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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Hyperchaos and Multistability in a Four-Dimensional Financial Mathematical Model

Journal of Applied Nonlinear Dynamics 10(2) (2021) 211--218 | DOI:10.5890/JAND.2021.06.002

Paulo C.Rech

Departamento de F'{i}sica, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Brazil

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Abstract

In this paper we report on hyperchaos and multistability in a four-dimensional nonlinear dynamical system, namely a financial system modeled by a set of four first order ordinary differential equations, whose dynamical behavior is defined by five control parameters. An arbitrarily chosen cross-section of its five-dimensional parameter-space is used to prove numerically the occurrence of both phenomena, multistability and hyperchaos, in the system. Basins of attraction of periodic and chaotic attractors are presented, as well as some typical phase-space portraits.

References

  1. [1]  Yu, H., Cai, G.~and Li, Y.~(2012), Dynamic analysis and control of a new hyperchaotic finance system, Nonlinear Dynamics, 67, 2171-2182.
  2. [2]  Jahanshahi, H., Yousefpour, A., Wei, Z., Alcaraz, R., and Bekiros, S.~(2019), A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization, Chaos Solitons Fractals, 97, 66-77.
  3. [3]  Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A.~(1985), Determining Lyapunov exponents from a time series, Physica D, 16, 285-317.
  4. [4]  R\"ossler, O.E.~(1979), An equation for hyperchaos, Physics Letters A, 71, 155-157.
  5. [5]  Perez, G. and Cerdeira, H.A.~(1995), Extracting messages masked by chaos, Physical Review Letters, 74, 1970-1973.
  6. [6]  Li, Q.D. and Yang, X.S.~(2008), Hyperchaos from two coupled Wien-bridge oscillators, International Journal of Circuit Theory and Applications, 36, 19-29.
  7. [7]  Bao, B.C., Bao, H., Wang, N., Chen, M., and Xu, Q.~(2017), Hidden extreme multistability in memristive hyperchaotic system, Chaos Solitons Fractals, 94, 102-111.
  8. [8]  Li, C.D., Liao, X.F., and Wong, K.W.~(2005), Lag synchronization of hyperchaos with application to secure communications, Chaos Solitons Fractals, 23, 183-193.
  9. [9]  Vafamand, N., Khorshidi, S., and Khayatian, A.~(2018), Secure communication for non-ideal channel via robust TS fuzzy observer-based hyperchaotic synchronization, Chaos Solitons Fractals, 112, 116-124.
  10. [10]  Aguilar-Bustos, A.Y. and Cruz-Hern\andez, C.~(2009), Synchronization of discrete-time hyperchaotic systems: An application in communications, Chaos Solitons Fractals, 41, 1301-1310.
  11. [11]  Mahmoud, E.E. and Abo-Dahab, S.M.~(2018), Dynamical properties and complex anti synchronization with applications to secure communications for a novel chaotic complex nonlinear model, Chaos Solitons Fractals, 106, 273-284.
  12. [12]  Vicente, R., Dauden, J., Colet, P., and Toral, R.~(2005), Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop, IEEE Journal of Quantum Electronics, 41, 541-548.
  13. [13]  Mahmoud, E.E. and AL-Harthi, B.H.~(2020), A hyperchaotic detuned laser model with an infinite number of equilibria existing on a plane and its modified complex phase synchronization with time lag, Chaos Solitons Fractals, 130, 109442.
  14. [14]  Cenys, A., Tamasevicius, A., Baziliauskas, A., Krivickas, R., and Lindberg, E.~(2003), Hyperchaos in coupled Colpitts oscillators, Chaos Solitons Fractals, 17, 349-353.
  15. [15]  Yan, Z.~and Yu, P.~(2008), Hyperchaos synchronization and control on a new hyperchaotic attractor, Chaos Solitons Fractals, 35, 333-345.
  16. [16]  Zhu, C.~(2010), Controlling hyperchaos in hyperchaotic Lorenz system using feedback controllers, Applied Mathematics and Computation, 216, 3126-3132.
  17. [17]  Gao, Y., Liang, C., Wu, Q.,~and Yuan, H.~(2015), A new fractional-order hyperchaotic system and its modified projective synchronization, Chaos Solitons Fractals, 76, 190-204.
  18. [18]  Jajarmi, A., Hajipour, M., and Baleanu, D.~(2017), New aspects of the adaptive synchronization and hyperchaos suppression of a financial model, Chaos Solitons Fractals, 99, 285-296.
  19. [19]  Correia, M.J.~and Rech, P.C.~ (2012), Hyperchaotic states in the parameter-space, Applied Mathematics and Computation, 218, 6711-6715.
  20. [20]  Ramos, J.C.~and Rech, P.C. (2019), Delimitation of Hyperchaotic Regions in Parameter Planes of a Four-Dimensional Dynamical System, Discontinuity, Nonlinearity, and Complexity, 8, 459-465.
  21. [21]  Feudel, U.~and Grebogi, C.~(1997), Multistability and the control of complexity, Chaos, 7, 597-604.
  22. [22]  Xu, Q., Lin, Y., Bao. B.~and Chen, M.~(2016), Multiple attractors in a non-ideal active voltage-controlled memristor based Chuas circuit, Chaos Solitons Fractals, 83, 186-200.
  23. [23]  Ngouonkadi, E.B.M., Fotsin, H.B., Fotso, P.L., Tamba, V.K., and Cerdeira, H.A.~(2016), Bifurcations and multistability in the extended Hindmarsh-Rose neuronal oscillator, Chaos Solitons Fractals, 85, 151-163.
  24. [24]  Wiggers, V.~and Rech, P.C.~(2017), Multistability and organization of periodicity in a Van der Pol-Duffing oscillator, Chaos Solitons Fractals, 103, 632-637.
  25. [25]  Munmuangsaen, B.~and Srisuchinwong, B.~(2018), A hidden chaotic attractor in the classical Lorenz system, Chaos Solitons Fractals, 107, 61-66.
  26. [26]  Njitacke, Z.T., Kengne, J., Tapche, R.W., and Pelap, F.B.~(2018), Uncertain destination dynamics of a novel memristive 4D autonomous system, Chaos Solitons Fractals, 107, 177-185.
  27. [27]  da Silva, A.~and Rech, P.C.~(2018), Numerical investigation concerning the dynamics in parameter planes of the Ehrhard-M\"uller system, Chaos Solitons Fractals, 110, 152-157.
  28. [28]  Naimzada, A.~and Pireddu, M.~(2018), An evolutive discrete exchange economy model with heterogeneous preferences, Chaos Solitons Fractals, 111, 35-43.
  29. [29]  Arecchi, F.T., Meucci, R., Puccioni, G., and Tredicce, J.~(1982), Experimental evidence of sub-harmonic bifurcations, multistability, and turbulence in a q-switched gas laser, Physical Review Letters, 49, 1217-1219.
  30. [30]  Pisarchik, A.N.~and Goswami, B.K.~(2000), Annihilation of one of the coexisting attractors in a bistable system, Physical Review Letters, 84, 1423-1426.
  31. [31]  Chizhevsky, V.N.~(2001), Multistability in dynamical systems induced by weak periodic perturbations, Physical Review E, 64, 036223.
  32. [32]  Gadaleta, S.~and Dangelmayr, G.~(2001), Learning to control a complex multistable system, Physical Review E, 63, 036217.
  33. [33]  Pisarchik, A.N.~(2001), Controlling the multistability of nonlinear systems with coexisting attractors, Physical Review E, 64, 046203. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%