Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Delimitation of Hyperchaotic Regions in Parameter Planes of a Four-Dimensional Dynamical System

Discontinuity, Nonlinearity, and Complexity 8(4) (2019) 459--465 | DOI:10.5890/DNC.2019.12.009

Juliane C. Ramos, Paulo C. Rech

Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Brazil

Download Full Text PDF



We report results of a numerical investigation on parameter planes of a set of four autonomous first-order nonlinear ordinary differential equations. More specifically, here are reported ten numerically computed parameter plane diagrams for a five-parameter four-dimensional system, where the dynamical behavior of each point is characterized by using the related Lyapunov exponents spectrum. Each of these parameter plane diagrams indicates parameter values for which hyperchaos, chaos, quasiperiodicity, and periodicity may be found, i.e., each of these diagrams shows delimited regions for each of these dynamical behaviors.


The authors thank Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, and Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina-FAPESC, Brazilian Agencies, for financial support.


  1. [1]  Rössler, O.E. (1979), An equation for hyperchaos, Physics Letters A, 71, 155-157.
  2. [2]  Perez, G. and Cerdeira, H.A. (1955), Extracting messages masked by chaos, Physical Review Letters, 74, 1970-1973.
  3. [3]  Takahashi, Y., Nakano, H. and Saito, T. (2004), A simple hyperchaos generator based on impulsive switching, IEEE Transactions on Circuits and Systems II, 51, 468-472.
  4. [4]  Li, Q.D. and Yang, X.S. (2008), Hyperchaos from two coupledWien-bridge oscillators, International Journal of Circuit Theory and Applications, 36, 19-29.
  5. [5]  Liu, M., Feng, J., and Tse, C.K. (2010), A new hyperchaotic system and its circuit implementation, International Journal of Bifurcation and Chaos, 20, 1201-1208.
  6. [6]  Goedgebuer, J.P., Levy, P., Larger, L., Chen, C.C., and Rhodes,W.T. (2002), Optical communicationwith synchronized hyperchaos generated electrooptically, IEEE Journal of Quantum Electronics, 38, 1178-1183.
  7. [7]  Li, C.D., Liao, X.F., andWong, K.W. (2005), Lag synchronization of hyperchaos with application to secure communications, Chaos Solitons & Fractals, 23, 183-193.
  8. [8]  Gangadhar, C. and Deergha, R. (2009), Hyperchaos based image encryption, International Journal of Bifurcation and Chaos, 19, 3833-3839.
  9. [9]  Zhang, S.H. and Shen, K. (2003), Controlling hyperchaos in erbium-doped fibre laser, Chinese Physics, 12, 149-153.
  10. [10]  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.
  11. [11]  Cenys, A., Tamasevicius, A., Baziliauskas, A., Krivickas, R., and Lindberg, E. (2003), Hyperchaos in coupled Colpitts oscillators, Chaos Solitons & Fractals, 17, 349-353.
  12. [12]  Li, Y., Tang, W.K.S., and Chen, G. (2005), Generating hyperchaos via state feedback control, International Journal of Bifurcation and Chaos, 15, 3367-3375.
  13. [13]  Zhu, C. (2010), Controlling hyperchaos in hyperchaotic Lorenz system using feedback controllers, Applied Mathematics and Computation, 216, 3126-3132.
  14. [14]  Sun, K., Liu, X., Zhu, C., and Sprott, J.C. (2012), Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system, Nonlinear Dynamics, 69, 1383-1391.
  15. [15]  Yu, H., Cai, G., and Li, Y. (2012), Dynamic analysis and control of a new hyperchaotic finance system, Nonlinear Dynamics, 67, 2171-2182.
  16. [16]  Yan, Z. and Yu, P. (2007), Globally exponential hyperchaos (lag) synchronization in a family of modified hyperchaotic Rössler systems, International Journal of Bifurcation and Chaos, 2007, 1759-1774.
  17. [17]  Vidal, G. and Mancini, H. (2010), Hyperchaotic synchronization, International Journal of Bifurcation and Chaos, 20, 885-896.
  18. [18]  Njah, A.N. (2010), Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques, Nonlinear Dynamics, 61, 1-9.
  19. [19]  Cafagna, D. and Grassi, G. (2012), Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems, Nonlinear Dynamics, 68, 117-128.
  20. [20]  Sun, Z., Si, G., Min, F., and Zhang, Y. (2012), Adaptive modified function projective synchronization and parameter identification of uncertain hyperchaotic (chaotic) systems with identical or non-identical structures, Nonlinear Dynamics, 68, 471-486.
  21. [21]  Cai, L., Ma, X.K., and Wang, S. (2003), Study of hyperchaotic behavior in quantum cellular neural networks, Acta Physica Sinica, 52, 3002-3006.
  22. [22]  Li, X. and Yan, Z.Y. (2015), Hopf bifurcation in a new four-dimensional hyperchaotic system, Communications in Theoretical Physics, 64, 197-202.
  23. [23]  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.
  24. [24]  Wiggins, S. (2003), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer: New York.
  25. [25]  Correia, M.J. and Rech, P.C. (2012), Hyperchaotic states in the parameter-space, Applied Mathematics and Computation, 218, 6711-6715.