Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays

Abstract

a b s t r a c tBy using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, the global exponential stability and periodicity are investigated for a class of delayed high-order Hopfield neural networks (HHNNs) with impulses, which are new and complement previously known results. Finally, an example with numerical simulation is given to show the effectiveness of the proposed method and results. The numerical simulation shows that our models can occur in many forms of complexities including periodic oscillation and the Gui chaotic strange attractor.

Notice of Retraction: New Strange Attractor for a Delayed Nonautonomous Predator-prey Model Controlled by Impulse

“…Gui [4] has researched periodic solution for nonautonomous bidirectional associative memory neural networks with impulses. During his research, He found that the model is capable of exhibiting gui strange attractor(see [11,12]. In this paper, we find new strange attractor again.…”

“…Periodic oscillation of system (1) will be destroyed by impulses effect (seeFigs.4-7). Numeric results show that system (1) has a gui chaotic strange attractor in[11],[12](seeFig.6-7). Every solution of system (1) will finally tend to the new chaotic strange attractor.…”

Exponential Synchronization of the Hopfield Neural Networks with New Chaotic Strange Attractor

“…The Hopfield neural networks with impulse effect are studied, where the criteria on the existence, uniqueness and global stability of periodic solution are obtained. Further, Gui chaos strange attractor was also found [6][7][8][9][10].…”

“…As is known to all that (1) can exhibit chaotic phenomenons [6][7][8][9][10] . In order to show it clearly, we give the following example.…”

“…Taking (2) as the drive system in Example 1. The response system is constructed as follows: Then the error system of drive system (2) and respond system (9) is constructed as follows: Numeric results show that system (9) still has a Gui chaotic strange attractor [6][7][8][9][10] . The phase plot of Gui chaotic strange attractor of response system is shown in Fig.…”