Advertisement

A Unique Jerk System with Abundant Dynamics: Symmetric or Asymmetric Bistability, Tristability, and Coexisting Bubbles

  • Ying Li
  • Yicheng ZengEmail author
  • Jingfang Zeng
Statistical

Abstract

Finding and revealing new features and behaviors of simple chaotic systems have always been an important and attractive research topic. This paper aims to introduce a new 3D autonomous jerk chaotic system and explore its rich dynamical behaviors including period-doubling bifurcation and reverse period-doubling bifurcation routes to chaos, crisis and internal crisis, multiple symmetric coexisting attractors, and antimonotonicity. Especially, the phenomena of asymmetric bistability (e.g., coexistence of a point attractor and chaotic attractor or coexistence of a point attractor and period-5 limit cycle), tristability (e.g., coexistence of a point attractor and a pair of symmetric chaotic or periodic attractors), and coexisting bubbles are found, which have been rarely reported before. By using standard nonlinear analysis methods such as bifurcation diagrams, the largest Lyapunov exponent, phase portraits, Poincaré sections, 0–1 test chart and the basin of attraction for an attractor, and the complex dynamical behaviors of the system are investigated in detail. Furthermore, a corresponding hardware electronic circuit is designed to verify the numerical simulations.

Keywords

Jerk chaotic system Multiple symmetric attractors Tristability Antimonotonicity Coexisting bubble 

Notes

Funding Information

The work was supported by the National Natural Science Foundations of China under Grant No. 61471310.

