doi: 10.3934/dcdsb.2020003

Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

1. 

Institute of Nonlinear Analysis and Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, China

2. 

Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, 341000, China

* Corresponding author: Haijun Wang

Received  November 2018 Revised  March 2019 Published  December 2019

Fund Project: The first author is supported by NSF of Zhejiang Province (grant: LQ18A010001)

In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small $ b > 0 $, i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-12, 12, 23,-1, -1, 1, 2.1, -6, -0.2) $, consisting of occurrence of degenerate pitchfork bifurcation at $ S_{z} $, the change in the stability index of the saddle at the origin as $ b $ crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.

Citation: Haijun Wang, Fumin Zhang. Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020003
References:
[1]

R. Barboza and G. R. Chen, On the global boundedness of the Chen system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 21 (2011), 3373-3385.  doi: 10.1142/S021812741103060X.  Google Scholar

[2]

B. Cannas and S. Cincotti, Hyperchaotic behaviour of two bi-directionally Chua's circuits, Int. J. Circ. Theor. Appl., 30 (2002), 625-637.  doi: 10.1002/cta.213.  Google Scholar

[3]

A. ČenysA. TamaševičiusA. BaziliauskasR. Krivickas and E. Lindberg, Hyperchaos in coupled Colpitts oscillators, Chaos Solitons Fractals, 17 (2003), 349-353.   Google Scholar

[4]

Y. M. Chen, The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system, Nonlinear Dyn., 87 (2017), 1445-1452.  doi: 10.1007/s11071-016-3126-1.  Google Scholar

[5]

Y. M. Chen and Q. G. Yang, Dynamics of a hyperchaotic Lorenz-type system, Nonlinear Dyn., 77 (2014), 569-581.  doi: 10.1007/s11071-014-1318-0.  Google Scholar

[6]

E. Tlelo-Cuautle, J. Rangel-Magdaleno and L. G. de la Fraga, Engineering Applications of FPGAs: Chaotic Systems, Artificial Neural Networks, Random Number Generators, and Secure Communication Systems, Springer, Switzerland, 2016. Google Scholar

[7]

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

[8]

G. Grassi and S. Mascolo, A system theory approach for designing cryptosystems based on hyperchaos, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 46 (1999), 1135-1138.  doi: 10.1109/81.788815.  Google Scholar

[9]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[10]

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

[11]

T. Kapitaniak and L. O. Chua, Hyperchaotic attractor of unidirectionally coupled Chua's circuit, Int. J. Bifurc. Chaos Appl. Sci. Eng., 4 (1994), 477-482.  doi: 10.1142/S0218127494000356.  Google Scholar

[12]

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

[13]

Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[14]

G. A. Leonov, A criterion for the existence of four limit cycles in quadratic systems, J. Appl. Math. Mech., 74 (2010), 135-143.  doi: 10.1016/j.jappmathmech.2010.05.002.  Google Scholar

[15]

Y. X. LiX. Z. LiuG. R. Chen and X. X. Liao, A new hyperchaotic Lorenz-type system: Generation, analysis, and implementation, Int. J. Circ. Theor. Appl., 39 (2011), 865-879.  doi: 10.1002/cta.673.  Google Scholar

[16]

X. Y. Li and Q. J. Ou, Dynamical properties and simulation of a new Lorenz-like chaotic system, Nonlinear Dyn., 65 (2011), 255-270.  doi: 10.1007/s11071-010-9887-z.  Google Scholar

[17]

T. C. LiG. T. Chen and G. R. Chen, On homoclinic and heteroclinic orbits of the Chen's system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 3035-3041.  doi: 10.1142/S021812740601663X.  Google Scholar

[18]

D. M. LiX. Q. Wu and J.-a. Lu, Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system, Chaos Solitons Fractals, 39 (2009), 1290-1296.  doi: 10.1016/j.chaos.2007.06.038.  Google Scholar

[19]

X. Y. Li and H. J. Wang, Homoclinic and heteroclinic orbits and bifurcations of a new Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 21 (2011), 2695-2712.  doi: 10.1142/S0218127411030039.  Google Scholar

