Abstract
The main goal of this paper is to propose the adaptive nonsingular terminal sliding mode controllers for complete synchronization (CS) and anti-synchronization (AS) between two identical Φ 6 Van der Pol or Duffing oscillators with presentations of system uncertainties and external disturbances. Unlike directly eliminating the nonlinear items of synchronized error system for sliding mode control schemes in the literature, the proposed adaptive controllers can tackle the nonlinear dynamics without active cancellation. The controllers can be implemented without known bounds of system uncertainties and external disturbances. Meanwhile, the feedback gains are not determined in advance but updated by the adaptive rules. Sufficient conditions are given based on the Lyapunov stability theorem and numerical simulations are performed to verify the effectiveness of presented schemes. The results show that the chaotic synchronization can be achieved accurately by the chattering free control.
Similar content being viewed by others
References
Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
Njah, A.N., Vincent, U.E.: Chaos synchronization between single and double wells Duffing–Van der Pol oscillators using active control. Chaos Solitons Fractals 37, 1356–1361 (2008)
Idowu, B.A., Vincent, U.E., Njah, A.N.: Synchronization of chaos in non-identical parametrically excited systems. Chaos Solitons Fractals 39, 2322–2331 (2009)
Njah, A.N.: Synchronization via active control of identical and non-identical Φ 6 chaotic oscillators with external excitation. J. Sound Vib. 327, 322–332 (2009)
Njah, A.N.: Synchronization via active control of parametrically and externally excited Φ 6 Van der Pol and Duffing oscillators and application to secure communications. J. Vib. Control 17, 493–504 (2011)
Harb, A.M., Zaher, A.A., Al-Qaisia, A.A., Zohdy, M.A.: Recursive backstepping control of chaotic Duffing oscillators. Chaos Solitons Fractals 34, 639–645 (2007)
Bowong, S.: Adaptive synchronization of chaotic systems with unknown bounded uncertainties via backstepping approach. Nonlinear Dyn. 49, 59–70 (2007)
Njak, A.N.: Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn. 61, 1–9 (2010)
Peng, C.C., Hsue, A.W.J., Chen, C.L.: Variable structure based robust backstepping controller design for nonlinear systems. Nonlinear Dyn. 63, 253–262 (2011)
Li, R., Xu, W., Li, S.: Chaos control and synchronization of the Φ 6 Van der Pol system driven by external and parametric excitations. Nonlinear Dyn. 53, 261–271 (2008)
Lei, Y., Yung, K.L., Xu, Y.: Chaos synchronization and parameter estimation of single-degree-of-freedom oscillators via adaptive control. J. Sound Vib. 329, 973–979 (2010)
Yang, C.C.: Adaptive control and synchronization of identical new chaotic flows with unknown parameters via single input. Appl. Math. Comput. 216, 1316–1324 (2010)
Li, X.F., Leung, A.C.S., Liu, X.J., Han, X.P., Chu, Y.D.: Adaptive synchronization of identical chaotic and hyper-chaotic systems with uncertain parameters. Nonlinear Anal., Real World Appl. 11, 2215–2223 (2010)
Yang, C.C.: Adaptive synchronization of Lü hyperchaotic system with uncertain parameters based on single-input controller. Nonlinear Dyn. 63, 447–454 (2011)
Li, X.F., Leung, A.C.S., Han, X.P., Liu, X.J., Chu, Y.D.: Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlinear Dyn. 63, 263–275 (2011)
Li, S.Y., Ge, Z.M.: Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters via nonlinear control. Nonlinear Dyn. 64, 77–87 (2011)
Yau, H.T.: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fractals 22, 341–347 (2004)
Zribi, M., Smaoui, N., Salim, H.: Synchronization of the unified chaotic systems using sliding mode controller. Chaos Solitons Fractals 42, 3197–3209 (2010)
Feng, J.W., He, L., Xu, C., Francis, A., Wu, G.: Synchronizing the noise perturbed Genesio chaotic system by sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 15, 2546–2551 (2010)
Li, W., Liu, Z., Miao, J.: Adaptive synchronization for a unified chaotic system with uncertainty. Commun. Nonlinear Sci. Numer. Simul. 15, 3015–3021 (2010)
Etemadi, S., Alasty, A., Salarieh, H.: Synchronization of chaotic systems with parameter uncertainties via variable structure control. Phys. Lett. A 357, 17–21 (2005)
Pourmahmood, M., Khanmohammadi, S., Alizadeh, G.: Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Commun. Nonlinear Sci. Numer. Simul. 16, 2853–2868 (2011)
Aghababa, P.M., Khanmohammadi, S., Alizadeh, G.: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model. 35, 3080–3091 (2011)
Yahyazadeh, M., Noei, A.R., Ghaderi, R.: Synchronization of chaotic systems with known and unknown parameters using a modified active sliding mode control. ISA Trans. 50, 262–267 (2011)
Ling, B.W., Lu, H.H., Lam, H.K.: Control of Chaos in Nonlinear Circuits and Systems. World Scientific, Singapore (2009)
Man, Z., Palinski, A.P., Wu, H.: A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE Trans. Autom. Control 39, 2464–2468 (1994)
Feng, Y., Yu, X., Man, Z.: Non-singular terminal sliding mode control of rigid manipulators. Automatica 38, 2159–2167 (2002)
Yu, S., Yu, X., Shirinzadeh, B., Man, Z.: Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41, 1957–1964 (2005)
Siewe Siewe, M., Moukam Kakmeni, F.M., Tchawoua, C.: Resonant oscillation and homoclinic bifurcation, in a Φ 6 Van der Pol oscillator. Chaos Solitons Fractals 21, 841–853 (2004)
Tchoukuegno, R., Nana Nbendjo, B.R., Woafo, P.: Resonant oscillations and fractal basin boundaries of a particle in a Φ 6 potential. Physica A 304, 362–368 (2002)
Tchoukuegno, R., Nana Nbendjo, B.R., Woafo, P.: Linear feedback and parametric controls of vibrations and chaotic escape in a Φ 6 potential. Int. J. Non-Linear Mech. 38, 531–541 (2003)
Jing, Z., Yang, Z., Jiang, T.: Complex dynamics in Duffing–Van der Pol equation. Chaos Solitons Fractals 27, 722–747 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, CC. Adaptive nonsingular terminal sliding mode control for synchronization of identical Φ 6 oscillators. Nonlinear Dyn 69, 21–33 (2012). https://doi.org/10.1007/s11071-011-0243-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0243-8