References

  1. 1.
    J.C. Sprott, Some simple chaotic flows [J]. Phys Rev E 50(2), R647 (1994).  https://doi.org/10.1103/physreve.50.r647 ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    K.H. Sun, J.C. Sprott, A simple jerk system with piecewise exponential nonlinearity [J]. Int J Nonlin Sci Num 10(11–12), 1443–1450 (2009).  https://doi.org/10.1515/IJNSNS.2009.10.11-12.1443
  3. 3.
    K. Rajagopal, V.T. Pham, F.R. Tahir, A. Akgul, H.R. Abdolmohammadi, S. Jafari, A chaotic jerk system with non-hyperbolic equilibrium: dynamics, effect of time delay and circuit realization [J]. Pramana 90(4), 52–58 (2018).  https://doi.org/10.1007/s12043-018-1545-x ADSCrossRefGoogle Scholar
  4. 4.
    C. Wang, H. Xia, L. Zhou, A memristive hyperchaotic multiscroll jerk system with controllable scroll numbers [J]. Int J Bifurcat Chaos 27(6), 1750091 (2017).  https://doi.org/10.1142/S0218127417500912 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    A.R. Elsonbaty, A.M.A. El-Sayed, Further nonlinear dynamical analysis of simple jerk system with multiple attractors [J]. Nonlinear Dynam 87(2), 1169–1186 (2017).  https://doi.org/10.1007/s11071-016-3108-3 CrossRefGoogle Scholar
  6. 6.
    J. Ma, X. Wu, R. Chu, et al., Selection of multi-scroll attractors in jerk circuits and their verification using Pspice [J]. Nonlinear Dynam 76(4), 1951–1962 (2014).  https://doi.org/10.1007/s11071-014-1260-1 CrossRefGoogle Scholar
  7. 7.
    X. Xia, Y. Zeng, Z. Li, Coexisting multiscroll hyperchaotic attractors generated from a novel memristive jerk system [J]. Pramana 91(6), 82–14 (2018).  https://doi.org/10.1007/s12043-018-1657-3 ADSCrossRefGoogle Scholar
  8. 8.
    S. Fang, Z. Li, X. Zhang, Y. Li, Hidden extreme multistability in a novel no-equilibrium fractional-order chaotic system and its synchronization control [J]. Braz J Phys 49, 1–13 (2019).  https://doi.org/10.1007/s13538-019-00705-1 CrossRefGoogle Scholar
  9. 9.
    K. Rajagopal, S. Jafari, A. Akgul, A. Karthikeyan, Modified jerk system with self-exciting and hidden flows and the effect of time delays on existence of multi-stability [J]. Nonlinear Dyn 93, 1087–1108 (2018).  https://doi.org/10.1007/s11071-018-4247-5 CrossRefGoogle Scholar
  10. 10.
    T.V. Kamdoum, K.S. Takougang, K.G. Fautso, et al., Coexistence of attractors in autonomous Van der Pol-Duffing jerk oscillator: analysis, chaos control and synchronisation in its fractional-order form [J]. Pramana 91(1), 12–11 (2018).  https://doi.org/10.1007/s12043-018-1586-1 CrossRefGoogle Scholar
  11. 11.
    W. Zhen, A. Akgul, V.T. Pham, et al., Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors [J]. Nonlinear Dyn 89(3), 1877–1887 (2017).  https://doi.org/10.1007/s11071-017-3558-2 CrossRefGoogle Scholar
  12. 12.
    S. Vaidyanathan, A. Akgul, S. Kaçar, U. Çavuşoğlu, A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography [J]. Eur Phys J Plus 133(2), 46–18 (2018).  https://doi.org/10.1140/epjp/i2018-11872-8
  13. 13.
    K. Rajagopal, A. Akgul, S. Jafari, et al., Chaotic chameleon: dynamic analyses, circuit implementation, FPGA design and fractional-order form with basic analyses [J]. Chaos, Solitons Fractals 103, 476–487 (2017).  https://doi.org/10.1016/j.chaos.2017.07.007 ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    H.P. Ren, C. Bai, K. Tian, et al., Dynamics of delay induced composite multi-scroll attractor and its application in encryption [J]. Int J Nonlin Mech 94, 334–342 (2017).  https://doi.org/10.1016/j.ijnonlinmec.2017.04.014 ADSCrossRefGoogle Scholar
  15. 15.
    Ü. Çavuşoğlu, S. Panahi, A. Akgül, S. Jafari, S. Kaçar, A new chaotic system with hidden attractor and its engineering applications: analog circuit realization and image encryption [J]. Analog Integr Circ S 2019, 98(1):85-99.  https://doi.org/10.1007/s10470-018-1252-z CrossRefGoogle Scholar
  16. 16.
    J. Sun, Y. Shen, Q. Yin, et al., Compound synchronization of four memristor chaotic oscillator systems and secure communication [J]. Chaos 23(1), 013140 (2013).  https://doi.org/10.1063/1.4794794 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    T.C. Lin, F.Y. Huang, Z. Du, et al., Synchronization of fuzzy modeling chaotic time delay memristor-based Chua’s circuits with application to secure communication [J]. Int J Fuzzy Syst 17(2), 206–214 (2015).  https://doi.org/10.1007/s40815-015-0024-5 MathSciNetCrossRefGoogle Scholar
  18. 18.
    S.T. Pan, C.C. Lai, Identification of chaotic systems by neural network with hybrid learning algorithm [J]. Chaos, Solitons Fractals 37(1), 233–244 (2008).  https://doi.org/10.1016/j.chaos.2006.08.037 ADSCrossRefGoogle Scholar
  19. 19.
    S.P. Adhikari, H. Kim, R.K. Budhathoki, et al., A circuit-based learning architecture for multilayer neural networks with memristor bridge synapses [J]. IEEE Trans Circ Syst I, Reg Pap 62(1), 215–223 (2015).  https://doi.org/10.1109/TCSI.2014.2359717 CrossRefGoogle Scholar
  20. 20.
    E.K. Lee, H.S. Pang, J.D. Park, et al., Bistability and chaos in an injection-locked semiconductor laser [J]. Phys Rev A 47(1), 736–739 (1993).  https://doi.org/10.1103/PhysRevA.47.736 ADSCrossRefGoogle Scholar
