An averaging model for chaotic system with periodic time-varying parameter

In this paper, we provide a mathematical justification to explain the dynamics of chaotic system with periodic time-varying parameter which have been illustrated by some of us in a previous paper [1]. Based on an equivalent averaging model, it is proved that such a parametric time-varying system follows the same trajectory of its averaging model, provided that the parameter is varied periodically with a sufficiently high frequency. Some other observations related with this class of chaotic systems are also remarked in this paper.

[1]  Guanrong Chen,et al.  A switching scheme for synthesizing attractors of dissipative chaotic systems , 2008, Appl. Math. Comput..

[2]  Marco Squassina,et al.  Deformation from symmetry for nonhomogeneous Schrodinger equations of higher order on unbounded domains , 2003 .

[3]  M. Porfiri,et al.  Fast-switching pulse synchronization of chaotic oscillators , 2008, 2008 American Control Conference.

[4]  John Tsinias,et al.  Local Asymptotic Stability via Averaging for Time-Varying Systems With Unbounded Dynamics With Respect to Time , 2009, IEEE Transactions on Automatic Control.

[5]  Jun Zhao,et al.  Synchronization of Complex Dynamical Networks with Switching Topology: a Switched System Point of View , 2008 .

[6]  L. Chua,et al.  The double scroll family , 1986 .

[7]  Martin Hasler,et al.  Classes of stochastically switched (blinking) systems , 2007, 2007 IEEE International Symposium on Circuits and Systems.

[8]  Jeffrey J. DaCunha,et al.  Transition matrix and generalized matrix exponential via the Peano-Baker series , 2005 .

[9]  Zhendong Sun,et al.  A Note on Marginal Stability of Switched Systems , 2007, IEEE Transactions on Automatic Control.

[10]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[11]  On the averaging method for differential equations with delay. , 2002 .

[12]  BRAD LEHMAN,et al.  Averaging Theory for Delay Difference Equations with Time-Varying Delays , 1999, SIAM J. Appl. Math..

[13]  Dirk Aeyels,et al.  Exponential Stability of Nonlinear Time-Varying Differential Equations and Partial Averaging , 2002, Math. Control. Signals Syst..

[14]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[15]  Marius-F. Danca,et al.  Random parameter-switching synthesis of a class of hyperbolic attractors. , 2008, Chaos.

[16]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[17]  Marius-F. Danca.  Finding stable attractors of a class of dissipative dynamical systems by numerical parameter switching , 2010 .