[20]

X. Y. Li and P. Wang, Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system, Nonlinear Dyn., 73 (2013), 621-632.  doi: 10.1007/s11071-013-0815-x.  Google Scholar

[21]

Y. J. Liu and W. Pang, Dynamics of the general Lorenz family, Nonlinear Dyn., 67 (2012), 1595-1611.  doi: 10.1007/s11071-011-0090-7.  Google Scholar

[22]

X. Z. Liu, Estimation for globally exponentially attractive set of a hyperchaotic Lorenz-type systems and its application, 12th World Congress on Intelligent Control and Automation, IEEE, 24 (2016), 2884-2889.  doi: 10.1109/WCICA.2016.7578835.  Google Scholar

[23]

Y. J. Liu, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system, Nonlinear Anal. Real World Appl., 13 (2012), 2466-2475.  doi: 10.1016/j.nonrwa.2012.02.011.  Google Scholar

[24]

Y. J. Liu and Q. G. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal. Real World Appl., 11 (2010), 2563-2572.  doi: 10.1016/j.nonrwa.2009.09.001.  Google Scholar

[25]

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

[26]

A. NeimanX. PeiD. RussellW. WojtenekL. WilkensF. MossH. A. BraunM. T. Huber and K. Voigt, Synchronization of the noisy electrosensitive cells in the paddlefish, Phys. Rev. Lett., 82 (1999), 660-663.  doi: 10.1103/PhysRevLett.82.660.  Google Scholar

[27]

C. Z. Ning and H. Haken, Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations, Phys. Rev. A, 41 (1990), 3826-3837.  doi: 10.1103/PhysRevA.41.3826.  Google Scholar

[28]

J. Pathak, B. Hunt, M. Girvan, Z. Lu and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Lett., 120 (2018), 024102, 5 pp. doi: 10.1103/PhysRevLett.120.024102.  Google Scholar

[29]

O. E. Rössler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155-157.  doi: 10.1016/0375-9601(79)90150-6.  Google Scholar

[30]

R. M. Rubinger, A. W. M. Nascimento, L. F. Mello, C. P. L. Rubinger, N. M. Filho and H. A. Albuquerque, Inductorless Chua's circut: Experimental time series analysis, Math.Probl. Eng., 2007 (2007), 83893, 16 pp. Google Scholar

[31]

S. J. SchiffK. JergerD. H. DuongT. ChangM. L. Spano and W. L. Ditto, Controlling chaos in the brain, Nature, 370 (1994), 615-620.  doi: 10.1038/370615a0.  Google Scholar

[32]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics. Part Ⅱ. World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812798558_0001.  Google Scholar

[33]

K. ThamilmaranM. Lakshmanan and A. Venkatesan, Hyperchaos in a modified canonical Chua's circuit, Int. J. Bifurc. Chaos Appl. Sci. Eng., 14 (2004), 221-243.  doi: 10.1142/S0218127404009119.  Google Scholar

[34]

G. Tigan and D. Constantinescu, Heteroclinic orbits in the $T$ and the Lü system, Chaos Solitons Fractals, 42 (2009), 20-23.  doi: 10.1016/j.chaos.2008.10.024.  Google Scholar

[35]

G. Tigan and J. Llibre, Heteroclinic, homoclinic and closed orbits in the Chen system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 26 (2016), 1650072, 6 pp. doi: 10.1142/S0218127416500723.  Google Scholar

[36]

V. S. UdaltsovJ. P. GoedgebuerL. LargerJ. B. CuenotP. Levy and W. T. Rhodes, Communicating with hyperchaos: The dynamics of a DNLF emitter and recovery of transmitted information, Opt. Spectrosc., 95 (2003), 114-118.  doi: 10.1134/1.1595224.  Google Scholar

[37]

R. VicenteJ. DaudenP. Colet and R. Toral, Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop, IEEE Journal of Quantum Electronics, 41 (2005), 541-548.  doi: 10.1109/JQE.2005.843606.  Google Scholar

