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Infinite coexisting attractors in an autonomous hyperchaotic megastable oscillator and linear quadratic regulator-based control and synchronization

  • Regular Article - Statistical and Nonlinear Physics
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Abstract

The coexistence of different attractors with fixed parameters affords versatility in the performance of the dynamical system. As a result, recent research has focused on defining nonlinear oscillators with infinite coexisting attractors, the vast majority of which are non-autonomous systems. In this study, we present an infinite number of equilibriums achieved with the simplest autonomous megastable oscillator. To begin, we explore the symmetry property of distinct coexisting attractors, such as periodic, quasi-periodic, and chaotic attractors. The dynamical properties of the proposed system are further investigated using phase portraits, Lyapunov exponents spectrum, and bifurcation diagram using the local maxima of the state variables. The detailed investigation reveals that the proposed system has a wide range of dynamical properties ranging from periodic oscillations to hyperchaos. Linear quadratic regulator (LQR) control-based controllers are designed to control the chaos in the proposed system and also achieve synchronization between proposed systems that have different initial conditions. The effectiveness of the controller designed based on LQR is presented by simulation results. It is shown that the proposed system converges asymptotically to the desired equilibrium point, and the synchronization between the slave and the master systems is achieved by converging the error to zero after controllers are activated.

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References

  1. A.H. Pisarchik, A.N. Pisarchik, A.E. Hramov, Multistability in Physical and Living Systems: Characterization and Applications (Springer, Berlin, 2022)

    MATH  Google Scholar 

  2. Ostrovskii, V. Y., Tutueva, A. V., Rybin, V. G., Karimov, A. I., & Butusov, D. N. (2020). Continuation analysis of memristor-based modified Chua's circuit. In 2020 International Conference Nonlinearity, Information and Robotics (NIR) (pp. 1–5). IEEE.

  3. K. Sathiyadevi, D. Premraj, T. Banerjee, M. Lakshmanan, Additional complex conjugate feedback-induced explosive death and multistabilities. Phys. Rev. E 106(2), 024215 (2022)

    MathSciNet  ADS  Google Scholar 

  4. K. Sathiyadevi, V.K. Chandrasekar, M. Lakshmanan, Emerging chimera states under nonidentical counter-rotating oscillators. Phys. Rev. E 105(3), 034211 (2022)

    MathSciNet  ADS  Google Scholar 

  5. K. Sathiyadevi, S. Karthiga, V.K. Chandrasekar, D.V. Senthilkumar, M. Lakshmanan, Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings. Phys. Rev. E 95(4), 042301 (2017)

    ADS  Google Scholar 

  6. X. Wang, N.V. Kuznetsov, G. Chen, Chaotic Systems with Multistability and Hidden Attractors Vol 40 (Springer, Berlin, 2021), pp.149–150

    MATH  Google Scholar 

  7. P. Kruse, M. Stadler (eds.), Ambiguity in Mind and Nature: Multistable Cognitive Phenomena Vol 64 (Springer, Berlin, 2012)

    Google Scholar 

  8. C. Li, J.C. Sprott, W. Hu, Y. Xu, Infinite multistability in a self-reproducing chaotic system. Int. J. Bifurc. Chaos 27(10), 1750160 (2017)

    MathSciNet  MATH  Google Scholar 

  9. C. Li, W. Hu, J.C. Sprott, X. Wang, Multistability in symmetric chaotic systems. Eur. Phys. J. Spec. Top. 224(8), 1493–1506 (2015)

    Google Scholar 

  10. C. Li, J.C. Sprott, Multistability in the Lorenz system: a broken butterfly. Int. J. Bifurc. Chaos 24(10), 1450131 (2014)

    MathSciNet  MATH  Google Scholar 

  11. B.C. Bao, H. Bao, N. Wang, M. Chen, Q. Xu, Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94, 102–111 (2017)

    MathSciNet  MATH  ADS  Google Scholar 

  12. H. Jahanshahi, K. Rajagopal, A. Akgul, N.N. Sari, H. Namazi, S. Jafari, Complete analysis and engineering applications of a megastable nonlinear oscillator. Int. J. Non-Linear Mech. 107, 126–136 (2018)

    ADS  Google Scholar 

  13. S. Jafari, K. Rajagopal, T. Hayat, A. Alsaedi, V.T. Pham, Simplest megastable chaotic oscillator. Int. J. Bifurc. Chaos 29(13), 1950187 (2019)

    MathSciNet  MATH  Google Scholar 

  14. J.C. Sprott, S. Jafari, A.J.M. Khalaf, T. Kapitaniak, Megastability: coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. Eur. Phys. J. Spec. Top. 226(9), 1979–1985 (2017)

    Google Scholar 

  15. B. Chen, K. Rajagopal, I.I. Hamarash, A. Karthikeyan, I. Hussain, Simple megastable oscillators with different types of attractors; tori, chaotic and hyperchaotic ones. Eur. Phys. J. Spec. Top. 229(6), 1155–1161 (2020)

    Google Scholar 

  16. P. Prakash, K. Rajagopal, J.P. Singh, B.K. Roy, Megastability, multistability in a periodically forced conservative and dissipative system with signum nonlinearity. Int. J. Bifurc. Chaos 28(09), 1830030 (2018)

