ON THE MATHEMATICAL CLARIFICATION OF THE SNAP-BACK-REPELLER IN HIGH-DIMENSIONAL SYSTEMS AND CHAOS IN A DISCRETE NEURAL NETWORK MODEL

2002 ◽  
Vol 12 (05) ◽  
pp. 1129-1139 ◽  
Author(s):  
WEI LIN ◽  
JIONG RUAN ◽  
WEIRUI ZHAO
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Fixed Point ◽  
Numerical Simulations ◽  
Neural Network Model ◽  
Jacobian Matrix ◽  
Nonlinear Dynamical System ◽  
High Dimensional ◽  
Nonlinear Dynamical

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.

scholarly journals Krasnoselskii N-Tupled Fixed Point Theorem with Applications to Fractional Nonlinear Dynamical System

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Tamer Nabil
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Evolution Equations ◽  
Nonlinear Dynamical System ◽  
Nonlinear Dynamical ◽  
Fractional Dynamical System ◽  
Theoretical Results

In this paper, N-tupled fixed point theorems for two monotone nondecreasing mappings in complete normed linear space are established. The extension of Krasnoseskii fixed point theorem for a version of N-tupled fixed point is given. Our theoretical results are applied to prove the existence of a mild solution of the system of N-nonlinear fractional evolution equations. Finally, an example of a nonlinear fractional dynamical system is given to illustrate the results.

BIFURCATION IN A NONLINEAR DYNAMICAL SYSTEM ARISING FROM SEEKING STEADY STATES OF A NEURAL NETWORK

2010 ◽  
Vol 20 (08) ◽  
pp. 2585-2588 ◽  
Author(s):  
GEN-QIANG WANG ◽  
SUI SUN CHENG
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Network Model ◽  
Nonlinear Dynamical System ◽  
Steady States ◽  
Real Parameter ◽  
Dynamic Neural Network ◽  
Nonlinear Dynamical ◽  
The Real ◽  
Bifurcation Phenomenon

We show that in an artificial dynamic neural network that depends on a real parameter μ, steady states do not exist for μ ≤ -2, and positive and negative steady states exist for μ > -2. We hope that such a bifurcation phenomenon in our network model may explain some of the real observations in nature.

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