[38]

J. WangG. ChenT. QinW. Ni and X. Wang, Synchronizing spatiotemporal chaos in coupled map lattices via active-passive decomposition, Phys. Rev. E, 58 (1998), 3017-3021.   Google Scholar

[39]

H. J. Wang, C. Li and X. Y. Li, New heteroclinic orbits coined, Int. J. Bifurc. Chaos Appl. Sci. Eng., 26 (2016), 1650194, 13 pp. doi: 10.1142/S0218127416501947.  Google Scholar

[40]

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

[41]

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

[42]

P. WangD. M. Li and Q. L. Hu, Bounds of the hyperchaotic Lorenz-Stenflo system, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2514-2520.  doi: 10.1016/j.cnsns.2009.09.015.  Google Scholar

[43]

P. WangD. M. LiX. Q. WuJ. H. Lü and X. H. Yu, Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems, Int. J. Bifurc. Chaos Appl. Sci. Eng., 21 (2011), 2679-2694.  doi: 10.1142/S0218127411030027.  Google Scholar

[44]

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

[45]

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

[46]

P. WangY. H. ZhangS. L. Tan and L. Wan, Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating the Hausdorff dimension, Nonlinear Dyn., 74 (2013), 133-142.  doi: 10.1007/s11071-013-0773-3.  Google Scholar

[47]

S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003.  Google Scholar

[48]

Q. G. YangG. R. Chen and K. F. Huang, Chaotic attractors of the conjugate Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 17 (2007), 3929-3949.  doi: 10.1142/S0218127407019792.  Google Scholar

[49]

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

[50]

Z. ZhangG. R. Chen and S. M. Yu, Hyperchaotic signal generation via DSP for efficient perturbations to liquid mixing, Int. J. Circ. Theor. Appl., 37 (2009), 31-41.  doi: 10.1002/cta.470.  Google Scholar

[51]

F. H. ZhangY. L. ShuH. L. Yang and X. W. Li, Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization, Chaos Solitons Fractals, 44 (2011), 137-144.  doi: 10.1016/j.chaos.2011.01.001.  Google Scholar

[52]

F. C. ZhangY. H. Li and C. L. Mu, Bounds of solutions of a kind of hyperchaotic systems and application, J. Math. Res. Appl., 33 (2013), 345-352.   Google Scholar

show all references

References:
[1]

R. Barboza and G. R. Chen, On the global boundedness of the Chen system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 21 (2011), 3373-3385.  doi: 10.1142/S021812741103060X.  Google Scholar

[2]

B. Cannas and S. Cincotti, Hyperchaotic behaviour of two bi-directionally Chua's circuits, Int. J. Circ. Theor. Appl., 30 (2002), 625-637.  doi: 10.1002/cta.213.  Google Scholar

[3]

A. ČenysA. TamaševičiusA. BaziliauskasR. Krivickas and E. Lindberg, Hyperchaos in coupled Colpitts oscillators, Chaos Solitons Fractals, 17 (2003), 349-353.   Google Scholar

[4]

Y. M. Chen, The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system, Nonlinear Dyn., 87 (2017), 1445-1452.  doi: 10.1007/s11071-016-3126-1.  Google Scholar

[5]

Y. M. Chen and Q. G. Yang, Dynamics of a hyperchaotic Lorenz-type system, Nonlinear Dyn., 77 (2014), 569-581.  doi: 10.1007/s11071-014-1318-0.  Google Scholar

[6]

E. Tlelo-Cuautle, J. Rangel-Magdaleno and L. G. de la Fraga, Engineering Applications of FPGAs: Chaotic Systems, Artificial Neural Networks, Random Number Generators, and Secure Communication Systems, Springer, Switzerland, 2016. Google Scholar

[7]

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

[8]

G. Grassi and S. Mascolo, A system theory approach for designing cryptosystems based on hyperchaos, IEEE Transactions on Circuits and Systems-Ⅰ: Fundamental Theory and Applications, 46 (1999), 1135-1138.  doi: 10.1109/81.788815.  Google Scholar

