Dynamics for Discrete-Time Cellular Neural Networks

This presentation investigates the dynamics of discrete-time cellular neural networks (DT-CNN). In contrast to classical neural networks that are mostly gradient-like systems, DT-CNN possesses both complete stability and chaotic behaviors as different parameters are considered. An energy-like function which decreases along orbits of DT-CNN as well as the existence of a globally attracting set are derived. Complete stability can then be concluded, with further analysis on the sets on which the energy function is constant. The formations of saturated stationary patterns for DT-CNN are shown to be analogous to the ones in continuous-time CNN. Thus, DT-CNN shares similar properties with continuous-time CNN. By confirming the existence of snap-back repellers, hence transversal homoclinic orbits, we also conclude that DT-CNN with certain parameters exhibits chaotic dynamics, according to the theorem by Marotto.

[1]  Cheng-Hsiung Hsu,et al.  Existence and multiplicity of traveling waves in a lattice dynamical system , 2000 .

[2]  Chih-Wen Shih,et al.  Cycle-symmetric matrices and convergent neural networks , 2000 .

[3]  Chih-Wen Shih,et al.  Pattern Formation and Spatial Chaos for Cellular Neural Networks with Asymmetric Templates , 1998 .

[4]  F. R. Marotto Snap-back repellers imply chaos in Rn , 1978 .

[5]  Josef A. Nossek,et al.  An analog implementation of discrete-time cellular neural networks , 1992, IEEE Trans. Neural Networks.

[6]  Chih-Wen Shih,et al.  Complete stability for a Class of Cellular Neural Networks , 2001, Int. J. Bifurc. Chaos.

[7]  Shyan-Shiou Chen,et al.  Transversal homoclinic orbits in a transiently chaotic neural network. , 2002, Chaos.

[8]  J. P. Lasalle The stability of dynamical systems , 1976 .

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

[10]  Leon O. Chua,et al.  Local Activity Criteria for Discrete-Map CNN , 2002, Int. J. Bifurc. Chaos.

[11]  Chih-Wen Shih,et al.  Complete Stability for Standard Cellular Neural Networks , 1999 .

[12]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[13]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[14]  S. A. Robertson,et al.  NONLINEAR OSCILLATIONS, DYNAMICAL SYSTEMS, AND BIFURCATIONS OF VECTOR FIELDS (Applied Mathematical Sciences, 42) , 1984 .

[15]  Jack K. Hale,et al.  Convergence in gradient-like systems with applications to PDE , 1992 .

[16]  Chih-Wen Shih Influence of Boundary Conditions on Pattern Formation and Spatial Chaos in Lattice Systems , 2000, SIAM J. Appl. Math..

[17]  Chih-Wen Shih,et al.  On the Templates Corresponding to Cycle-Symmetric Connectivity in Cellular Neural Networks , 2002, Int. J. Bifurc. Chaos.

[18]  Jonq Juang,et al.  Cellular Neural Networks: Mosaic Pattern and Spatial Chaos , 2000, SIAM J. Appl. Math..

[19]  George S. Moschytz,et al.  Unifying results in CNN theory using delta operator , 1999, ISCAS'99. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSI (Cat. No.99CH36349).

[20]  B. Fiedler,et al.  A class of convergent neural network dynamics , 1998 .

[21]  Ioannis G. Kevrekidis,et al.  International Journal of Bifurcation and Chaos in Applied Sciences and Engineering: Editorial , 2005 .

[22]  F. R. Marotto,et al.  Chaotic behavior in the Hénon mapping , 1979 .