International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Global λ, µ stability of delayed recurrent neural
network
Neeraj Sahu1, Poonam Sinha2, A. K. Verma3
1
Research scholar “Department of Mathematics S.M.S Government Model
Science College Gwalior(M.P)”, India
2
Department of Mathematics S.M.S Government Model
Science College, Gwalior (M.P),India
3
Department of Mathematical, Sciences and Computer Applications.
Bundelkhand University Jhansi (U.P.),India
Abstract
Here we investigate dynamical system with time varying delays for unique equilibrium point. A new thought of stability, global
stability is planned. For some situation we prove that the dynamical systems with unbounded time varying delayed are
globally
stable.
1. Introduction
Neural network has improved technology for solving formerly unsolvable problem and also improve system performance.
This time delay occurs in signal transmission among the neurons which affect the stability of neural network. Therefore
the performance of recurrent neural network using analogy circuit may face problem due to complex dynamic action such
as periodic oscillation, bifurcation or chaos. In this way can say that the time delays affect the performance of recurrent
neural network.
The various others [1],[2],[4]-[6],[8] has studies the stability analysis of delay recurrent neural network, various sufficient
conditions have been proved for global stability of with and without time delay. Hu, Wang [3] derived several sufficient
conditions to ascertain the existence of unique equilibrium, global asymptotic stability, and global exponential stability of
delayed complex-valued recurrent neural networks with two classes of complex-valued activation function. Huaguang and
his coauthors [9] develop an innovative method for stability problem for a class of neural networks with time-varying
delay. Here we define a new model for recurrent neural network with time delays. This model will investigate dynamical
system theory. The model defines some assumption for unique and globally
stability.
Model description and preliminaries
If periodic input vector function
and
with period
then we drive the following recurrent neural network with time delays describe by delayed differential equation
,
ci > 0
i=1,2,3.....n
dj > 0
j=1,2,3.....m
(1)
Where m, n indicate the number of neurons xi(t), yj(t) indicate the state of ith , jth neurons at time t: let A=[aij], P=[pij] be
the feedback matrix B=[bij],Q=[qij] be delayed feedback matrix and c= dig(c1,c2,c3 ....cn) and d=dig(d1,d2 ,d3.....dm).
If any constant input vector
We assume that
the activation function fj (j=1,2,3....m) are globally Lipschitz continuous that is there exits a constant µi > 0, ηi > 0, such
that
(j=1,2,3.....m) for any
and
(i=1,2,3.....n) for
any
. We know that the neural network model includes the well known Hopfield neural networks, cellular neural
networks and bidirectional associative memory networks as its special cases.
The initial condition of the recurrent neural network (1) is assume to be
;
1 ≤ i, j ≤ m
;
1 ≤ i, j ≤ m
Where i=1,2,3.....n, j=1,2,3....m, in which
are continuous function.
Let c = c
, d=d
, be the Banach space of continuous function which maps
into Rn with the topology of uniform convergence. For any
we define
Volume 2, Issue 9, September 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Where r1 > 1, r2 > 1 is constant.
Define
Then
Let
be a unique equilibrium of model (1) then we have
Where r1 > 1, r2 > 1 is constant.
To be more general we investigate the following dynamical system.
Where
initial value is
is the activation function
(2)
represents the time-varying delays i=1,2,3.....n, j=1,2,3....m, the
(3)
This general model include many existing neural networks as special case. For example
Let
(4)
Then system (2) reduces to usual time varying delayed Hopfield neural network (1) in the outcome we will address the
stability topic of delayed dynamical (2) when
but unbounded.
Definition:Assumption:-
-norm
= matrix1,2….n
, where
i=1,2….n
fj(x1,x2,……….xn,y1 ,y2……….ym) and gi(y1,y2 ……….ym x1,x2,……….xn)is absolutely continuous
exist almost everywhere. Suppose that there exist constants F1ij, F2ij and g1ij, g2ij such that
Hold for all i=1,2,3…….n,j=1,2,3……m, k=1,2,3……n,1,2……m
In fj(x1,x2,……….xn,y1,y2……….ym) and gi(y1 ,y2……….ym x1,x2 ,……….xn) is Lipschitz continous then fj, gi satisfied
Assumption.
Definition:- suppose that
when
if there exist a scalar M > 0, N > 0 such that
Volume 2, Issue 9, September 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Then the system (2) is said to be
stable.
Theorem :- suppose that assumption 1 is satisfied
; the function µ(t) is non decreasing and satisfies.
(5)
Where
is a nonnegative scalar moreover, if there are positive constant
. Such that
(6)
Hold for i=1,2,.......n, j=1,2,.......m then the dynamical system (2)has an equilibrium point v* by[7] which is unique and
globally λ, µ stable.
Proof:-By leema.1, [7] we know that dynamical system (2) has an equilibrium point v*. Now will prove its global λ, µ
stability and uniqueness. By (5) and (6) one can find a sufficiently large T > 0 such that for all t > T. It holds
(7)
for i=1,2,.......n, j=1,2,.......m, without loss of generality in the following derivation we always assume that t > T
Define
and
We claim that M(t), N(t) is bounded which implies that
T there are two following cases
Case – I
Such that
. In this case there exists a
and
Case-II
differentiation
is bounded too. In fact or any t0 >
-neighbourhood of to
.
.In this case let let it0 = it0 (t0) be such an index that
we get
Volume 2, Issue 9, September 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Then there exist
such that
In summary we conclude that
for
for all t.>T. Which implies
.
(8)
There fore
is the unique equilibrium and isl λ, µ - globally stable.
Conclusion
Recurrent neural network with time delays define many new sufficient condition for ascertaining global exponential
stability and periodicity. These new criteria are easily verifiable and control many adjustable parameters, which yield
applicable flexibility for the design and analysis of globally λ, µ stable delayed recurrent neural network. Here we describe
a new model for recurrent neural network with delays. This model has a unique equilibrium point. The stability topic of
delayed dynamical is unbounded and the system of model has an equilibrium point v*. Which is unique and globally λ, µ
stable.
References
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Transactions on Volume:24 , Issue: 4 pp. 513 - 521April 2013.
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