[9]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[10]

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

[11]

T. Kapitaniak and L. O. Chua, Hyperchaotic attractor of unidirectionally coupled Chua's circuit, Int. J. Bifurc. Chaos Appl. Sci. Eng., 4 (1994), 477-482.  doi: 10.1142/S0218127494000356.  Google Scholar

[12]

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

[13]

Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[14]

G. A. Leonov, A criterion for the existence of four limit cycles in quadratic systems, J. Appl. Math. Mech., 74 (2010), 135-143.  doi: 10.1016/j.jappmathmech.2010.05.002.  Google Scholar

[15]

Y. X. LiX. Z. LiuG. R. Chen and X. X. Liao, A new hyperchaotic Lorenz-type system: Generation, analysis, and implementation, Int. J. Circ. Theor. Appl., 39 (2011), 865-879.  doi: 10.1002/cta.673.  Google Scholar

[16]

X. Y. Li and Q. J. Ou, Dynamical properties and simulation of a new Lorenz-like chaotic system, Nonlinear Dyn., 65 (2011), 255-270.  doi: 10.1007/s11071-010-9887-z.  Google Scholar

[17]

T. C. LiG. T. Chen and G. R. Chen, On homoclinic and heteroclinic orbits of the Chen's system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 3035-3041.  doi: 10.1142/S021812740601663X.  Google Scholar

[18]

D. M. LiX. Q. Wu and J.-a. Lu, Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system, Chaos Solitons Fractals, 39 (2009), 1290-1296.  doi: 10.1016/j.chaos.2007.06.038.  Google Scholar

[19]

X. Y. Li and H. J. Wang, Homoclinic and heteroclinic orbits and bifurcations of a new Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 21 (2011), 2695-2712.  doi: 10.1142/S0218127411030039.  Google Scholar

[20]

X. Y. Li and P. Wang, Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system, Nonlinear Dyn., 73 (2013), 621-632.  doi: 10.1007/s11071-013-0815-x.  Google Scholar

[21]

Y. J. Liu and W. Pang, Dynamics of the general Lorenz family, Nonlinear Dyn., 67 (2012), 1595-1611.  doi: 10.1007/s11071-011-0090-7.  Google Scholar

[22]

X. Z. Liu, Estimation for globally exponentially attractive set of a hyperchaotic Lorenz-type systems and its application, 12th World Congress on Intelligent Control and Automation, IEEE, 24 (2016), 2884-2889.  doi: 10.1109/WCICA.2016.7578835.  Google Scholar

[23]

Y. J. Liu, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system, Nonlinear Anal. Real World Appl., 13 (2012), 2466-2475.  doi: 10.1016/j.nonrwa.2012.02.011.  Google Scholar

[24]

Y. J. Liu and Q. G. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal. Real World Appl., 11 (2010), 2563-2572.  doi: 10.1016/j.nonrwa.2009.09.001.  Google Scholar

[25]

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

[26]

A. NeimanX. PeiD. RussellW. WojtenekL. WilkensF. MossH. A. BraunM. T. Huber and K. Voigt, Synchronization of the noisy electrosensitive cells in the paddlefish, Phys. Rev. Lett., 82 (1999), 660-663.  doi: 10.1103/PhysRevLett.82.660.  Google Scholar

[27]

C. Z. Ning and H. Haken, Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations, Phys. Rev. A, 41 (1990), 3826-3837.  doi: 10.1103/PhysRevA.41.3826.  Google Scholar

[28]

J. Pathak, B. Hunt, M. Girvan, Z. Lu and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Lett., 120 (2018), 024102, 5 pp. doi: 10.1103/PhysRevLett.120.024102.  Google Scholar

[29]

O. E. Rössler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155-157.  doi: 10.1016/0375-9601(79)90150-6.  Google Scholar

[30]

R. M. Rubinger, A. W. M. Nascimento, L. F. Mello, C. P. L. Rubinger, N. M. Filho and H. A. Albuquerque, Inductorless Chua's circut: Experimental time series analysis, Math.Probl. Eng., 2007 (2007), 83893, 16 pp. Google Scholar

