On Global Dynamic Behavior of Weakly Connected Oscillatory Networks

The global dynamics of weakly connected oscillatory networks is investigated: as a case study, one-dimensional arrays of third-order oscillators are considered. Through the joint application of the describing function technique and Malkin's Theorem a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. The total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. The proposed technique significantly extends the results available in the literature and can be applied to almost all complex networks of oscillators. In particular two-dimensional, space variant and fully connected networks can be dealt with here.

[1]  Marco Gilli,et al.  A Spectral Approach to the Study of Propagation Phenomena in CNNs , 1996, Int. J. Circuit Theory Appl..

[2]  A. Michel Dynamics of feedback systems , 1984, Proceedings of the IEEE.

[3]  Fernando Corinto,et al.  Periodic oscillations and bifurcations in cellular nonlinear networks , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[4]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[5]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[6]  P. Thiran,et al.  An approach to information propagation in 1-D cellular neural networks. II. Global propagation , 1998 .

[7]  E. Izhikevich,et al.  Oscillatory Neurocomputers with Dynamic Connectivity , 1999 .

[8]  Leon O. Chua,et al.  The CNN paradigm , 1993 .

[9]  Leon O. Chua,et al.  ON PERIODIC ORBITS AND HOMOCLINIC BIFURCATIONS IN CHUA’S CIRCUIT WITH A SMOOTH NONLINEARITY , 1993 .

[10]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[11]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[12]  Leon O. Chua,et al.  Turing patterns in CNNs. II. Equations and behaviors , 1995 .

[13]  Alberto Tesi,et al.  Distortion control of Chaotic Systems: the Chua's Circuit , 1993, Chua's Circuit.

[14]  Marco Gilli,et al.  Analysis of periodic oscillations in finite-dimensional CNNs through a spatio-temporal harmonic balance technique , 1997, Int. J. Circuit Theory Appl..

[15]  Grigory V. Osipov,et al.  Chaos and structures in a chain of mutually-coupled Chua's circuits , 1995 .

[16]  Lin-Bao Yang,et al.  Cellular neural networks: theory , 1988 .

[17]  Tamás Roska,et al.  The CNN universal machine: an analogic array computer , 1993 .

[18]  Patrick Thiran,et al.  An approach to information propagation in cellular neural networks - Part I : Local diffusion , 1998 .

[19]  Rabinder N Madan,et al.  Chua's Circuit: A Paradigm for Chaos , 1993, Chua's Circuit.

[20]  Leon O. Chua,et al.  Cellular Neural Networks and Visual Computing , 2002 .

[21]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[22]  Leon O. Chua,et al.  Methods for image processing and pattern formation in Cellular Neural Networks: a tutorial , 1995 .

[23]  Marco Gilli,et al.  Investigation of chaos in large arrays of Chua's circuits via a spectral technique , 1995 .

[24]  Leon O. Chua,et al.  Autonomous cellular neural networks: a unified paradigm for pattern formation and active wave propagation , 1995 .