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 he extended to synchronization in arbitrary coupled arrays of nonlinear systems. In particular, we use several well-known stability conditions to obtain synchronization criteria in arbitrarily coupled arrays: the passivity criterion, the circle criterion and a result on observer design of Lipschitz nonlinear systems. We also study how these synchronization criteria depend on the topology of the coupled networks. In particular, we show that synchronization is improved by using nonlocal connections or introducing random connections.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[26]  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).

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

[28]  Chai Wah Wu,et al.  Global synchronization in coupled map lattices , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[29]  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.

[30]  Chai Wah Wu.  Synchronization in Arrays of Chaotic Circuits Coupled via Dynamic Coupling Elements , 2000, Int. J. Bifurc. Chaos.

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