Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses

Abstract

By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness, and global exponential stability of periodic solution for shunting inhibitory cellular neural networks with impulses, dx(ij)dt=-a(ij)x(ij)- summation operator(C(kl)inN(r)(i,j))C(ij) (kl)f(ij)[x(kl)(t)]x(ij)+L(ij)(t), t>0,t not equal t(k); Deltax(ij)(t(k))=x(ij)(t(k) (+))-x(ij)(t(k) (-))=I(k)[x(ij)(t(k))], k=1,2,...] . Furthermore, the numerical simulation shows that our system can occur in many forms of complexities, including periodic oscillation and chaotic strange attractor. To the best of our knowledge, these results have been obtained for the first time. Some researchers have introduced impulses into their models, but analogous results have never been found.

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.…”

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

“…Recently, a new type of system has been introduced: dynamic systems with impulses, which are neither purely continuous time nor purely discrete time and exhibit a combination of continuous time and discrete time characteristics. Furthermore, the concept of chaos and strange attractors is very attractive for ecologists now (see [6,7]). For example, Gui [6,7] found that some systems are capable of exhibiting strange attractors.…”

“…Furthermore, the concept of chaos and strange attractors is very attractive for ecologists now (see [6,7]). For example, Gui [6,7] found that some systems are capable of exhibiting strange attractors. In the present paper, we not only proved the existence and global exponential stability of periodic solutions for the model, but also found a new strange attractor which is caused by impulsive perturbations.…”