}) = (1.1, 1.2, -0.1) $

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Figure 4. The period cycle of system (1) located in the globally exponentially attractive set $ \Psi_{1}^{2} $ with $ (a, d, f, e, c) = (-1, 0.5, -0.2, 1.2, -0.3) $ and $ (x_{0}, y_{0}, z_{0}) = (0.8, 1.9, -0.7) $

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Figure 5. The period cycle of system (1) located in the globally exponentially attractive set $ \Psi_{1}^{3} $ with $ (a, d, f, e, c) = (-1, 0.4, -0.2, 1.2, -0.08) $ and $ (x_{0}, y_{0}, z_{0}) = (1.3, 2.2, -0.9) $

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Figure 6. The chaotic attractor of system (1) located in the globally exponentially attractive set $ \Psi_{2}^{1} $ with $ (a, d, f, e, c) = (2, -0.2, 1.5, 0.2, -0.4) $ and $ (x_{0}, y_{0}, z_{0}) = (1.618, -1.618, 1.618) $

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Figure 7. Phase portraits of system (1) with $ (a, d, f, e, c) = (1.68, 0.4, 1, 0.70) $ and $ (x_{0}^{1, 2}, y_{0}^{1, 2}) = (\pm1.382 \pm1.618)\times1e-6 $, ($ E_{z}^{1} $) $ z_{0}^{1} = 55 $ and ($ E_{z}^{5} $) $ z_{0}^{5} = -30 $

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Figure 8. Phase portraits of system (1) with $ (a, d, f, e, c) = (1.68, 0.4, 1, 0.70) $ and $ E_{z}^{2} = (x_{0}^{1, 2}, y_{0}^{1, 2}, z_{0}^{2}) = (\pm1.382\times1e-6, \pm1.618\times1e-6, 5) $

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Figure 9. Phase portraits of system (1) with $ (a, d, f, e, c) = (1.68, 0.4, 1, 0.70) $ and $ (x_{0}^{1, 2}, y_{0}^{1, 2}) = (\pm1.382 \pm1.618)\times1e-6 $, ($ E_{z}^{3} $) $ z_{0}^{3} = 1.2 $ and ($ E_{z}^{4} $) $ z_{0}^{4} = 0.81 $

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Figure 10. When $ (a, d, f, e, c) = (11, -0.4187, 11, -0.2, -1.2) $, phase portraits of system (1) for the unstable Hopf bifurcation points $ E_{\pm}^{'} = (\pm3.4388, \pm5.5157, 18.9231) $ with the initial values $ (x_{0}^{1, 2}, y_{0}^{1, 2}, z_{0}^{1}) = (\pm3.569, \pm6.163, 19.37) $, and coexisting chaotic attractors with $ (x_{0}^{3, 4}, y_{0}^{3, 4}, z_{0}^{2}) = (\pm1.569, \pm1.163, 2.37) $

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Figure 11. When $ (a, d, f, e, c) = (11, -0.4187, 11, -0.2, -1.25) $, $ (x_{0}^{1, 2}, y_{0}^{1, 2}, z_{0}^{1}) = (\pm3.4749, \pm5.9136, 19.5225) $, $ (x_{0}^{3, 4}, y_{0}^{3, 4}, z_{0}^{2}) = (\pm2.1353, \pm3.7210, 16.4246) $, $ (x_{0}^{5, 6}, y_{0}^{5, 6}, z_{0}^{3}) = (\pm2.6, \pm3.7210, 16.4246) $ and $ (x_{0}^{7, 8}, y_{0}^{7, 8}, z_{0}^{4}) = (\pm2.8, \pm4.7210, 15.4246) $, four unstable limit cycles coexisting one chaotic attractor, the saddle $ E_{0} $ and the stable $ E_{\pm} $ of system (1)

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Figure 12. When $ (a, d, f, e, c) = (11, -0.425, 11, -0.2, -1.2) $, $ (x_{0}^{9, 10}, y_{0}^{9, 10}, z_{0}^{5}) = (\pm3.5021, \pm6.0812, 19.3265) $, $ (x_{0}^{11, 12}, y_{0}^{11, 12}, z_{0}^{6}) = (\pm2.1353, \pm2.3210, 13.4246) $, and $ (x_{0}^{13, 14}, y_{0}^{13, 14}, z_{0}^{7}) = (\pm1.569, \pm1.163, 2.37) $, two unstable limit cycles coexisting one chaotic attractor, the saddle $ E_{0} $ and the stable $ E_{\pm} $ of system (1)

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Table 1. The distribution of equilibrium of system (1)

$ c $	$ a+fd $	$ cf[e(a + fd) - a] $	distribution of equilibrium
$ = 0 $			$ E_{z} $
$ \neq0 $	$ = 0 $		$ E_{0} $
$ \neq0 $	$ \neq0 $	$ \leq 0 $	$ E_{0} $
$ \neq0 $	$ \neq0 $	$> 0 $	$ E_{0} $, $ E_{\pm} $

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Table 2. The dynamics of $ E_{z} $ with $ (a, d, f, e) = (1.68, 0.4, 1, 0.70) $ and the value of $ z $ varies

$ z $	$ [54.5775, \infty) $	$ (1.7, 54.5775) $	$ 1.7 $	$ (0.8225, 1.7) $
$ E_{z} $	unstable node	unstable focus	fold-Hopf bifurcation	stable focus

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Table 3. The dynamics of $ E_{z} $ with $ (a, d, f, e) = (1.68, 0.4, 1, 0.70) $ and the value of $ z $ varies

$ z $	$ (\frac{21}{26}, 0.8225] $	$ \frac{21}{26} $	$ (-\infty, \frac{21}{26}) $
$ E_{z} $	stable node	a 1D $ W_{loc}^{s} $ and a 2D $ W_{loc}^{c} $	saddle

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Table 4. The dynamics of $ E_{z}^{i} $, $ i = 1, 2, \cdots, 5 $

$ E_{z}^{i} $	classification	eigenvalues
$ E_{z}^{1}=(0, 0, 55) $	unstable node	$ 11.6169, 9.7031, 0 $
$ E_{z}^{2}=(0, 0, 5) $	unstable focus	$ 0.66\pm2.8783i, 0 $
$ E_{z}^{3}=(0, 0, 1.2) $	stable focus	$ -0.1\pm 0.8978i, 0 $
$ E_{z}^{4}=(0, 0, 0.81) $	stable node	$ -0.34, -0.014 0 $
$ E_{z}^{5}=(0, 0, -30) $	saddle	$ -16.5515, 3.8715, 0 $

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