Homoclinical Structure of Retarded SICNNs with Rectangular Input Currents

Abstract

The dynamics of retarded shunting inhibitory cellular neural networks (SICNNs) with rectangular input currents is investigated from the asymptotic point of view. It is rigorously proved that such networks possess homoclinic and heteroclinic outputs under certain conditions. Illustrative examples that support the theoretical results are provided. Moreover, the extension of the homoclinical structure is numerically demonstrated for unidirectionally coupled retarded SICNNs.

Appendix

The proof of Lemma 2.1 is as follows.

Proof of Lemma 2.1

Let us denote by $\mathcal {B}_0$ the set of continuous and uniformly bounded functions $v(t)=\left\{ v_{ij}(t)\right\}$, $i=1,2,\ldots ,m$, $j=1,2,,\ldots ,n$, which are defined on ${\mathbb {R}}$, such that $\left\| v\right\| _{\infty } \le K_0$, where $\left\| v\right\| _{\infty }= \sup _{t\in {\mathbb {R}}}\left\| v(t)\right\|$.

We fix a solution $\zeta =\left\{ \zeta _k\right\} _{k\in {\mathbb {Z}}}$ of the map (2.2) and define an operator $\Phi$ on the set $\mathcal {B}_0$ by

$$\begin{aligned} \left( \Phi v(t)\right) _{ij}= - \displaystyle \int _{-\infty }^t e^{-a_{ij}(t-s)} \bigg [ \sum _{C_{hl}\in N_r(i,j)} C_{ij}^{hl} f(v_{hl}(s-\tau )) v_{ij}(s) - L_{ij}(s) -P_{ij}(s,\zeta ) \bigg ]ds, \end{aligned}$$

where $\Phi v(t)=\left\{ \left( \Phi v(t)\right) _{ij}\right\}$.

If $v(t) \in \mathcal {B}_0$, then

$$\begin{aligned} \left| \left( \Phi v(t)\right) _{ij}\right| \le \displaystyle \frac{1}{a_{ij}} \left( M_f K_0 \sum _{C_{hl}\in N_r(i,j)} C_{ij}^{hl} + M_{ij} + M_F\right) . \end{aligned}$$

The last inequality yields

$$\begin{aligned} \left\| \Phi v \right\| \le \displaystyle M_f K_0 \delta + \max _{(i,j)} \frac{M_{ij}+M_F}{a_{ij}}= K_0. \end{aligned}$$

Therefore, we have $\Phi (\mathcal {B}_0)\subseteq \mathcal {B}_0$.

On the other hand, if v(t) and $\overline{v}(t)$ belong to $\mathcal {B}_0$, then one can confirm that

$$\begin{aligned} \left| \left( \Phi v(t)\right) _{ij}-\left( \Phi \overline{v}(t)\right) _{ij}\right|\le & {} \displaystyle \int _{-\infty }^t e^{-a_{ij}(t-s)} \sum _{C_{hl}\in N_r(i,j)} C_{ij}^{hl} M_f \left| v_{ij}(s) - \overline{v}_{ij}(s)\right| ds \\&+ \displaystyle \int _{-\infty }^t e^{-a_{ij}(t-s)} \sum _{C_{hl}\in N_r(i,j)} C_{ij}^{hl} K_0 L_f \left| v_{hl}(s-\tau ) - \overline{v}_{hl}(s-\tau )\right| ds \\&\le (M_f+K_0L_f) \frac{\sum _{C_{hl}\in N_r(i,j)} C_{ij}^{hl}}{a_{ij}} \left\| v-\overline{v}\right\| _{\infty }. \end{aligned}$$

Hence, the inequality $\left\| \Phi v - \Phi \overline{v}\right\| _{\infty } \le (M_f + K_0 L_f) \delta \left\| v-\overline{v}\right\| _{\infty }$ is valid, and accordingly the operator $\Phi$ is a contraction since $(M_f + K_0 L_f)\delta <1$.

Consequently, for each orbit $\zeta =\left\{ \zeta _k\right\} _{k\in {\mathbb {Z}}}$ of (2.2), SICNN (2.1) possesses a unique bounded solution $\phi _{\zeta }(t) = \left\{ \phi _{\zeta }^{ij}(t)\right\}$ such that $\sup _{t\in {\mathbb {R}}} \left\| \phi _{\zeta }(t)\right\| \le K_0.$$\square$