On the Mathematical Clarification of the SNAP-Back-Repeller in High-Dimensional Systems and Chaos in a Discrete Neural Network Model

We investigate the differences among several definitions of the snap-back-repeller, which is always regarded as an inducement to produce chaos in nonlinear dynamical system. By analyzing the norms in different senses and the illustrative examples, we clarify why a snap-back-repeller in the neighborhood of the fixed point, where all eigenvalues of the corresponding variable Jacobian Matrix are absolutely larger than 1 in norm, might not imply chaos. Furthermore, we theoretically prove the existence of chaos in a discrete neural networks model in the sense of Marotto with some parameters of the systems entering some regions. And the following numerical simulations and corresponding calculation, as concrete examples, reinforce our theoretical proof.

[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]  Peter E. Kloeden,et al.  Chaotic difference equations in Rn , 1981 .

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

[5]  J. Rinzel,et al.  INTEGRATE-AND-FIRE MODELS OF NERVE MEMBRANE RESPONSE TO OSCILLATORY INPUT. , 1981 .

[6]  Toshimitsu Ushio,et al.  Chaos in non-linear sampled-data control systems , 1983 .

[7]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[8]  James A. Yorke,et al.  No division implies chaos , 1982 .

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

[10]  James A. Yorke,et al.  Ergodic transformations from an interval into itself , 1978 .

[11]  J J Hopfield,et al.  Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire neurons. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

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

[13]  Guanrong Chen,et al.  Feedback anticontrol of discrete chaos , 1998 .

[14]  James A. Yorke,et al.  When the long-time behavior is independent of the initial density , 1996 .

[15]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[16]  Guanrong Chen,et al.  Chaotification via arbitrarily Small Feedback Controls: Theory, Method, and Applications , 2000, Int. J. Bifurc. Chaos.

[17]  Abraham Boyarsky,et al.  Snap-back repellers and scrambled sets in general topological spaces , 2001 .

[18]  Phil Diamond.  Chaotic behaviour of systems of difference equations , 1976 .

[19]  Masahiro Kurata,et al.  A generalization of a theorem of marotto , 1981, Nagoya Mathematical Journal.

[20]  Toshimitsu Ushio,et al.  Chaotic behavior in piecewise-linear sampled-data control systems , 1985 .

[21]  Akitaka Dohtani,et al.  Occurrence of chaos in higher-dimensional discrete-time systems , 1992 .

[22]  Kazuyuki Aihara,et al.  Strange attractors in chaotic neural networks , 2000 .