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Four-dimensional autonomous dynamical systems with conservative flows: two-case study

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Abstract

Conservative chaotic systems are rare, especially autonomous smooth dynamical systems. This paper reports two four-dimensional (4D) autonomous conservative systems. The conservation of these two systems has been verified using the trace of Jacobian matrix, perpetual point theory and Hamiltonian energy theory. Numerical analyses, including phase portrait, Poincaré section, Lyapunov exponent spectrum and bifurcation diagram, verify the existence of the chaotic and quasiperiodic flows. Moreover, a electronic circuit in Multisim is built to demonstrate their chaotic dynamics, whose circuit experimental results agree well with the numerical results.

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References

  1. Arecchi, F.T., Giacomelli, G., Ramazza, P.L., Residori, S.: Vortices and defect statistics in two-dimensional optical chaos. Phys. Rev. Lett. 67(27), 3749 (1991)

    Article  Google Scholar 

  2. Argyris, A., Syvridis, D., Larger, L., Annovazzi-Lodi, V., Colet, P., Fischer, I., Garcia-Ojalvo, J., Mirasso, C.R., Pesquera, L., Shore, K.A.: Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438(7066), 343–346 (2005)

    Article  Google Scholar 

  3. Azar, A.T., Vaidyanathan, S.: Chaos Modeling and Control Systems Design. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  4. Cang, S., Wu, A., Wang, Z., Wang, Z., Chen, Z.: A general method for exploring three-dimensional chaotic attractors with complicated topological structure based on the two-dimensional local vector field around equilibriums. Nonlinear Dyn. 83(1–2), 1069–1078 (2015)

    MathSciNet  Google Scholar 

  5. Cang, S.J., Qi, G.Y., Chen, Z.Q.: A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system. Nonlinear Dyn. 59(3), 515–527 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cang, S.J., Wang, Z.H., Chen, Z.Q., Jia, H.Y.: Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures. Nonlinear Dyn. 75(4), 745–760 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cang, S.J., Wu, A.G., Wang, Z.H., Chen, Z.Q.: Distinguishing Lorenz and Chen systems based upon hamiltonian energy theory. Int. J. Bifurc. Chaos 27(2), 1750024 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cang, S.J., Wu, A.G., Wang, Z.H., Xue, W., Chen, Z.Q.: Birth of one-to-four-wing chaotic attractors in a class of simplest three-dimensional continuous memristive systems. Nonlinear Dyn. 83(4), 1987–2001 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, D., Wu, C., Iu, H.H.C., Ma, X.: Circuit simulation for synchronization of a fractional-order and integer-order chaotic system. Nonlinear Dyn. 73(3), 1671–1686 (2013)

    Article  MathSciNet  Google Scholar 

  10. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(07), 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, L., Chen, G.: Controlling chaos in an economic model. Phys. A 374(1), 349–358 (2007)

    Article  Google Scholar 

  12. Chua, L.O.: Chua’s circuit: an overview ten years later. J. Circuits Syst. Comput. 4(02), 117–159 (1994)

    Article  Google Scholar 

  13. Degn, H., Holden, A.V., Olsen, L.F.: Chaos in Biological Systems, vol. 138. Springer, New York (2013)

    Google Scholar 

  14. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guan, Z.H., Huang, F., Guan, W.: Chaos-based image encryption algorithm. Phys. Lett. A 346(1), 153–157 (2005)

    Article  MATH  Google Scholar 

  16. Hoover, W.G.: Remark on “some simple chaotic flows”. Phys. Rev. E 51(1), 759 (1995)

    Article  Google Scholar 

  17. Jafari, S., Nazarimehr, F., Sprott, J.C., Golpayegani, S.M.R.H.: Limitation of perpetual points for confirming conservation in dynamical systems. Int. J. Bifurc. Chaos 25(13), 1550182 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kennedy, M.P., Kolumbn, G.: Digital communications using chaos. Signal Process. 80(7), 1307–1320 (2000)

    Article  MATH  Google Scholar 

  19. Li, F., Yao, C.G.: The infinite-scroll attractor and energy transition in chaotic circuit. Nonlinear Dyn. 84(4), 2305–2315 (2016)

    Article  MathSciNet  Google Scholar 

  20. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  Google Scholar 

  21. Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12(03), 659–661 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lü, J., Chen, G., Cheng, D., Celikovsky, S.: Bridge the gap between the Lorenz system and the chen system. Int. J. Bifurc. Chaos 12(12), 2917–2926 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, J., Li, A.B., Pu, Z.S., Yang, L.J., Wang, Y.Z.: A time-varying hyperchaotic system and its realization in circuit. Nonlinear Dyn. 62(3), 535–541 (2010)