    MathSciNet  MATH  Google Scholar 

  17. G.D. Leutcho, T.F. Fozin, A.N. Negou, Z.T. Njitacke, V.T. Pham, J. Kengne, S. Jafari, A novel megastable hamiltonian system with infinite hyperbolic and nonhyperbolic equilibria. Complexity 20, 20 (2020)

    MATH  Google Scholar 

  18. R. Li, E. Dong, J. Tong, S. Du, A new autonomous memristive megastable oscillator and its Hamiltonian-energy-dependent megastability. Chaos Interdiscip. J. Nonlinear Sci. 32(1), 013127 (2022)

    MathSciNet  Google Scholar 

  19. K. Zhang, M.D. Vijayakumar, S.S. Jamal, H. Natiq, K. Rajagopal, S. Jafari, I. Hussain, A novel megastable oscillator with a strange structure of coexisting attractors: design, analysis, and FPGA implementation. Complexity 20, 21 (2021)

    Google Scholar 

  20. B. Aguirre-Hernández, E. Campos-Cantón, J.A. López-Renteria, E.D. González, A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems. Chaos Solitons Fractals 71, 100–106 (2015)

    MathSciNet  MATH  ADS  Google Scholar 

  21. E. Campos-Cantón, R. Femat, G. Chen, Attractors generated from switching unstable dissipative systems. Chaos Interdiscip. J. Nonlinear Sci. 22(3), 033121 (2012)

    MathSciNet  MATH  Google Scholar 

  22. J.R. Pulido-Luna, J.A. López-Rentería, N.R. Cazarez-Castro, E. Campos, A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator. Integration 81, 34–42 (2021)

    Google Scholar 

  23. R.D.J. Escalante-González, E. Campos, Multistable systems with hidden and self-excited scroll attractors generated via piecewise linear systems. Complexity 20, 20 (2020)

    MATH  Google Scholar 

  24. L.J. Ontañón-García, E. Campos-Cantón, Widening of the basins of attraction of a multistable switching dynamical system with the location of symmetric equilibria. Nonlinear Anal. Hybrid Syst. 26, 38–47 (2017)

    MathSciNet  MATH  Google Scholar 

  25. B. Ramakrishnan, A. Ahmadi, F. Nazarimehr, H. Natiq, S. Jafari, I. Hussain, Oyster oscillator: a novel mega-stable nonlinear chaotic system. Eur. Phys. J. Spec. Top. 20, 1–9 (2021)

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    MathSciNet  MATH  ADS  Google Scholar 

  28. 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)

    MathSciNet  MATH  ADS  Google Scholar 

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

    MathSciNet  MATH  ADS  Google Scholar 

  30. R.J. Escalante-González, E. Campos, Hyperchaotic attractors through coupling of systems without equilibria. Eur. Phys. J. Spec. Top. 229(6), 1309–1318 (2020)

    Google Scholar 

  31. S. Jafari, J.C. Sprott, V.T. Pham, C. Volos, C. Li, Simple chaotic 3D flows with surfaces of equilibria. Nonlinear Dyn. 86(2), 1349–1358 (2016)

    Google Scholar 

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

    MathSciNet  MATH  ADS  Google Scholar 

  33. T. Gotthans, J.C. Sprott, J. Petrzela, Simple chaotic flow with circle and square equilibrium. Int. J. Bifurc. Chaos 26(08), 1650137 (2016)

    MathSciNet  MATH  Google Scholar 

  34. C. Li, J.C. Sprott, Y. Mei, An infinite 2-D lattice of strange attractors. Nonlinear Dyn. 89(4), 2629–2639 (2017)

    MathSciNet  Google Scholar 

  35. C. Li, J.C. Sprott, An infinite 3-D quasiperiodic lattice of chaotic attractors. Phys. Lett. A 382(8), 581–587 (2018)

    MathSciNet  MATH  ADS  Google Scholar 

  36. X. Wang, G. Chen, A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2012)

    MathSciNet  ADS  Google Scholar 

  37. Z. Wang, Z. Wei, K. Sun, S. He, H. Wang, Q. Xu, M. Chen, Chaotic flows with special equilibria. Eur. Phys. J. Spec. Top. 229(6), 905–919 (2020)

    Google Scholar 

  38. S. Jafari, J.C. Sprott, M. Molaie, A simple chaotic flow with a plane of equilibria. Int. J. Bifurc. Chaos 26(06), 1650098 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Y.X. Tang, A.J.M. Khalaf, K. Rajagopal, V.T. Pham, S. Jafari, Y. Tian, A new nonlinear oscillator with infinite number of coexisting hidden and self-excited attractors. Chin. Phys. B 27(4), 040502 (2018)

    ADS  Google Scholar 

  40. N. Wang, G. Zhang, N.V. Kuznetsov, H. Bao, Hidden attractors and multistability in a modified Chua’s circuit. Commun. Nonlinear Sci. Numer. Simul. 92, 105494 (2021)

