Biomedical Engineering Reference
In-Depth Information
We can write now the mathematical model of
the Hopfield neural network, which is representa-
tive for RNN
“good behavior solution” of some natural physical
systems (Rǎsvan, 1998). For instance, such sys-
tems there exist in the fields of biology, economics,
recurrent neural networks, chemical reactions,
and electrical machines. The standard stability
properties (Lyapunov, asymptotic and exponential
stability) are defined for a single equilibrium. Their
counterparts for several equilibria are: mutability,
global asymptotics and gradient behavior.
The basic concepts of the theory of systems
with several equilibria are introduced by Kalman
(1957) and Moser (1967) and have been developed
by Gelig, Leonov & Yakubovich (1978, 2004),
Hirsch (1988) and Leonov, Reitmann & Smirnova
(1992). From the last reference we introduce here
the basic concepts of the field.
Consider the system of
n
ordinary differential
equations
m
∑
=
x
=
−
a
x
−
c
f
(
x
)
+
I
,
i
=
1
m
i
i
i
ij
j
j
i
j
1
(5)
and identify from (1) and (5) the nonlinear func-
tions
m
∑
h
( )
x
= −
a x
−
c f
(
x
)
+
I
.
i
i
i
ij
j
j
i
j
=
1
It is important to say here that the presence of
many nonlinear characteristics
j
1=
leads
to the existence of several equilibria, whereas the
Lipschitz sufficient conditions (3) and (4) ensure
the existence and the uniqueness of the solution of
system (5) i.e. for a given initial condition there is
only one way that the RNN can evolve in time (the
reader is sent to any topic which treats ordinary
differential equations or nonlinear systems; see
for instance: Hartman, 1964; Khalil, 1992).
B.
Dynamical systems with several equilibria
occur in many fields of science and engineering
where the stable equilibria are possible “operating
points” of the man-made systems or represent a
,
f
(⋅
)
m
j
x
=
h
(
t
,
x
)
(6)
n
n
with
h
:
×
→
continuous, at least in the
+
first argument.
Definition 1:
a.
Any constant solution of (6) is called equi-
librium. The set of equilibria
S
is called
stationary set.
Figure 1. Sigmoidal function
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