  21. 21.
    C. Li, J.C. Sprott, W. Thio, Bistability in a hyperchaotic system with a line equilibrium [J]. J Exp Theor Phys 118(3), 494–500 (2014).  https://doi.org/10.1134/S1063776114030121 ADSCrossRefGoogle Scholar
  22. 22.
    J.C. Sprott, S. Jafari, V.T. Pham, et al., A chaotic system with a single unstable node [J]. Phys Lett A 379(36), 2030–2036 (2015).  https://doi.org/10.1016/j.physleta.2015.06.039 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J.R.M. Pone, V.K. Tamba, G.H. Kom, et al., Period-doubling route to chaos, bistability and antimononicity in a jerk circuit with quintic nonlinearity [J]. International Journal of Dynamics and Control 7(1), 1–22 (2019).  https://doi.org/10.1007/s40435-018-0431-1 CrossRefGoogle Scholar
  24. 24.
    S. Lynch, Multistability, bistability and chaos control [J]. Nonlinear Anal-Theor 47(7), 4501–4512 (2001).  https://doi.org/10.1016/S0362-546X(01)00563-6 MathSciNetCrossRefGoogle Scholar
  25. 25.
    X.Z. He, K. Li, C. Wang, Time-varying economic dominance in financial markets: a bistable dynamics approach. [J]. Chaos 28(5), 055903 (2018).  https://doi.org/10.1063/1.5021141 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    S.P. Dawson, C. Grebogi, J.A. Yorke, et al., Antimonotonicity: inevitable reversals of period-doubling cascades [J]. Phys Lett A 162(3), 249–254 (1992).  https://doi.org/10.1016/0375-9601(92)90442-o ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Kengne, V.R.F. Signing, J.C. Chedjou, et al., Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors [J]. Int J Dyn Control 6(11), 1–18 (2017).  https://doi.org/10.1007/s40435-017-0318-6 MathSciNetCrossRefGoogle Scholar
  28. 28.
    J. Kengne, S.M. Njikam, V.R.F. Signing, A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity [J]. Chaos, Solitons Fractals 106, 201–213 (2018).  https://doi.org/10.1016/j.chaos.2017.11.027 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    L. Zhou, C. Wang, X. Zhang, et al., Various attractors, coexisting attractors and antimonotonicity in a simple fourth-order memristive twin-T oscillator [J]. Int J Bifurcation Chaos 28(04), 1850050 (2018).  https://doi.org/10.1142/S0218127418500505 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    S. Zhang, Y. Zeng, A simple jerk-like system without equilibrium: asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees [J]. Chaos, Solitons Fractals 120, 25–40 (2019).  https://doi.org/10.1016/j.chaos.2018.12.036 ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    H.P.W. Gottlieb, Question 38. What is the simplest jerk function that gives chaos [J]. Am J Phys 64, 525–525 (1996).  https://doi.org/10.1119/1.18276 ADSCrossRefGoogle Scholar
  32. 32.
    G.A. Gottwald, I. Melbourne, On the validity of the 0-1 test for chaos [J]. Nonlinearity 22(6), 1367 (2009).  https://doi.org/10.1088/0951-7715/22/6/006 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A. Wolf, J.B. Swift, H.L. Swinney, et al., Determining Lyapunov exponents from a time series [J]. Physica D 16(3), 285–317 (1985).  https://doi.org/10.1016/0167-2789(85)90011-9 ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    G. Manai, F. Delogu, M. Rustici, Onset of chaotic dynamics in a ball mill: attractors merging and crisis induced intermittency [J]. Chaos 12(3), 601–609 (2002).  https://doi.org/10.1063/1.1484016 ADSCrossRefGoogle Scholar
  35. 35.
    C. Li, J.C. Sprott, Multistability in the Lorenz system: a broken butterfly [J]. Int J Bifurcation Chaos 24(10), 1450131 (2014).  https://doi.org/10.1142/S0218127414501314 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    L. Kocarev, K.S. Halle, K. Eckert, et al., Experimental observation of antimonotonicity in Chua’s circuit [J]. Int J Bifurcation Chaos 3, 1051–1055 (1993).  https://doi.org/10.1142/s0218127493000878 CrossRefzbMATHGoogle Scholar
  37. 37.
    B. Bocheng, X. Li, W. Ning, et al., Third-order RLCM-four-elements-based chaotic circuit and its coexisting bubbles [J]. AEU Int J Electron Commun 94, 26–35 (2018).  https://doi.org/10.1016/j.aeue.2018.06.042 CrossRefGoogle Scholar
  38. 38.
    S. Zhang, Y. Zeng, Z. Li, et al., Hidden extreme multistability, antimonotonicity and offset boosting control in a novel fractional-order hyperchaotic system without equilibrium [J]. Int J Bifurcation Chaos 28(13), 1850167 (2018).  https://doi.org/10.1142/S0218127418501675 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    K. Srinivasan, K. Thamilmaran, A. Venkatesan, Effect of nonsinusoidal periodic forces in Duffing oscillator: numerical and analog simulation studies [J]. Chaos, Solitons Fractals 40(1), 319–330 (2009).  https://doi.org/10.1016/j.chaos.2007.07.090 ADSCrossRefzbMATHGoogle Scholar
  40. 40.
    I.M. Kyprianidis, I.N. Stouboulos, P. Haralabidis, et al., Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit. [J]. Int J Bifurcation Chaos 10(08), 1903–1915 (2000).  https://doi.org/10.1142/S0218127400001171 CrossRefGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2019

Authors and Affiliations

  1. 1.School of Physics and Optoelectric EngineeringXiangtan UniversityXiangtanChina

Personalised recommendations