May  2022, 27(5): 2891-2915. doi: 10.3934/dcdsb.2021165

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 Published  May 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (5) : 2891-2915. doi: 10.3934/dcdsb.2021165
References:
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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.

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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.

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F. S. DiasL. 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.

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J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos Appl. Sci. Eng., 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.

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M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. Phys. A, 42 (2009), 115101, 18 pp. doi: 10.1088/1751-8113/42/11/115101.

[23]

L. Minati, L. V. Gambuzza, W. J. Thio, J. C. Sprott and M. Frasca, A chaotic circuit based on a physical memristor, Chaos, Solitons and Fractals, 138 (2020), 109990, 9 pp. doi: 10.1016/j.chaos.2020.109990.

[24]

L. PanW. N. Zhou and J. Fang, On dynamics analysis of a novel three-scroll chaotic attractor, J. Franklin Inst., 347 (2010), 508-522.  doi: 10.1016/j.jfranklin.2009.10.018.

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[33]

J. C. Sprott, Elegant Fractals: Automated Generation of Computer Art, World Scientific Publishing, Singapore, 2019. doi: 10.1142/10906.

[34]

J. C. Sprott, Do we need more chaos examples?, Chaos Theory and Applications, 2 (2020), 1-2. 

[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.

[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.

[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.

[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.

[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.

[40]

H. J. Wang and X. Y. Li, Some new insights into a known Chen-like system, Math. Methods Appl. Sci., 39 (2016), 1747-1764.  doi: 10.1002/mma.3599.

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H. J. Wang and X. Y. Li, Infinitely many heteroclinic orbits of a complex Lorenz system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 27 (2017), 1750110, 14 pp. doi: 10.1142/S0218127417501103.

[42]

H. J. Wang and X. Y. Li, Hopf Bifurcation and new singular orbits coined in a Lorenz-like system, J. Appl. Anal. Comput., 8 (2018), 1037-1025.  doi: 10.11948/2018.1307.

[43]

H. J. Wang and X. Y. Li, A novel hyperchaotic system with infinitely many heteroclinic orbits coined, Chaos, Solitons and Fractals, 106 (2018), 5-15.  doi: 10.1016/j.chaos.2017.10.029.

[44]

H. J. Wang and G. L. Dong, New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system, Appl. Math. Comput., 346 (2018), 272-286.  doi: 10.1016/j.amc.2018.10.006.

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H. J. Wang, On singular orbits and global exponential attractive set of a Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 29 (2019), 1950082, 11 pp. doi: 10.1142/S0218127419500822.

[46]

H. J. Wang and F. M. Zhang, Bifurcations, ultimate boundedness and singular orbits in a {unified hyperchaotic Lorenz-type} system, Discr. Contin. Dyn. Syst. Ser. B, 25 (2020), 1791-1820.  doi: 10.3934/dcdsb.2020003.

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Y. H. Xu, Z. Y. Ke, W. N. Zhou and C. R. Xie, Dynamic evolution analysis of stock price fluctuation and its control, Complexity, 2018 (2018), 5728090, 10 pp. doi: 10.1155/2018/5728090.

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Y. H. Xu and Y. L. Wang, A new chaotic system without linear term and its impulsive synchronization, Optik, 125 (2014), 2526-2530. doi: 10.1016/j.ijleo.2013.10.123.

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Y. H. Xu, B. Li, Y. L. Wang, W. N. Zhou and J. A. Fang, A new four-scroll chaotic attractor consisted of transient chaotic two-scroll and ultimate chaotic two-scroll, Math. Probl. Eng., 2012 (2012), 438328, 12 pp. doi: 10.1155/2012/438328.

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Y. H. Xu, W. N. Zhou, J. A. Fang and Y. L. Wang, Generating the new chaotic attractor by feedback controlling method, Math. Meth. Appl. Sci., 34 (2011), 2159-2166. doi: 10.1002/mma.1513.

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W. N. Zhou, Y. H. Xu, H. Q. Lu and L. Pan, On dynamics analysis of a new chaotic attractor, Phys. Lett. A, 372 (2008), 5773-5777. doi: 10.1016/j.physleta.2008.07.032.

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Q. G. Yang and Y. M. Chen, Complex dynamics in the unified Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 24 (2014), 1450055, 30 pp. doi: 10.1142/S0218127414500552.

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W. N. ZhouL. PanZ. Li and W. A. Halang, Non-linear feedback control of a novel chaotic system, Int. J. Control Autom., 7 (2009), 939-944.  doi: 10.1016/j.chaos.2005.12.059.

show all references

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.

[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.

[3]

V. BraginV. VagaitsevN. 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.

[4] G. R. Chen and J. H. Lü, Dynamical Analysis, Control and Synchronization of Lorenz Families, Chinese Science Press, Beijing, 2003. 
[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.

[6]

F. S. DiasL. 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.

[7]

J. K. Hale, Ordinary Diferential Equations, Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience, New York-London-Sydney, 1969.

[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.

[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.

[10]

N. V. KuznetsovG. A. LeonovT. N. MokaevA. 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.

[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.

[12]

G. A. LeonovN. 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.

[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.

[14]

D. L. Li, A three-scroll chaotic attractor, Phys. Lett. A, 372 (2008), 387-393.  doi: 10.1016/j.physleta.2007.07.045.

[15]

X. X. LiC. 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.

[16]

X. X. LiaoP. YuS. 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.

[17] X. X. Liao, New Research on Some Mathematical Problems of Lorenz Chaotic Family, Huazhong University of Science & Technology Press, Wuhan, 2017. 
[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.