[31]

S. J. SchiffK. JergerD. H. DuongT. ChangM. L. Spano and W. L. Ditto, Controlling chaos in the brain, Nature, 370 (1994), 615-620.  doi: 10.1038/370615a0.  Google Scholar

[32]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics. Part Ⅱ. World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812798558_0001.  Google Scholar

[33]

K. ThamilmaranM. Lakshmanan and A. Venkatesan, Hyperchaos in a modified canonical Chua's circuit, Int. J. Bifurc. Chaos Appl. Sci. Eng., 14 (2004), 221-243.  doi: 10.1142/S0218127404009119.  Google Scholar

[34]

G. Tigan and D. Constantinescu, Heteroclinic orbits in the $T$ and the Lü system, Chaos Solitons Fractals, 42 (2009), 20-23.  doi: 10.1016/j.chaos.2008.10.024.  Google Scholar

[35]

G. Tigan and J. Llibre, Heteroclinic, homoclinic and closed orbits in the Chen system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 26 (2016), 1650072, 6 pp. doi: 10.1142/S0218127416500723.  Google Scholar

[36]

V. S. UdaltsovJ. P. GoedgebuerL. LargerJ. B. CuenotP. Levy and W. T. Rhodes, Communicating with hyperchaos: The dynamics of a DNLF emitter and recovery of transmitted information, Opt. Spectrosc., 95 (2003), 114-118.  doi: 10.1134/1.1595224.  Google Scholar

[37]

R. VicenteJ. DaudenP. Colet and R. Toral, Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop, IEEE Journal of Quantum Electronics, 41 (2005), 541-548.  doi: 10.1109/JQE.2005.843606.  Google Scholar

[38]

J. WangG. ChenT. QinW. Ni and X. Wang, Synchronizing spatiotemporal chaos in coupled map lattices via active-passive decomposition, Phys. Rev. E, 58 (1998), 3017-3021.   Google Scholar

[39]

H. J. Wang, C. Li and X. Y. Li, New heteroclinic orbits coined, Int. J. Bifurc. Chaos Appl. Sci. Eng., 26 (2016), 1650194, 13 pp. doi: 10.1142/S0218127416501947.  Google Scholar

[40]

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

[41]

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

[42]

P. WangD. M. Li and Q. L. Hu, Bounds of the hyperchaotic Lorenz-Stenflo system, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2514-2520.  doi: 10.1016/j.cnsns.2009.09.015.  Google Scholar

[43]

P. WangD. M. LiX. Q. WuJ. H. Lü and X. H. Yu, Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems, Int. J. Bifurc. Chaos Appl. Sci. Eng., 21 (2011), 2679-2694.  doi: 10.1142/S0218127411030027.  Google Scholar

[44]

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

[45]

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

[46]

P. WangY. H. ZhangS. L. Tan and L. Wan, Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating the Hausdorff dimension, Nonlinear Dyn., 74 (2013), 133-142.  doi: 10.1007/s11071-013-0773-3.  Google Scholar

[47]

S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003.  Google Scholar

[48]

Q. G. YangG. R. Chen and K. F. Huang, Chaotic attractors of the conjugate Lorenz-type system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 17 (2007), 3929-3949.  doi: 10.1142/S0218127407019792.  Google Scholar

[49]

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

[50]

Z. ZhangG. R. Chen and S. M. Yu, Hyperchaotic signal generation via DSP for efficient perturbations to liquid mixing, Int. J. Circ. Theor. Appl., 37 (2009), 31-41.  doi: 10.1002/cta.470.  Google Scholar

[51]

F. H. ZhangY. L. ShuH. L. Yang and X. W. Li, Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization, Chaos Solitons Fractals, 44 (2011), 137-144.  doi: 10.1016/j.chaos.2011.01.001.  Google Scholar

[52]

F. C. ZhangY. H. Li and C. L. Mu, Bounds of solutions of a kind of hyperchaotic systems and application, J. Math. Res. Appl., 33 (2013), 345-352.   Google Scholar

