Skip Navigation Links
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


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:

Synchronization Dynamics of Modified Relay-coupled Chaotic Systems

Journal of Applied Nonlinear Dynamics 7(1) (2018) 11--24 | DOI:10.5890/JAND.2018.03.002

Patrick Louodop$^{1}$,$^{2}$, Elie B.Megam Ngouonkadi$^{2}$, Paulsamy Muruganandam$^{3}$, Hilda A. Cerdeira$^{1}$

$^{1}$ Instituto de Física Teórica - UNESP, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil

$^{2}$ Laboratory of Electronics and Signal Processing, Faculty of Sciences, Department of Physics, University of Dschang, P.O. Box 67 Dschang, Cameroon

$^{3}$ Department of Physics, Bharathidasan University, Tiruchirapalli 620024, India

Download Full Text PDF



In this manuscript, we study the dynamics of a modified relay-coupled chaotic systems. The modification consists on the fact that the relay unit is modeled to lead the entire network to a desired dynamics. Then we achieve finite-time synchronization indirectly through a linear combination of the three systems. Further, we consider the existence of a switch on time of the coupling from the relay unit to the outer systems. It appears some interesting behaviors such as bifurcations, alternation of crisis and phases transitions when varied the switch on time. An open result is also found. In our scheme and for the selected changeable initial conditions, it seems that the appearance or disappearance of coexistence of attractors is linked to the type of synchronization we are dealing with. Mathematical demonstrations are given to sustain our theory while numerical simulations show its effectiveness.


H.A Cerdeira, P. Louodop and P.Muuganandam thank the ICTP-SAIFR and FAPESP grant 2011/11973-4 for partial financial support. P. L. acknowledges support by FAPESP grant 2014/13272-1.