    Article  MATH  Google Scholar 

  24. Martyna, G.J., Klein, M.L., Tuckerman, M.: Nosé–Hoover chains: the canonical ensemble via continuous dynamics. J. Chem. Phys. 97(4), 2635–2643 (1992)

    Article  Google Scholar 

  25. Matsumoto, T., Chua, L.O., Kobayashi, K.: Hyper chaos: laboratory experiment and numerical confirmation. IEEE Trans. Circuits Syst. 33(11), 1143–1147 (1986)

    Article  Google Scholar 

  26. Pradeepkumar, D., Ravi, V.: FOREX rate prediction using chaos and quantile regression random forest. In: 2016 3rd International Conference on Recent Advances in Information Technology (RAIT), pp. 517–522. IEEE

  27. Prasad, A.: Existence of perpetual points in nonlinear dynamical systems and its applications. Int. J. Bifurc. Chaos 25(02), 1530005 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ramesh, M., Narayanan, S.: Chaos control by nonfeedback methods in the presence of noise. Chaos Solitons Fractals 10(9), 1473–1489 (1999)

    Article  MATH  Google Scholar 

  29. Rohrlich, F.: The validity of the Helmholtz theorem. Am. J. Phys. 72, 412–413 (2004)

    Article  Google Scholar 

  30. Sarasola, C., DÀnjou, A., Torrealdea, F.J., Moujahid, A., Graña, M.: Energy-like functions for some dissipative chaotic systems. Int. J. Bifurc Chaos 15(8), 2507–2521 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sarasola, C., Torrealdea, F.J., DÀnjou, A., Moujahid, A., Graña, M.: Energy balance in feedback synchronization of chaotic systems. Phys. Rev. E 69(1), 011606 (2004)

    Article  Google Scholar 

  32. Song, X.L., Jin, W.Y., Ma, J.: Energy dependence on the electric activities of a neuron. Chin. Phys. B 24(12), 128710 (2015)

  33. Sprott, J.: Some simple chaotic flows. Phys. Rev. E 50(2), R647 (1994)

    Article  MathSciNet  Google Scholar 

  34. Vaidyanathan, S.: Global chaos control of 3-cells cellular neural network attractor via integral sliding mode control. Int. J. PharmTech Res. 8(8), 211–221 (2015)

    MathSciNet  Google Scholar 

  35. Wang, C.N., Wang, Y., Ma, J.: Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem. Nonlinear Dyn. 65(24), 240501 (2016)

    Google Scholar 

  36. Wang, Y., Wong, K.W., Liao, X., Chen, G.: A new chaos-based fast image encryption algorithm. Appl. Soft Comput. 11(1), 514–522 (2011)

    Article  Google Scholar 

  37. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, New York (2003)

    MATH  Google Scholar 

  38. Zhang, F., Liao, X., Zhang, G.: Some new results for the generalized Lorenz system. Qual. Theory Dyn. Syst. 1–11 (2016). doi:10.1007/s12346-016-0206-z

  39. Zhang, M., Liu, T., Li, P., Wang, A., Zhang, J., Wang, Y.: Generation of broadband chaotic laser using dual-wavelength optically injected Fabry–Perot laser diode with optical feedback. IEEE Photonics Technol. Lett. 23(24), 1872–1874 (2011)

    Article  Google Scholar 

  40. Zhong, G.Q., Tang, K.S., Chen, G.R., Man, K.F.: Bifurcation analysis and circuit implementation of a simple chaos generator. Latin Am. Appl. Res. 31(3), 227–232 (2001)

    Google Scholar 

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Acknowledgements

This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 61573199, 61403274), the Application Base and Frontier Technology Research Project of Tianjin of China (Grant No. 13JCQNJC03600) and South African National Research Foundation Incentive Grant (No. 81705).

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Authors and Affiliations

  1. School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, People’s Republic of China

    Shijian Cang & Aiguo Wu

  2. Department of Product Design, School of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin, 300457, People’s Republic of China

    Shijian Cang

  3. Department of Electrical and Mining Engineering, University of South Africa, Florida, 1710, South Africa

    Zenghui Wang

  4. College of Computer and Control Engineering, Nankai University, Tianjin, 300071, People’s Republic of China

    Zengqiang Chen

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  1. Shijian Cang
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  2. Aiguo Wu
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  3. Zenghui Wang
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Correspondence to Shijian Cang.

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Cang, S., Wu, A., Wang, Z. et al. Four-dimensional autonomous dynamical systems with conservative flows: two-case study. Nonlinear Dyn 89, 2495–2508 (2017). https://doi.org/10.1007/s11071-017-3599-6

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