# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021165
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## 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

Received  October 2020 Revised  April 2021 Early access June 2021

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

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.

Citation: Haijun Wang, Hongdan Fan, Jun Pan. A true three-scroll chaotic attractor coined. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021165
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##### References:
Phase portraits of system (1) with $(a, d, f, e, c) = (168, 0.4, 100, 0.70, 11)$ and the initial value $(x_{0}, y_{0}, z_{0}) = (1.618, -1.618, 1.618)$
Poincaré cross-sections of system (1) with $(a, d, f, e, c) = (168, 0.4, 100, 0.70, 11)$ and $(x_{0}, y_{0}, z_{0}) = (1.618, -1.618, 1.618)$
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 $(x_{0}, y_{0}, z_{0}) = (1.1, 1.2, -0.1)$
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 $(x_{0}, y_{0}, z_{0}) = (0.8, 1.9, -0.7)$
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 $(x_{0}, y_{0}, z_{0}) = (1.3, 2.2, -0.9)$
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 $(x_{0}, y_{0}, z_{0}) = (1.618, -1.618, 1.618)$
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 ($E_{z}^{5}$) $z_{0}^{5} = -30$
Phase portraits of system (1) with $(a, d, f, e, c) = (1.68, 0.4, 1, 0.70)$ and $E_{z}^{2} = (x_{0}^{1, 2}, y_{0}^{1, 2}, z_{0}^{2}) = (\pm1.382\times1e-6, \pm1.618\times1e-6, 5)$
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 ($E_{z}^{4}$) $z_{0}^{4} = 0.81$
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 $(x_{0}^{3, 4}, y_{0}^{3, 4}, z_{0}^{2}) = (\pm1.569, \pm1.163, 2.37)$
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)
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)
The distribution of equilibrium of system (1)
 $c$ $a+fd$ $cf[e(a + fd) - a]$ distribution of equilibrium $= 0$ $E_{z}$ $\neq0$ $= 0$ $E_{0}$ $\neq0$ $\neq0$ $\leq 0$ $E_{0}$ $\neq0$ $\neq0$ $> 0$ $E_{0}$, $E_{\pm}$
 $c$ $a+fd$ $cf[e(a + fd) - a]$ distribution of equilibrium $= 0$ $E_{z}$ $\neq0$ $= 0$ $E_{0}$ $\neq0$ $\neq0$ $\leq 0$ $E_{0}$ $\neq0$ $\neq0$ $> 0$ $E_{0}$, $E_{\pm}$
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
 $z$ $[54.5775, \infty)$ $(1.7, 54.5775)$ $1.7$ $(0.8225, 1.7)$ $E_{z}$ unstable node unstable focus fold-Hopf bifurcation stable focus
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
 $z$ $(\frac{21}{26}, 0.8225]$ $\frac{21}{26}$ $(-\infty, \frac{21}{26})$ $E_{z}$ stable node a 1D $W_{loc}^{s}$ and a 2D $W_{loc}^{c}$ saddle
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 $-16.5515, 3.8715, 0$
 $E_{z}^{i}$ 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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