Figure 1.  Phase portrait of the system (1) in the projection space $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, c, d, e, f, p, q) = (1,-0.3,0.8,-1.25,1,0.5,0.02,0.02) $
Figure 2.  Phase portrait of the system (1) in the projection spaces (a) $ x $-$ y $-$ z $, (b) $ y $-$ w $-$ z $ with $ (a_{1}, a_{2}, b, c, d, e, f, p, q) $$ = (1,-0.3,0.07,0.8,-1.25,1,0.5,0.02,0.02) $
Figure 3.  Phase portrait of the system (1) in the projection space $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, c, d, e, f, p, q) = (-0.3, 0.3, 0.575, -0.025, -1, 0.025, -0.15, -0.005) $ and $ (x_{0}, y_{0}, w_{0}) = (\pm1.618\times1e-4, \pm3.14\times1e-4, \pm0.618\times1e-4) $, ($ A_{1} $) $ z_{0}^{1} = 0.3 $, ($ A_{2} $) $ z_{0}^{2} = 0.35 $, ($ A_{3} $) $ z_{0}^{3} = 0.4 $
Figure 4.  The $ x $-coordinate $ (t, x(t)) $ with the parameter $ (a_{1}, a_{2}, c, d, e, f, b, p, q) $ $ = (-0.3, 0.3, 0.575, -0.025, -1, 0.025, 0.0525, -0.15, -0.005) $, initial conditions $ (1.618\times1e-4, 3.14\times1e-4, 0.3, 0.618\times1e-4) $ and $ (1.618\times1e-4, 3.14\times1e-4, 0.31, 0.618\times1e-4) $, showing the sensitive dependence on initial conditions
Figure 5.  Lyapunov exponents of the system (1) with $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-0.3, 0.3, 0.575, -0.025, -1, 0.025, 0.0525, -0.15, -0.005) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (1.618\times1e-4, 3.14\times1e-4, 0.3, 0.618\times1e-4) $
Figure 6.  Phase portrait of the system (1) in the projection space $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-0.3, 0.3, 0.575, -0.025, -1, 0.025, 0.0525, -0.15, -0.005) $ and $ (x_{0}, y_{0}, w_{0}) = (\pm1.618\times1e-4, \pm3.14\times1e-4, \pm0.618\times1e-4) $, ($ B_{1} $) $ z_{0}^{1} = 0.425 $, ($ B_{2} $) $ z_{0}^{2} = 0.465 $, ($ B_{3} $) $ z_{0}^{3} = 0.525 $
Figure 7.  The $ x $-coordinate $ (t, x(t)) $ with the parameter $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-0.3, 0.3, 0.575, -0.025, -1, 0.025, 0.0525, -0.15, -0.005) $, initial conditions $ (1.618\times1e-4, 3.14\times1e-4, 0.525, 0.618\times1e-4) $ and $ (1.618\times1e-4, 3.14\times1e-4, 0.515, 0.618\times1e-4) $, showing the sensitive dependence on initial conditions
Figure 8.  Lyapunov exponents of the system (1) with $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-0.3, 0.3, 0.575, -0.025, -1, 0.025, 0.0525, -0.15, -0.005) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (1.618\times1e-4, 3.14\times1e-4, 0.525, 0.618\times1e-4) $
Figure 9.  Phase portrait of the system (1) in the projection spaces (a) $ y $-$ w $-$ z $ and (b) $ y $-$ x $-$ z $ with $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (1,-0.3,0.8,-1.25,1,0.5,0.00,0.02,0.02) $ and $ (x_{0}, y_{0}, w_{0}) = (\pm1.618\times1e-4, \pm3.14\times1e-4, \pm0.618\times1e-4) $, ($ C_{1} $) $ z_{0}^{1} = 3.3 $, ($ C_{2} $) $ z_{0}^{2} = 3.4 $, ($ C_{3} $) $ z_{0}^{3} = 3.5 $