  1. [1]  Gutiérrez, R., Sevilla-Escoboz, R., Piedrahita, P., Finke, C., Feudel, U., Buldú, J.M., Huerta-Cuellar, G., Jaimes-Reátegui, R., Moreno, Y., and Boccaletti, S. (2013), Generalized synchronization in relay systems with instantaneous coupling, Physical Review E, 88(052908), 1-6.
  2. [2]  Camacho, E., Rand, R., and Howland, H. (2004), Dyanamics of two Van der pol oscillators coupled via a bath, International Journal of Solids and Structures, 41, 2133-43.
  3. [3]  Quintero-Quiroz, C. and Cosenza, M.G. (2015), Collective behavior of chaotic oscillators with environmental coupling, Chaos, Solitons and Fractals, 71, 41-5.
  4. [4]  Sharma, A., Dev Shrimali, M., Prasad, A., Ramaswamy, R., and Feudel, U. (2011), Phase-flip transition in relay-coupled nonlinear oscillators, Physical Review E, 84(016226), 1-5.
  5. [5]  Nishikawa, I., Tsukamoto, N., and Aihara, K. (2009), Switching phenomenon induced by breakdown of chaotic phase synchronization, Physica D, 238, 1197-1202.
  6. [6]  Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., and Zhou. C.S. (2002), The synchronization of chaotic systems, Physics Reports, 366, 1-101.
  7. [7]  Pecora, L.M. and Carroll, T.L. (2015), Synchronization of chaotic systems, Chaos, 25(097611); doi: 10.1063/1.4917383.
  8. [8]  Louodop, P., Fotsin, H., and Bowong, S. (2012), A strategy for adaptive synchronization of an electrical chaotic circuit based on nonlinear control, Physica Scripta, 85(025002), 1-6.
  9. [9]  Padmanaban, E., Saha, S., Vigneshwaran, M., and Syamal Dana, K. (2015), Amplified response in coupled chaotic oscillators by induced heterogeneity, Physical Review E, 92(062916), 1-7.
  10. [10]  Louodop, P., Fotsin, H., Bowong, S., and Kammogne, A.S.T. (2014), Adaptive time-delay synchronization of uncertain chaotic systems with disturbances using a nonlinear feedback coupling, Journal of Vibration and Control, 20(6), 815-26.
  11. [11]  Megam Ngouonkadi, E.B., Fotsin, H.B., and Louodop Fotso, P. (2014), Implementing a Memristive Van Der Pol oscillator coupled to a linear oscillator: synchronization and application to secure communication, Phys. Scr., 89(035201), 1-14.
  12. [12]  Du, H., He, Y., and Cheng, Y. (2014) Finite-time synchronization of a class of second-order nonlinear multiagent systems using output feedback control, Ieee Transactions on Circuits and Systems-I: Regular papers, 61(6), 1778-88.
  13. [13]  Louodop, P., Kountchou, Fotsin H.M., and Bowong, S. (2014), Practical finite-time synchronization of jerk systems: Theory and experiment, Nonlinear Dynamics, 78(1), 597-607.
  14. [14]  Yang, W., Xia, Z., Dong, Y., and Zheng, S. (2010), Finite time synchronization between two different chaotic systems with uncertain parameters, Computer and Information Science, 3, 174-179.
  15. [15]  Yang, X., Wu, Z., and Cao, J. (2013), Finite-time synchronization of complex networks with nonidentical discontinuous nodes, Nonlinear Dynamics, 73, 2313-27.
  16. [16]  Aghababa, M.P. (2012), Design of an adaptive finite-time controller for synchronization of two identical/ different non-autonomous chaotic flywheel governor systems, Chin. Phys. B, 21(030502), 1-12.
  17. [17]  Cai, J. and Lin, M. (2010), Finite-time synchronization of non-autonomous chaotic systems with unknown parameters, 2010 International Workshop on Chaos-Fractal Theory and its Applications, IEEE, DOI 10.1109/IWCFTA.2010.28
  18. [18]  Oza, H.B., Spurgeon, S.K., and Valeyev, N.V. (2014), Non-Lipschitz growth functions as a natural way of modelling finite time behaviour in auto-immune dynamics, The International Federation of Automatic Control Cape Town, South Africa, 24-29.
  19. [19]  Murray, J.D. (2002), Mathematical Biology I. An Introduction Third Edition, Berlin Heidelberg: Springer- Verlag.
  20. [20]  Eriksson, K., Estep, D., and Johnson, C. (2004), Applied Mathematics: Body and Soul, 1, Derivatives and Geometry in IR3: Springer.
  21. [21]  Louodop, P., Fotsin, H., Kountchou, M., Megam Ngouonkadi, E.B., Cerdeira, H.A., and Bowong, S. (2014), Finite-time synchronization of tunnel diode based chaotic oscillators, Phys. Rev. E., 89(032921), 1-11.
  22. [22]  Grosu, I., Padmanaban, E., Roy, P.K., and Dana, S.K. (2008), Designing coupling for synchronization and amplification of chaos, Phys. Rev. Lett., 100(234102), 1-4.
  23. [23]  Grebogi, C., Ott, E., and Yorke, J.A. (1983), Crises, sudden changes in chaotic attractors and transient chaos, Physica D, 7, 181-200.
  24. [24]  Pomeau, Y. and Manneville, P. (1980), Intermittent transition to turbulence in dissipative dynamical systems, 74, 189-97.
  25. [25]  Kengne, J., Tabekoueng Njitacke, Z., and Fotsin, H.B. (2016), Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing-Holmes type chaotic oscillators, Communication in Nonlinear Science Numerical Simulation, 36, 29-44.
  26. [26]  Da Silva, I.G., Buldú, J.M., Mirasso, C.R., and García-Ojalvo, J.J. (2006), Synchronization by dynamical relaying in electronic circuit arrays, Chaos, 16(4), 043113-1-8.
  27. [27]  Moukam Kakmeni, F.M., Bowong, S., Senthikumar, D.V., and Kurths, J. (2010), Practical time-delay synchronization of a periodically modulated self-excited oscillators with uncertainties, Chaos, 043121, 1-9.
  28. [28]  Hens, C., Dana, S.K., and Feudel, U. (2015), Extreme multistability: Attractor manipulation and robustness, Chaos, 25(053112), 1-8.
  29. [29]  Megam Ngouonkadi, E.B., Fotsin, H.B., Louodop Fotso, P., Kamdoum Tamba, V., and Cerdeira, H.A. (2016), Bifurcation and multistability in the extended Hindmarsh-Rose neuronal Oscillator, Chaos Solitons and Fractals, 85, 151-163.
  30. [30]  Rosenblum, M.G., Pikovsky, A.S., and Kurths, J. (1996), Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76(11), 1804-1807.
  31. [31]  Megam Ngouonkadi, E.B., Fotsin, H.B., and Louodop Fotso, P. (2014), The combined effect of dynamic chemical and electrical synapses in time-delay-induced phase-transition to synchrony in coupled bursting neurons, Int. J. of Bifurc. and Chaos, 24, 1450069-1-16.
  32. [32]  Yang, M., Cai, C., and Cai, G. (2010), Projective synchronization of a modified three-dimensional chaotic finance system, International Journal of Nonlinear Science, 10(1), 32-38.
  33. [33]  Sun, Z. (2013), Function projective synchronization of two four-scroll hyperchaotic systems with unknown parameters, Cent. Eur. J. Phys., 11(1), 89-95.
  34. [34]  Li, C. and Sprott, J.C. (2014), Coexisting hidden attractors in a 4-D simplified lorenz system, International Journal of Bifurcation and Chaos, 24(3), 1450034-1-12.