# 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:
 [1] G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 1: Theory, Meccanica, 15 (1980), 9-20. doi: 10.1007/bf02128236.  Google Scholar [2] G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application, Meccanica, 15 (1980), 21-30. doi: 10.1007/bf02128237.  Google Scholar [3] V. Bragin, V. Vagaitsev, N. Kuznetsov and G. Leonov, Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits, J. Comput. Syst. Sci. Int., 50 (2011), 511-543.  doi: 10.1134/s106423071104006x.  Google Scholar [4] G. R. Chen and J. H. Lü, Dynamical Analysis, Control and Synchronization of Lorenz Families, Chinese Science Press, Beijing, 2003.   Google Scholar [5] G. R. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurc. Chaos Appl. Sci. Eng., 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.  Google Scholar [6] F. S. Dias, L. F. Mello and J.-G. Zhang, Nonlinear analysis in a Lorenz-like system, Nonlinear Anal. Real World Appl., 11 (2010), 3491-3500.  doi: 10.1016/j.nonrwa.2009.12.010.  Google Scholar [7] J. K. Hale, Ordinary Diferential Equations, Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience, New York-London-Sydney, 1969.  Google Scholar [8] N. V. Kuznetsov and G. A. Leonov, International Conference on Physics and Control, PhysCon 2005, Proceedings, IEEE 2005, Saint Petersburg, Russia, 2005,596-599. Google Scholar [9] H. Kokubu and R. Roussarie, Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences. I, J. Dyn. Differ. Equ., 16 (2004), 513-557.  doi: 10.1007/s10884-004-4290-4.  Google Scholar [10] N. V. Kuznetsov, G. A. Leonov, T. N. Mokaev, A. Prasad and M. D. Shrimali, Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285.  doi: 10.1142/S0218127417501152.  Google Scholar [11] N. V. Kuznetsov, T. Alexeeva and G. A. Leonov, Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations, Nonlinear Dyn., 85 (2016), 195-201. doi: 10.1007/s11071-016-2678-4.  Google Scholar [12] G. A. Leonov, N. V. Kuznetsov and T. N. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion, The European Physical Journal Special Topics, 224 (2015), 1421-1458.  doi: 10.1016/j.cnsns.2015.04.007.  Google Scholar [13] G. A. Leonov and N. V. Kuznetsov, Time-varying linearization and the Perron effects, Int. J. Bifurc. Chaos Appl. Sci. Eng., 17 (2007), 1079-1107.  doi: 10.1142/S0218127407017732.  Google Scholar [14] D. L. Li, A three-scroll chaotic attractor, Phys. Lett. A, 372 (2008), 387-393.  doi: 10.1016/j.physleta.2007.07.045.  Google Scholar [15] X. X. Li, C. Li and H. J. Wang, Complex dynamics of a simple 3D autonomous chaotic system with four-wing, J. Appl. Anal. Comput., 7 (2017), 745-769.  doi: 10.11948/2017047.  Google Scholar [16] X. X. Liao, P. Yu, S. L. Xie and Y. Fu, Study on the global property of the smooth Chua's system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 2815-2841.  doi: 10.1142/S0218127406016483.  Google Scholar [17] X. X. Liao, New Research on Some Mathematical Problems of Lorenz Chaotic Family, Huazhong University of Science & Technology Press, Wuhan, 2017.   Google Scholar [18] J. Llibre, M. Messias and P. R. Silva, On the global dynamics of the Rabinovich system, J. Phys. A: Math. Theor., 41 (2008), 275210, 21 pp. doi: 10.1088/1751-8113/41/27/275210.  Google Scholar [19] R. 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A, 57 (1976), 397-398.  doi: 10.1016/0375-9601(76)90101-8.  Google Scholar [29] O. E. Rössler, On the Rössler attractor, Chaos Theory and Applications, 2 (2020), 49-51.   Google Scholar [30] T. Shimizu and N. Morioka, On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Phys. Lett. A, 76 (1980), 201-204.  doi: 10.1016/0375-9601(80)90466-1.  Google Scholar [31] J. C. Sprott, Some simple chaotic flows, Phys. Rev. E, 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647.  Google Scholar [32] J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific Publishing, Singapore, 2010. doi: 10.1142/7183.  Google Scholar [33] J. C. Sprott, Elegant Fractals: Automated Generation of Computer Art, World Scientific Publishing, Singapore, 2019. doi: 10.1142/10906.  Google Scholar [34] J. C. Sprott, Do we need more chaos examples?, Chaos Theory and Applications, 2 (2020), 1-2.   Google Scholar [35] X. Wang and G. R. Chen, A chaotic system with only one stable equilibrium, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 1264-1272.  doi: 10.1016/j.cnsns.2011.07.017.  Google Scholar [36] H. J. Wang and X. Y. Li, More dynamical properties revealed from a 3D Lorenz-like system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 24 (2014), 1450133, 29 pp. doi: 10.1142/S0218127414501338.  Google Scholar [37] H. J. Wang and X. Y. Li, On singular orbits and a given conjecture for a 3D Lorenz-like system, Nonlinear Dyn., 80 (2015), 969-981.  doi: 10.1007/s11071-015-1921-8.  Google Scholar [38] H. J. Wang and X. Y. Li, New results to a three-dimensional chaotic system with two different kinds of non-isolated equilibria, J. Comput. Nonlinear Dyn., 10 (2015), 061021, 14 pp. doi: 10.1115/1.4030028.  Google Scholar [39] H. J. Wang and X. Y. Li, New route of chaotic behavior in a 3D chaotic system, Optik, 126 (2015), 2354-2361.  doi: 10.1016/j.ijleo.2015.05.142.  Google Scholar [40] H. J. Wang and X. Y. 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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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