Abstract
The study focuses on the chaotic behavior of a three-dimensional Hopfield neural network with time delay. We find the aspecific coefficient matrix and the initial value condition of the system and use MATLAB software to draw its graph. The result shows that their shape is very similar to the figure of Roslerʼs chaotic system. Furthermore, we analyzed the divergence, the eigenvalue of the Jacobian matrix for the equilibrium point, and the Lyapunov exponent of the system. These properties prove that the system does have chaotic behavior. This result not only confirms that there is chaos in the neural networks but also that the chaotic characteristics of the system are very similar to those of Roslerʼs chaotic system under certain conditions. This discovery provides useful information that can be applied to other aspects of chaotic Hopfield neural networks, such as chaotic synchronization and control.
Highlights
Chaotic systems are nonlinear dynamical systems working in a stochastic process that is neither periodic nor convergent but are highly dependent on the initial value
In 1880, Poincare first studied the possibility of chaos [1]. He studied the three-body problem and became the first person to discover the deterministic system of chaos which showed an acyclic behavior dependent on initial conditions. is made long-term prediction impossible, thereby laying the foundation for modern chaos theory
Based on previous ideas and work of Sampath et al, in this paper, we focus on the chaotic behavior of a threedimensional Hopfield neural network with time delay
Summary
Chaotic systems are nonlinear dynamical systems working in a stochastic process that is neither periodic nor convergent but are highly dependent on the initial value. E revolutionary work on chaos was discovered from a three-dimensional chaotic system by an American meteorologist, Edward Lorenz, in 1963 while studying weather models. E Rossler system is the most famous chaotic system having simple asymmetric attractor substructure extracted from the Lorenz attractor by the German physical chemist, Rossler It plays a vital role in signal processing [15], secure communication [16], and other issues. Yang et al analyzed the Journal of Mathematics transient chaos in a chaos bifurcation problem of a class of simple chaotic Hopfield neural networks. In a study by Das et al, rich dynamic characteristics were revealed based on analysis of artificial neural networks composed of three neurons and they drew bifurcation and three-dimensional phase diagrams of the model [19]. To understand the graphical characteristics of this model, we used MATLAB to draw its phase diagram. e result shows that their shape is very similar to the figure of Roslers chaotic system. en, we analyzed the divergence, the eigenvalue of the Jacobian matrix for the equilibrium point, and the Lyapunov exponent of the system. ese properties prove that the system does have chaotic behavior. is research is a refreshing discovery
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