    MathSciNet  MATH  Google Scholar 

  41. J.C. Sprott, A new chaotic jerk circuit. IEEE Trans. Circ. Syst. II Express Briefs 58(4), 240–243 (2011)

    Google Scholar 

  42. F. Li, C. Tai, H. Bao, J. Luo, B. Bao, Hyperchaos, quasi-period and coexisting behaviors in second-order-memristor-based jerk circuit. Eur. Phys. J. Spec. Top. 229(6), 1045–1058 (2020)

    Google Scholar 

  43. F.Y. Dalkiran, J.C. Sprott, Simple chaotic hyperjerk system. Int. J. Bifurc. Chaos 26(11), 1650189 (2016)

    MathSciNet  Google Scholar 

  44. K.E. Chlouverakis, J.C. Sprott, Chaotic hyperjerk systems. Chaos Solitons Fractals 28(3), 739–746 (2006)

    MathSciNet  MATH  ADS  Google Scholar 

  45. G.D. Leutcho, J. Kengne, L.K. Kengne, A. Akgul, V.T. Pham, S. Jafari, A novel chaotic hyperjerk circuit with bubbles of bifurcation: mixed-mode bursting oscillations, multistability, and circuit realization. Phys. Scr. 95(7), 075216 (2020)

    ADS  Google Scholar 

  46. G.D. Leutcho, J. Kengne, A.N. Negou, T.F. Fozin, V.T. Pham, S. Jafari, A modified simple chaotic hyperjerk circuit: coexisting bubbles of bifurcation and mixed-mode bursting oscillations. Z. Nat. A 75(7), 593–607 (2020)

    Google Scholar 

  47. K. Rajagopal, J.P. Singh, A. Karthikeyan, B.K. Roy, Existence of metastable, hyperchaos, line of equilibria and self-excited attractors in a new hyperjerk oscillator. Int. J. Bifurc. Chaos 30(13), 2030037 (2020)

    MathSciNet  MATH  Google Scholar 

  48. D. Premraj, S. Kumarasamy, K. Thamilmaran, K. Rajagopal, Strange nonchaotic attractor in memristor-based van der Pol oscillator. Eur. Phys. J. Spec. Top. 20, 1–7 (2022)

    Google Scholar 

  49. P. Durairaj, S. Kanagaraj, T. Kathamuthu, K. Rajagopal, Strange nonchaotic attractors in memristor-based Shimizu–Morioka oscillator. Int. J. Bifurc. Chaos 32(09), 2230022 (2022)

    MathSciNet  MATH  Google Scholar 

  50. K. Rajagopal, H. Jahanshahi, M. Varan, I. Bayır, V.T. Pham, S. Jafari, A. Karthikeyan, A hyperchaotic memristor oscillator with fuzzy based chaos control and LQR based chaos synchronization. AEU-Int. J. Electron. Commun. 94, 55–68 (2018)

    Google Scholar 

  51. D.S. Naidu, Optimal Control Systems (CRC Press, Boca Raton, 2002)

    Google Scholar 

  52. S.K. Choudhary, LQR based optimal control of chaotic dynamical systems. Int. J. Model. Simul. 35(3–4), 104–112 (2015)

    Google Scholar 

  53. K.J. Åström, R.M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University Press, Princeton, 2021)

    MATH  Google Scholar 

  54. Y. Cao, Y. Li, G. Zhang, K. Jermsittiparsert, M. Nasseri, An efficient terminal voltage control for PEMFC based on an improved version of whale optimization algorithm. Energy Rep. 6, 530–542 (2020)

    Google Scholar 

  55. Y. Batmani, Chaos control and chaos synchronization using the state-dependent Riccati equation techniques. Trans. Inst. Meas. Control. 41(2), 311–320 (2019)

    Google Scholar 

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Acknowledgements

We gratefully acknowledge this work is funded by the Center for Nonlinear Systems, Chennai Institute of Technology (CIT), India, vide funding number CIT/CNS/2022/RP-016.

Author information

Authors and Affiliations

  1. Department of Electronics and Communication Engineering, Chennai Institute of Technology, Chennai, India

    Prasina Alexander

  2. Department of Electrical and Electronics Engineering, Faculty of Engineering, Sakarya University, Sakarya, Turkey

    Selçuk Emiroğlu

  3. Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai, Tamilnadu, 600 069, India

    Sathiyadevi Kanagaraj & Karthikeyan Rajagopal

  4. Department of Computer Engineering, Faculty of Engineering, Hitit University, Çorum, Turkey

    Akif Akgul

  5. Department of Electronics and Communications Engineering and University Centre for Research and Development, Chandigarh University, Mohali, Punjab, 140 413, India

    Karthikeyan Rajagopal

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  1. Prasina Alexander
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  3. Sathiyadevi Kanagaraj
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Correspondence to Karthikeyan Rajagopal.

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Alexander, P., Emiroğlu, S., Kanagaraj, S. et al. Infinite coexisting attractors in an autonomous hyperchaotic megastable oscillator and linear quadratic regulator-based control and synchronization. Eur. Phys. J. B 96, 12 (2023). https://doi.org/10.1140/epjb/s10051-022-00471-1

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