Full- and reduced-order synchronization of a class of time-varying systems containing uncertainties

Abstract Investigation on chaos synchronization of autonomous dynamical systems has been largely reported in the literature. However, synchronization of time-varying, or nonautonomous, uncertain dynamical systems has received less attention. The present contribution addresses full- and reduced-order synchronization of a class of nonlinear time-varying chaotic systems containing uncertain parameters. A unified framework is established for both the full-order synchronization between two completely identical time-varying uncertain systems and the reduced-order synchronization between two strictly different time-varying uncertain systems. The synchronization is successfully achieved by adjusting the determined algorithms for the estimates of unknown parameters and the linear feedback gain, which is rigorously proved by means of the Lyapunov stability theorem for nonautonomous differential equations together with Barbalat’s lemma. Moreover, the synchronization result is robust against the disturbance of noise. We illustrate the applicability for full-order synchronization using two identical parametrically driven pendulum oscillators and for reduced-order synchronization using the parametrically driven second-order pendulum oscillator and an additionally driven third-order Rossler oscillator.

[1]  O. Rössler.  An equation for continuous chaos , 1976 .

[2]  A. Winfree.  The geometry of biological time , 1991 .

[3]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[4]  Otsuka.  Self-induced phase turbulence and chaotic itinerancy in coupled laser systems. , 1990, Physical review letters.

[5]  P. Glendinning.  Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations , 1994 .

[6]  L. Tsimring,et al.  Generalized synchronization of chaos in directionally coupled chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Leon O. Chua,et al.  ON ADAPTIVE SYNCHRONIZATION AND CONTROL OF NONLINEAR DYNAMICAL SYSTEMS , 1996 .

[8]  S. Bishop,et al.  Zones of chaotic behaviour in the parametrically excited pendulum , 1996 .

[9]  J. Kurths,et al.  From Phase to Lag Synchronization in Coupled Chaotic Oscillators , 1997 .

[10]  G. Tao.  A simple alternative to the Barbalat lemma , 1997, IEEE Trans. Autom. Control..

[11]  J. Kurths,et al.  Heartbeat synchronized with ventilation , 1998, Nature.

[12]  N. Kopell,et al.  Dynamics of two mutually coupled slow inhibitory neurons , 1998 .

[13]  G. Du,et al.  CHAOS SYNCHRONIZATION OF TWO PARAMETRICALLY EXCITED PENDULUMS , 1999 .

[14]  Jean Bragard,et al.  Synchronization in Nonidentical Extended Systems , 1999 .

[15]  Voss,et al.  Anticipating chaotic synchronization , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Kurths,et al.  Synchronization of chaotic structurally nonequivalent systems , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[18]  Constructing nonautonomous differential equations from experimental time series. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[20]  R. Femat,et al.  Synchronization of chaotic systems with different order. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  H Henk Nijmeijer,et al.  Synchronization of Mechanical Systems , 2003 .

[22]  S. Bowong.  Stability analysis for the synchronization of chaotic systems with different order: application to secure communications , 2004 .

[23]  Grigory V. Osipov,et al.  Synchronization Analysis of Coupled Noncoherent Oscillators , 2006 .

[24]  A. Tamaševičius,et al.  Coupled Chaotic Colpitts Oscillators: Identical and Mismatched Cases , 2006 .

[25]  T. Chai,et al.  Adaptive synchronization between two different chaotic systems with unknown parameters , 2006 .