[19]

R. Lozi and A. N. Pchelintsev, A new reliable numerical method for computing chaotic solutions of dynamical systems: The Chen attractor case, Int. J. Bifurc. Chaos Appl. Sci. Eng., 25 (2015), 1550187, 10 pp. doi: 10.1142/S0218127415501874.

[20]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[21]

J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos Appl. Sci. Eng., 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.

[22]

M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. Phys. A, 42 (2009), 115101, 18 pp. doi: 10.1088/1751-8113/42/11/115101.

[23]

L. Minati, L. V. Gambuzza, W. J. Thio, J. C. Sprott and M. Frasca, A chaotic circuit based on a physical memristor, Chaos, Solitons and Fractals, 138 (2020), 109990, 9 pp. doi: 10.1016/j.chaos.2020.109990.

[24]

L. PanW. N. Zhou and J. Fang, On dynamics analysis of a novel three-scroll chaotic attractor, J. Franklin Inst., 347 (2010), 508-522.  doi: 10.1016/j.jfranklin.2009.10.018.

[25]

T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-3486-9.

[26]

A. S. PikovskiiM. I. Rabinovich and V. Y. Trakhtengerts, Onset of stochasticity in decay confinement of parametric instability, Sov. Phys. JETP, 47 (1978), 715-719. 

[27]

T. Rikitake, Oscillations of a system of disk dynamos, Proc. Camb. Phil. Soc., 54 (1958), 89-105.  doi: 10.1017/S0305004100033223.

[28]

O. E. Rössler, An equation for continuous chaos, Phys. Lett. A, 57 (1976), 397-398.  doi: 10.1016/0375-9601(76)90101-8.

[29]

O. E. Rössler, On the Rössler attractor, Chaos Theory and Applications, 2 (2020), 49-51. 

[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.

[31]

J. C. Sprott, Some simple chaotic flows, Phys. Rev. E, 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647.

[32]

J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific Publishing, Singapore, 2010. doi: 10.1142/7183.

[33]

J. C. Sprott, Elegant Fractals: Automated Generation of Computer Art, World Scientific Publishing, Singapore, 2019. doi: 10.1142/10906.

[34]

J. C. Sprott, Do we need more chaos examples?, Chaos Theory and Applications, 2 (2020), 1-2. 

[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.

[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.

[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.

[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.

[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.

[40]

H. J. Wang and X. Y. Li, Some new insights into a known Chen-like system, Math. Methods Appl. Sci., 39 (2016), 1747-1764.  doi: 10.1002/mma.3599.

[41]

H. J. Wang and X. Y. Li, Infinitely many heteroclinic orbits of a complex Lorenz system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 27 (2017), 1750110, 14 pp. doi: 10.1142/S0218127417501103.

[42]

H. J. Wang and X. Y. Li, Hopf Bifurcation and new singular orbits coined in a Lorenz-like system, J. Appl. Anal. Comput., 8 (2018), 1037-1025.  doi: 10.11948/2018.1307.

[43]

H. J. Wang and X. Y. Li, A novel hyperchaotic system with infinitely many heteroclinic orbits coined, Chaos, Solitons and Fractals, 106 (2018), 5-15.  doi: 10.1016/j.chaos.2017.10.029.

[44]

H. J. Wang and G. L. Dong, New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system, Appl. Math. Comput., 346 (2018), 272-286.  doi: 10.1016/j.amc.2018.10.006.

[45]

H. J. Wang, On singular orbits and global exponential attractive set of a Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 29 (2019), 1950082, 11 pp. doi: 10.1142/S0218127419500822.

[46]

H. J. Wang and F. M. Zhang, Bifurcations, ultimate boundedness and singular orbits in a {unified hyperchaotic Lorenz-type} system, Discr. Contin. Dyn. Syst. Ser. B, 25 (2020), 1791-1820.  doi: 10.3934/dcdsb.2020003.

[47]

A. WolfJ. B. SwiftH. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.

[48]

Y. H. Xu, Z. Y. Ke, W. N. Zhou and C. R. Xie, Dynamic evolution analysis of stock price fluctuation and its control, Complexity, 2018 (2018), 5728090, 10 pp. doi: 10.1155/2018/5728090.

[49]

Y. H. Xu and Y. L. Wang, A new chaotic system without linear term and its impulsive synchronization, Optik, 125 (2014), 2526-2530. doi: 10.1016/j.ijleo.2013.10.123.

[50]

Y. H. Xu, B. Li, Y. L. Wang, W. N. Zhou and J. A. Fang, A new four-scroll chaotic attractor consisted of transient chaotic two-scroll and ultimate chaotic two-scroll, Math. Probl. Eng., 2012 (2012), 438328, 12 pp. doi: 10.1155/2012/438328.

[51]

Y. H. Xu, W. N. Zhou, J. A. Fang and Y. L. Wang, Generating the new chaotic attractor by feedback controlling method, Math. Meth. Appl. Sci., 34 (2011), 2159-2166. doi: 10.1002/mma.1513.

[52]

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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) $
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) $
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) $
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) $
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) $
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) $
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 $
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) $
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 $
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) $
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)
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)
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} $
$ 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} $
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
$ 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
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
$ 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
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 $
$ 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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