Synchronization in arrays of coupled nonlinear systems: passivity circle criterion and observer design

It has been shown that synchronization between two nonlinear systems can be studied as a control theory problem. We show that this relationship can be extended to synchronization in coupled arrays of nonlinear systems. In particular, we extend several stability conditions to synchronization criteria in arbitrarily coupled arrays: the passivity criterion, the circle criterion and a result on observer design of Lipschitz nonlinear systems.

[1]  L. Chua,et al.  A qualitative analysis of the behavior of dynamic nonlinear networks: Stability of autonomous networks , 1976 .

[2]  吉沢 太郎 Stability theory by Liapunov's second method , 1966 .

[3]  Chai Wah Wu Synchronization in arrays of chaotic circuits coupled via hypergraphs: static and dynamic coupling , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[4]  L. Chua,et al.  Application of graph theory to the synchronization in an array of coupled nonlinear oscillators , 1995 .

[5]  Gade,et al.  Synchronization of oscillators with random nonlocal connectivity. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Bojan Mohar,et al.  Isoperimetric numbers of graphs , 1989, J. Comb. Theory, Ser. B.

[7]  L. Chua,et al.  Application of Kronecker products to the analysis of systems with uniform linear coupling , 1995 .

[8]  G. Hu,et al.  Instability and controllability of linearly coupled oscillators: Eigenvalue analysis , 1998 .

[9]  He,et al.  Analysis and synthesis of synchronous periodic and chaotic systems. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[10]  L. Chua,et al.  A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS , 1994 .

[11]  Endre Szemerédi,et al.  On the second eigenvalue of random regular graphs , 1989, STOC '89.

[12]  S. Mascolo,et al.  Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal , 1997 .

[13]  Henk Nijmeijer,et al.  An observer looks at synchronization , 1997 .

[14]  Johnson,et al.  Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  L. Chua,et al.  A qualitative analysis of the behavior of dynamic nonlinear networks: Steady-state solutions of nonautonomous networks , 1976 .

[16]  R. Rajamani Observers for Lipschitz nonlinear systems , 1998, IEEE Trans. Autom. Control..

[17]  Béla Bollobás,et al.  The Isoperimetric Number of Random Regular Graphs , 1988, Eur. J. Comb..

[18]  Cerdeira,et al.  Coupled maps on trees. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Leon O. Chua,et al.  On Chaotic Synchronization in a Linear Array of Chua's Circuits , 1993, J. Circuits Syst. Comput..

[20]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[21]  Bojan Mohar,et al.  Eigenvalues, diameter, and mean distance in graphs , 1991, Graphs Comb..

[22]  Carroll,et al.  Synchronous chaos in coupled oscillator systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Xiao Fan Wang,et al.  Synchronization in Small-World Dynamical Networks , 2002, Int. J. Bifurc. Chaos.

[24]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

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

[26]  M. Fiedler Algebraic connectivity of graphs , 1973 .

[27]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[28]  Carroll,et al.  Driving systems with chaotic signals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[29]  J. Suykens,et al.  Absolute stability theory and master-slave synchronization , 1997 .