Figure 10.  Phase portrait of the system (1) in the projection spaces (a) $ y $-$ w $-$ z $ and (b) $ y $-$ x $-$ z $ with $ (a_{1}, a_{2}, c, d, e, f, b, p, q)=(1,-0.3,0.8,-1.25,1,0.5,0.07,0.02,0.02) $ and $ (x_{0}, y_{0}, w_{0})=(\pm1.618\times1e-4, \pm3.14\times1e-4, \pm0.618\times1e-4) $, $ z_{0}^{1,2,3} = 3.3, 3.4, 3.5 $
Figure 11.  The $ x $-coordinate $ (t, x(t)) $ with the parameter $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (1,-0.3,0.8,-1.25,1,0.5,0.07,0.02,0.02) $, initial conditions $ (1.618\times1e-4, 3.14\times1e-4, 3.5, 0.618\times1e-4) $ and $ (1.618\times1e-4, 3.14\times1e-4, 3.51, 0.618\times1e-4) $, showing the sensitive dependence on initial conditions
Figure 12.  Lyapunov exponents of the system (1) with $ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (1,-0.3,0.8,-1.25,1,0.5,0.07,0.02,0.02) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (1.618\times1e-4, 3.14\times1e-4, 3.5, 0.618\times1e-4) $
Figure 13.  Phase portrait of the system (1) in projection spaces (a) $ y $-$ w $-$ z $, (b) $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, b, c, d, e, f, p, q) = (-1, -3,2.5, -5, 0.5,-2,-1,1,-0.5) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (\pm1.618, \pm3.14, \pm2.718, \pm0.618)\times1e-4 $. These figures illustrate that the system (1) has two heteroclinic orbits to $ S_{0} $ and $ S_{\pm} $ when $ a_{2}\neq 0 $, $ a_{1} < 0 $, $ 2a_{1} + b > 0 $, $ e < 0 $, $ a_{1} + d < 0 $, $ q < 0 $, $ fp(-a_{1} + q) > 0 $ and $ a_{2}cq-a_{1}(dq-fp) < 0 $
Figure 14.  Phase portrait of the system (1) in projection spaces (a) $ y $-$ w $-$ z $, (b) $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, b, c, d, e, f, p, q) = (-1, -3,2, -5, 0.5,-2,-1,1,-0.5) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (\pm1.618, \pm3.14, \pm2.718, \pm0.618)\times1e-4 $. These figures illustrate that the system (1) has two heteroclinic orbits to $ S_{0} $ and $ S_{\pm} $ when $ a_{2}\neq 0 $, $ a_{1} < 0 $, $ 2a_{1} + b = 0 $, $ e < 0 $, $ a_{1} + d < 0 $, $ q < 0 $, $ fp(-a_{1} + q) > 0 $ and $ a_{2}cq-a_{1}(dq-fp) < 0 $
Figure 15.  Phase portrait of the system (1) in projection spaces (a) $ y $-$ w $-$ z $, (b) $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, b, c, d, e, f, p, q) = (-9, 9, 0.01, 1, 1, -1, -1, 2, -5) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (\pm1.618, \pm3.14, \pm2.718, \pm0.618)\times1e-4 $. These figures illustrate that the system (1) has two heteroclinic orbits to $ S_{0} $ and $ S_{\pm} $ when $ (a_{1}, a_{2}, b, c, d, e, f, p, q)\in W_{1}^{3} $
Figure 16.  Phase portrait of the system (1) in projection spaces (a) $ y $-$ w $-$ z $, (b) $ x $-$ y $-$ z $ with $ (a_{1}, a_{2}, b, c, d, e, f, p, q) = (-2, 2, 1.2, 1, 1, -1, -1, 2, -5) $ and $ (x_{0}, y_{0}, z_{0}, w_{0}) = (\pm1.618, \pm3.14, \pm2.718, \pm0.618)\times1e-4 $. These figures illustrate that the system (1) has two heteroclinic orbits to $ S_{0} $ and $ S_{\pm} $ when $ (a_{1}, a_{2}, b, c, d, e, f, p, q)\in W_{1}^{3} $
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