May  2020, 25(5): 1791-1820. 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  May 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (5) : 1791-1820. doi: 10.3934/dcdsb.2020003
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[40]

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[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.

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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.

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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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[48]

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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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