Skip to main content
SpringerLink
Log in

Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. B. Blażejczyk-Okolewska, T. Kapitaniak, Co-existing attractors of impact oscillator. Chaos Solitons Fractals 9(8), 1439–1443 (1998)

    Article  MATH  Google Scholar 

  2. V. Bragin, V. Vagaitsev, N.V. Kuznetsov, G.A. Leonov, Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int 50(4), 511–543 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Caponetto, Fractional Order Systems: Modeling and Control Applications, vol. 72 (World Scientific, Singapore, 2010)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Chudzik, P. Perlikowski, A. Stefanski, T. Kapitaniak, Multistability and rare attractors in van der Pol–Duffing oscillator. Int. J. Bifurc. Chaos 21(07), 1907–1912 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Deng, Short memory principle and a predictor–corrector approach for fractional differential equations. J. Comput. Appl. Math. 206(1), 174–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. Circuits Syst. Mag. IEEE 10(4), 40–50 (2010)

    Article  Google Scholar 

  9. I.V. Ermakov, S.T. Kingni, V.Z. Tronciu, J. Danckaert, Chaotic semiconductor ring lasers subject to optical feedback: applications to chaos-based communications. Opt. Commun. 286, 265–272 (2013)

    Article  Google Scholar 

  10. S. Faraji, M. Tavazoei, The effect of fractionality nature in differences between computer simulation and experimental results of a chaotic circuit. Open Phys. 11(6), 836–844 (2013)

    Article  Google Scholar 

  11. M. Feki, An adaptive feedback control of linearizable chaotic systems. Chaos Solitons Fractals 15(5), 883–890 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Fichera, C. Losenno, A. Pagano, Clustering of chaotic dynamics of a lean gas-turbine combustor. Appl. Energy 69(2), 101–117 (2001)

    Article  Google Scholar 

  13. T.J. Freeborn, B. Maundy, A.S. Elwakil, Measurement of supercapacitor fractional-order model parameters from voltage-excited step response. Emerg. Sel. Top. Circuits Syst. IEEE J. 3(3), 1–10 (2013)

    Article  MATH  Google Scholar 

  14. T. Gotthans, J. Petržela, New class of chaotic systems with circular equilibrium. Nonlinear Dyn. 81(3), 1143–1149 (2015)

    Article  MathSciNet  Google Scholar 

  15. M. Inoue, A. Nagayoshi, A chaos neuro-computer. Phys. Lett. A 158(8), 373–376 (1991)

    Article  Google Scholar 

  16. S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377(9), 699–702 (2013)

    Article  MathSciNet  Google Scholar 

  17. S. Jafari, J.C. Sprott, Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79–84 (2013)

    Article  MathSciNet  Google Scholar 

  18. S.T. Kingni, L. Keuninckx, P. Woafo, G. Van der Sande, J. Danckaert, Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation. Nonlinear Dyn. 73(1–2), 1111–1123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Kingni, S. Jafari, H. Simo, P. Woafo, Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur. Phys. J. Plus 129(5), 1–16 (2014)

    Article  Google Scholar 

  20. S.T. Kingni, G.S.M. Ngueuteu, P. Woafo, Bursting generation mechanism in a three-dimensional autonomous system, chaos control, and synchronization in its fractional-order form. Nonlinear Dyn. 76(2), 1169–1183 (2014)

    Article  MathSciNet  Google Scholar 

  21. M.S. Krishna, S. Das, K. Biswas, B. Goswami, Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization. Electron Devices IEEE Trans. 58(11), 4067–4073 (2011)

    Article  Google Scholar 

  22. N.V. Kuznetsov, G.A. Leonov, S. Seledzhi, Hidden oscillations in nonlinear control systems. IFAC Proc. Vol. 18(1), 2506–2510 (2011)

    Google Scholar 

  23. G.A. Leonov, N.V. Kuznetsov, V. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. Vol. 4(1), 29–33 (2010)

    MathSciNet  Google Scholar 

  24. G.A. Leonov, N.V. Kuznetsov, Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems, in Doklady Mathematics, vol. 1 (Springer, Berlin, 2011), pp. 475-481

  25. G.A. Leonov, N.V. Kuznetsov, Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proc. Vol. 18(1), 2494–2505 (2011)

    Google Scholar 

  26. G.A. Leonov, N.V. Kuznetsov, Hidden oscillations in dynamical systems. 16 Hilbert’s problem, Aizerman’s and Kalman’s conjectures, hidden attractors in Chua’s circuits. J. Math. Sci. 201(5), 645–662 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. G.A. Leonov, N.V. Kuznetsov, M. Kiseleva, E. Solovyeva, A. Zaretskiy, Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77(1–2), 277–288 (2014)

    Article  Google Scholar 

  28. G.A. Leonov, N.V. Kuznetsov, V. Vagaitsev, Localization of hidden Chua’s attractors. Phys. Lett. A 375(23), 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. G.A. Leonov, N.V. Kuznetsov, V. Vagaitsev, Hidden attractor in smooth Chua systems. Phys. D Nonlinear Phenom. 241(18), 1482–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. G.A. Leonov, N.V. Kuznetsov, Analytical-numerical methods for hidden attractors’ localization: the 16th Hilbert problem, Aizerman and Kalman conjectures, and Chua circuits, in Numerical Methods for Differential Equations, Optimization, and Technological Problems (Springer, Berlin, 2013), pp. 41–64

  31. G.A. Leonov, N.V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos 23(01), 1330002 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. A.Y.T. Leung, X.-F. Li, Y.-D. Chu, X.-B. Rao, Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling. Nonlinear Dyn. 82(1), 185–199 (2015)

