doi: 10.3934/dcdsb.2021165
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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
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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

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

[28]

O. E. Rössler, An equation for continuous chaos, Phys. Lett. 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. Li, Some new insights into a known Chen-like system, Math. Methods Appl. Sci., 39 (2016), 1747-1764.  doi: 10.1002/mma.3599.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

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.  Google Scholar

[53]

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.  Google Scholar

[54]

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.  Google Scholar

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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

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

[28]

O. E. Rössler, An equation for continuous chaos, Phys. Lett. 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. Li, Some new insights into a known Chen-like system, Math. Methods Appl. Sci., 39 (2016), 1747-1764.  doi: 10.1002/mma.3599.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

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.  Google Scholar

[53]

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.  Google Scholar

[54]

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.  Google Scholar

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 $
[1]

G. Deugoué, J. K. Djoko, A. C. Fouape, A. Ndongmo Ngana. Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations. Communications on Pure &amp; Applied Analysis, 2020, 19 (3) : 1509-1535. doi: 10.3934/cpaa.2020076

[2]

Theodore Tachim Medjo. Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088

[3]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655

[4]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[5]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure &amp; Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[6]

O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449

[7]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060

[8]

Rudolf Zimka, Michal Demetrian, Toichiro Asada, Toshio Inaba. A three-country Kaldorian business cycle model with fixed exchange rates: A continuous time analysis. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021143

[9]

Veysel Fuat Hatipoğlu. A novel model for the contamination of a system of three artificial lakes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2261-2272. doi: 10.3934/dcdss.2020176

[10]

Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611

[11]

Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178

[12]

Hong-Zhi Wei, Chun-Rong Chen. Three concepts of robust efficiency for uncertain multiobjective optimization problems via set order relations. Journal of Industrial & Management Optimization, 2019, 15 (2) : 705-721. doi: 10.3934/jimo.2018066

[13]

Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479

[14]

Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021017

[15]

Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure &amp; Applied Analysis, 2017, 16 (4) : 1103-1120. doi: 10.3934/cpaa.2017053

[16]

Dongfeng Zhang, Junxiang Xu, Xindong Xu. Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2851-2877. doi: 10.3934/dcds.2018123

[17]

Yuming Qin, Yang Wang, Xing Su, Jianlin Zhang. Global existence of solutions for the three-dimensional Boussinesq system with anisotropic data. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1563-1581. doi: 10.3934/dcds.2016.36.1563

[18]

Bin Han, Na Zhao. Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system. Communications on Pure &amp; Applied Analysis, 2020, 19 (9) : 4455-4478. doi: 10.3934/cpaa.2020203

[19]

Jing Lu, Yifei Wu. Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension. Communications on Pure &amp; Applied Analysis, 2015, 14 (5) : 1641-1670. doi: 10.3934/cpaa.2015.14.1641

[20]

Ching-Lung Lin, Gunther Uhlmann, Jenn-Nan Wang. Optimal three-ball inequalities and quantitative uniqueness for the Stokes system. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1273-1290. doi: 10.3934/dcds.2010.28.1273

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (70)
  • HTML views (141)
  • Cited by (0)

Other articles
by authors

[Back to Top]