Existence of Periodic Solution for Fuzzy Cellular Neural Networks with Time-Varying Delays - Advanced Research on Computer Education, Simulation and Modeling - page 200

i ct denotes the rate
with which the i th neuron will reset its potential to the resting state in isolation when
disconnected from the network and external inputs;

i xt is the activations of the i th neuron at time t ,

() ()

For

∧ ∨

and denote the fuzzy

ij at denotes the strengths of connectivity between

()

AND and fuzzy OR operations.

cell i and cell j at time t .

i Ht are elements of fuzzy
feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN
template and fuzzy feed-forward MAX template between cell i and j at time t .
()

α

ij t

() β

ij t

()

ij Tt and

() ()

, ,

τ

ij t

corresponds to the time delay required in processing and transmitting a signal
from the j th cell to the i th cell at time t .

j ut and

()

i It denote the external

()

input, bias of the i th neurons at time t , respectively.

f () ⋅ is signal transmission

j

functions.
Throughout the paper, we give the following assumptions
(A1) |

fx pxq ()| ≤ | | +

for all xR

∈ j = 1, 2, , where , n

j p , j q are

,

j j j

nonnegative constants.

τ = max sup { τ

ij t

( )} ϕ

Let , For continuous functions defined on

1, ≤≤ ijn t ≥ 0 i

T

[, ] − i = 1, 2, ,we set , n Ψ= (, , , ϕϕ . Assume that system (1) is ϕ

,

12 n

supplemented with initial value of type

xt () = ϕ (), t −≤≤ t 0

.

i i

EK −

1

ρ ()1 K < Kk × = () ≥ 0 ( − ) ≥ 0

Lemma 1. [11] If for matrix , then ,

ij n n

where E denotes the identity matrix of size n .

Lemma 2. [2] Suppose x and y are two states of system (1), then we have

n

n n

∑

| ∧ α ( ) tfx ( ) − ∧ α ( ) tf y ( )| ≤ | α ( )|| t fx f y ( ) − ( )|

,

ij j ij j ij j j

j = 1 j = 1

j = 1

and

n

n n

∑

| ∨ β ( ) tfx ( ) − ∨ β ( ) tf y ( )| ≤ | β ( )|| t fx f y ( ) − ( )|

.

ij j ij j ij j j

j = 1 j = 1

j = 1

The remainder of this paper is organized as follows. In Section 2, we will give the
sufficient conditions to ensure the existence of periodic oscillatory solution for fuzzy
cellular neural networks with time-varying delays. In Section 3 we will give a general
conclusion.