    Article  MathSciNet  Google Scholar 

  33. C. Li, J.C. Sprott, Chaotic flows with a single nonquadratic term. Phys. Lett. A 378(3), 178–183 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. B. Maundy, A. Elwakil, S. Gift, On a multivibrator that employs a fractional capacitor. Analog Integr. Circuits Signal Process. 62(1), 99–103 (2010)

    Article  Google Scholar 

  37. M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Y. Nakamura, A. Sekiguchi, The chaotic mobile robot. Robot. Automa. IEEE Trans. 17(6), 898–904 (2001)

    Article  Google Scholar 

  39. G.M. Ngueuteu, P. Woafo, Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems. Mech. Res. Commun. 46, 20–25 (2012)

    Article  Google Scholar 

  40. V. Petrov, V. Gaspar, J. Masere, K. Showalter, Controlling chaos in the Belousov–Zhabotinsky reaction. Nature 361(6409), 240–243 (1993)

    Article  Google Scholar 

  41. P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems. Chaos Interdiscip. J. Nonlinear Sci. 20(1), 013111–013120 (2010)

    Article  MathSciNet  Google Scholar 

  42. V.-T. Pham, C. Volos, S. Jafari, Z. Wei, X. Wang, Constructing a novel no-equilibrium chaotic system. Int. J. Bifurc. Chaos 24(05), 1450073 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. A.G. Radwan, Stability analysis of the fractional-order \(\text{ RL }\beta \text{ C }\alpha \) circuit. J. Fract. Calc. Appl. 3(1), 1–15 (2012)

    Google Scholar 

  44. A.G. Radwan, K.N. Salama, Passive and active elements using fractional circuit. Circuits Syst. I Regul. Pap. IEEE Trans. 58(10), 2388–2397 (2011)

    Article  MathSciNet  Google Scholar 

  45. A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31(6), 1901–1915 (2012)

    Article  MathSciNet  Google Scholar 

  46. O.E. Rössler, An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    Article  Google Scholar 

  47. A. Silchenko, T. Kapitaniak, V. Anishchenko, Noise-enhanced phase locking in a stochastic bistable system driven by a chaotic signal. Phys. Rev. E 59(2), 1593 (1999)

    Article  Google Scholar 

  48. C.P. Silva, Shil’nikov’s theorem—a tutorial. Circuits Syst. I Fundam. Theory Appl. IEEE Trans. 40(10), 675–682 (1993)

    Article  MATH  Google Scholar 

  49. L.J. Sheu, A speech encryption using fractional chaotic systems. Nonlinear Dyn. 65(1–2), 103–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  51. J.C. Sprott, A new class of chaotic circuit. Phys. Lett. A 266(1), 19–23 (2000)

    Article  MathSciNet  Google Scholar 

  52. J.C. Sprott, Simplest dissipative chaotic flow. Phys. Lett. A 228(4), 271–274 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  53. M.S. Tavazoei, M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367(1), 102–113 (2007)

    Article  MATH  Google Scholar 

  54. M.C. Tripathy, K. Biswas, S. Sen, A design example of a fractional-order Kerwin–Huelsman–Newcomb biquad filter with two fractional capacitors of different order. Circuits Syst. Signal Process. 32(4), 1523–1536 (2013)

    Article  MathSciNet  Google Scholar 

  55. X. Wang, G. Chen, Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71(3), 429–436 (2013)

    Article  MathSciNet  Google Scholar 

  56. J. Wang, G. Feng, Bifurcation and chaos in discrete-time BVP oscillator. Int. J. Non Linear Mech. 45(6), 608–620 (2010)

    Article  Google Scholar 

  57. Y. Wang, X. Liao, T. Xiang, K.-W. Wong, D. Yang, Cryptanalysis and improvement on a block cryptosystem based on iteration a chaotic map. Phys. Lett. A 363(4), 277–281 (2007)

    Article  MATH  Google Scholar 

  58. Z. Wei, Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376(2), 102–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. S. Westerlund, L. Ekstam, Capacitor theory. Dielectr. Electr. Insul. IEEE Trans. 1(5), 826–839 (1994)

    Article  Google Scholar 

  60. T. Zhou, G. Chen, Classification of chaos in 3-D autonomous quadratic systems-I: basic framework and methods. Int. J. Bifurc. Chaos 16(09), 2459–2479 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

S.T.K. and G.R.K. thank Prof. Elisabeth Ngo Bum, the Director of the Institute of Mines and Petroleum Industries (University of Maroua, Cameroon), for creating a good environment with perfect balance between teaching and research time.

Author information

Authors and Affiliations

  1. Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni & Guy Richard Kol

  2. School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam

    Viet-Thanh Pham

  3. Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran, 15875-4413, Iran

    Sajad Jafari

  4. School of Geology and Mining Engineering, University of Ngaoundéré, P.O. Box 454, Ngaoundéré, Cameroon

    Guy Richard Kol

  5. Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP) and TWAS Research Unit, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    Sifeu Takougang Kingni & Paul Woafo

  6. Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Sifeu Takougang Kingni

Authors
  1. Sifeu Takougang Kingni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viet-Thanh Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sajad Jafari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guy Richard Kol
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Paul Woafo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sifeu Takougang Kingni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kingni, S.T., Pham, VT., Jafari, S. et al. Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form. Circuits Syst Signal Process 35, 1933–1948 (2016). https://doi.org/10.1007/s00034-016-0259-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-016-0259-x

Keywords

Search

Navigation