Transversal homoclinic orbits in a transiently chaotic neural network.

We study the existence of snap-back repellers, hence the existence of transversal homoclinic orbits in a discrete-time neural network. Chaotic behaviors for the network system in the sense of Li and Yorke or Marotto can then be concluded. The result is established by analyzing the structures of the system and allocating suitable parameters in constructing the fixed points and their pre-images for the system. The investigation provides a theoretical confirmation on the scenario of transient chaos for the system. All the parameter conditions for the theory can be examined numerically. The numerical ranges for the parameters which yield chaotic dynamics and convergent dynamics provide significant information in the annealing process in solving combinatorial optimization problems using this transiently chaotic neural network. (c) 2002 American Institute of Physics.

[1]  Sze-Bi Hsu,et al.  Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span , 1998 .

[2]  Kazuyuki Aihara,et al.  Chaotic simulated annealing by a neural network model with transient chaos , 1995, Neural Networks.

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

[4]  Carsten Peterson,et al.  A Mean Field Theory Learning Algorithm for Neural Networks , 1987, Complex Syst..

[5]  F. R. Marotto,et al.  Perturbations of stable and chaotic difference equations , 1979 .

[6]  Kazuyuki Aihara,et al.  Global searching ability of chaotic neural networks , 1999 .

[7]  K. Aihara,et al.  Chaos and asymptotical stability in discrete-time neural networks , 1997, chao-dyn/9701020.

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

[9]  D. Signorini,et al.  Neural networks , 1995, The Lancet.

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

[11]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[12]  K. Meyer,et al.  Stability of Skew Dynamical Systems , 1996 .

[13]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

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