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Universal approach to the problem of emergence of chaos in autonomous dynamical systems

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Abstract

A new method for studying the problem of emergence of chaos in dynamical systems, the right-hand sides of which are continuously differentiable functions, is proposed. The application of obtained results to the investigation of chaotic dynamical systems without homoclinic or heteroclinic orbits and also without use of the concept of equilibrium points (for example, it can be memristive systems) is shown. In addition, the offered method can be used as a tool for search of hidden chaotic attractors.

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References

  1. Luo, A.C.J.: Regularity and Complexity in Dynamical Systems. Springer, New York (2012)

    Book  Google Scholar 

  2. Magnitskii, N.A.: Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations. Commun. Nonlinear Sci. Numer. Simul. 13, 416–433 (2008)

    Article  MathSciNet  Google Scholar 

  3. Belozyorov, V.Y.: General method of construction of implicit discrete maps generating chaos in 3D quadratic systems of differential equations. Int. J. Bifurc. Chaos 24, 23 (2014) ID 1450025

  4. Belozyorov, V.Y.: Exponential-algebraic maps and chaos in 3D autonomous quadratic systems. Int. J. Bifurc. Chaos 25, 24 (2015) ID 1550048

  5. Belozyorov, V.Y.: Research of chaotic dynamics of 3D autonomous quadraic systems by their reduction to special 2D quadratic systems. Math. Probl. Eng. 2015, 15(2015) ID 271637

  6. Belozyorov, VYe, Volkova, S.A.: Role of logistic and Ricker’s maps in appearance of chaos in autonomous quadratic dynamical systems. Nonlinear Dyn 83, 719–729 (2016)

    Article  MathSciNet  Google Scholar 

  7. Magnitskii, N.A.: Universality of transition to chaos in all kinds of nonlinear differential equations. In: Nonlinearity, Bifurcation and Chaos-Theory and Applications, Chapter 6. Intech, pp. 133–174 (2012)

  8. Evstigneev, N.M., Magnitskii, N.A.: Numerical analysis of laminar-turbulent bifurcation scenarios in Kelvin-Helmholtz and Rayleigh-Taylor instabilities for compressible flow. Turbulence, Chapter 2. Intech, pp. 29 – 59 (2017)

  9. Magnitskii, N.A.: Bifurcation theory of dynamical chaos. Chaos Theory, Chapter 11. Intech, pp. 197 – 215 (2018)

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

    Article  MathSciNet  Google Scholar 

  11. Wang, Z., Cang, S., Ochola, E.O., Sun, Y.: A hyperchaotic system without equilibrium. Nonlinear Dyn. 69, 531–537 (2012)

    Article  MathSciNet  Google Scholar 

  12. Zhang, G., Wu, F., Wang, C., Ma, J.: Synchronization behaviors of coupled systems composed of hidden attractors. Int. J. Mod. Phys. B 31, 15 (2017). ID 1750180

    MathSciNet  MATH  Google Scholar 

  13. Wang, X., Chen, G.: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71, 429436 (2012)

    MathSciNet  Google Scholar 

  14. Wei, Z., Wang, R., Liu, A.: A new finding of the existence of hidden hyperchaotic attractors with no equilibria. Math. Comput. Simul. 100, 13–23 (2014)

    Article  MathSciNet  Google Scholar 

  15. Wei, Z., Yu, P., Zhang, W., Yao, M.: Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dyn. 82, 131–141 (2015)

    Article  MathSciNet  Google Scholar 

  16. Belozyorov, V.Ye: A novel search method of chaotic autonomous quadratic dynamical systems without equilibrium points. Nonlinear Dyn 86, 835–860 (2016)

    Article  MathSciNet  Google Scholar 

  17. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647–R650 (1994)

    Article  MathSciNet  Google Scholar 

  18. Hoang, D.V., Kingni, S.T., Pham, V.T.: A no-equilibrium hyperchaotic system and its fractional-order form. Math. Probl. Eng. 2017, 11 (2017) ID 3927184

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

    Article  MathSciNet  Google Scholar 

  20. Radwan, A.G., Azar, T.H., Vaidyanathan, S., Munoz-Pacheco, J. M., Ouannas, A.: Fractional-order and memristive nonlinear systems: advances and applications. Complexity 2017, 12(2017) ID 3760121

  21. Bao, B.C., Wu, P.Y., Bao, H., Xu, Q., Chen, M.: Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator. Chaos Solitons Fractals 106, 161170 (2018)

    Article  MathSciNet  Google Scholar 

  22. Belozyorov, V.Y.: Reduction method for search of chaotic attractors in generic autonomous quadratic dynamical systems. Int. J. Bifurc. Chaos 27, 26 (2017). ID 1750036

  23. Belozyorov, V.Ye: On novel conditions of chaotic attractors existence in autonomous polynomial dynamical systems. Nonlinear Dyn. 91, 2435–2452 (2018)

    Article  Google Scholar 

  24. Wei, Z., Moroz, I., Sprott, J.C., Akgul, A., Zhang, W.: Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo. Chaos 27, 10 (2017), ID 033101

  25. Wei, Z., Moroz, I., Sprott, J.C., Wang, Z., Zhang, W.: Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo. Int. J. Bifurc. Chaos 27, 19 (2017). ID 1730008

    MathSciNet  MATH  Google Scholar 

  26. Wei, Z.C., Zhang, W., Wang, Z., Yao, M.H.: Hidden attractors and dynamical behaviors in an extended Rikitake system. Int.J. Bifurc. Chaos 25, 11 (2015) ID 1550028

  27. Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice Hall, Upper Saddle River (1996)

    Google Scholar 

  28. Crownover, R.M.: Introduction to Fractals and Chaos. Jones and Bartlett Publishers/Boston, London (1995)

    Google Scholar 

  29. Belozyorov, V.Ye, Chernyshenko, S.V.: Quadratic model of inter-population interaction: investigation of stability areas. Appl. Math. Comput. 230, 43–56 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Wang, Z., Moroz, I., Wei, Z., Ren, H.: Dynamics at infinity and a Hopf bifurcation arising in a quadratic system with coexisting attractors. Pramana J. Phys. 90(12), 1–10 (2018)

    Google Scholar 

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Acknowledgements

The author is grateful to the anonymous referees for the brilliant comments which led to a significant improvement in the presentation.

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  1. Department of Applied Mathematics, Dnipro National University, Gagarin’s Avenue, 72, Dnipro, 49050, Ukraine

    Vasiliy Ye. Belozyorov

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  1. Vasiliy Ye. Belozyorov
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Correspondence to Vasiliy Ye. Belozyorov.

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Belozyorov, V.Y. Universal approach to the problem of emergence of chaos in autonomous dynamical systems. Nonlinear Dyn 95, 579–595 (2019). https://doi.org/10.1007/s11071-018-4584-4

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