Next, let us outline LaSalle invariant principle for non-autonomous difference equations. In
addition to the classical result by LaSalle there is a modified version for the theorem reported
in (Chen & Shih, 2004b). The alternative conditions derived therein is considered more
applicable and has been applied to study the convergence of the TCNN.
Let N be the set of positive integers. For a given continuous function F : N × R n −→ R n, we consider the non-autonomous difference equation
x( t + 1) = F( t, x( t)).
(10)
A sequence of points {x( t) }∞ in R n is a solution of (10) if x( t + 1) = F( t, x( t)), for all t ∈ N.
1
Let Ox = {x( t) | t ∈ N, x(1) = x }, be the orbit of x. We say that p is a ω-limit point of Ox if there exists a sequence of positive integers {tk} with tk → ∞ as k → ∞, such that
p = lim k→∞ x( tk). Denote by ω(x) the set of all ω-limit points of Ox.
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Discrete Time Systems
Let N i represent the set of all positive integers larger than ni, for some positive integer ni. Let
G be any set in R n and G be its closure. For a function V : N0 × G −→ R, define ˙
V( t, x) =
V( t + 1, F( t, x)) − V( t, x). If {x( t) } is a solution of (10), then ˙
V( t, x) = V( t + 1, x( t + 1)) −
V( t, x( t)). V is said to be a Lyapunov function for (10) if
(i) {V( t, ·) | t ∈ N0 } is equi-continuous, and
(ii) for each p ∈ G, there exists a neighborhood U of p such that V( t, x) is bounded
below for x ∈ U ∩ G and t ∈ N1, n 1 ≥ n 0, and
(iii) there exists a continuous function Q 0 : G → R such that ˙
V( t, x) ≤ −Q 0(x) ≤ 0
for all x ∈ G and for all t ∈ N2, n 2 ≥ n 1,
or
(iii) there exist a continuous function Q 0 : G → R and an equi-continuous family of
functions Q : N2 × G → R such that lim t→∞ |Q( t, x) − Q 0(x) | = 0 for all x ∈ G and
˙
V( t, x) ≤ −Q( t, x) ≤ 0 for all ( t, x) ∈ N2 × G, n 2 ≥ n 1.
Define
S 0 = {x ∈ G : Q 0(x) = 0 }.
Theorem (Chen & Shih, 2004a). Let V : N0 × G → R be a Lyapunov function for (10) and let
Ox be an orbit of (10) lying in G for all t ∈ N0. Then lim t→∞ Q( t, x( t)) = 0, and ω(x) ⊂ S 0.
This theorem with conditions (i), (ii), and (iii) has been given in (LaSalle, 1976). We quote
the proof for the second case reported in (Chen & Shih, 2004b). Let p ∈ ω(x). That is,
there exists a sequence {tk}∞
1 , tk → ∞ as k → ∞ and x( tk) → p as k → ∞.
Since
V( tk, x( tk)) is non-increasing and bounded below, V( tk, x( tk)) approaches a real number as k → ∞. Moreover, V( tk+1, x( tk+1)) − V( t 1, x( t 1)) ≤ − ∑ tk+1 − 1
t= t
Q( t, x( t)), by (iii) . Thus,
1
∑∞
t= t Q( t, x( t)) < ∞. Hence, Q( t, x( t)) → 0 as t → ∞, since Q( t, x( t)) ≥ 0. Notably, 1
Q( tk, x( tk)) → Q 0(x( tk)) as k → ∞. This can be justified by observing that
|Q( tk, x( tk)) − Q 0(x( tk)) |
≤ |Q( tk, x( tk)) + Q( tk, p) − Q( tk, p) + Q 0(p) − Q 0(p) − Q 0(x( tk)) |.
In addition, |Q 0(x( t)) | ≤ |Q( t, x( t)) | + |Q( t, x( t)) − Q 0(x( t)) |. It follows from (iii) that Q 0(x( tk)) → 0 as k → ∞. Therefore, Q 0(p) = 0, since Q 0 is continuous. Thus, p ∈ S 0.
If we further assume that V is bounded, then it is obvious that the proof can be much
simplified. In the investigations for the asymptotic behaviors of TCNN, condition (iii) is
more achievable.
We are interested in knowing whether if an orbit of the system (10) approaches an equilibrium
state or fixed point as time tends to infinity. The structure of ω-limit sets for the orbits provides
an important information toward this investigation. In discrete-time dynamical systems, the
ω-limit set of an orbit is not necessarily connected. However, the following proposition has
been proved by Hale and Raugel in 1992.
Proposition (Hale & Raugel, 1992). Let T be a continuous map on a Banach space X. Suppose
that the ω-limit set ω(x) is contained in the set of fixed points of T, and the closure of the orbit
Ox is compact. Then ω(x) is connected.
This proposition can be extended to non-autonomous systems for which there exist limiting
maps. Namely,
Multidimensional Dynamics: From Simple to Complicated
511
(A) There exists a continuous map F : R n → R n such that lim t→∞ F( t, x) − F(x) = 0, for all x ∈ R n.
Theorem (Chen & Shih, 2004b). Assume that (10) satisfies (A), the orbit Ox is bounded,
and ω(x), the ω-limit set of x, is contained in the set of fixed points of F. Then ω(x) is
connected. Under this circumstances, if F has only finitely many fixed points, then the orbit
Ox approaches some single fixed point of F, as t tends to infinity.
Let us represent the TCNN system (5)-(7) by the following time-dependent map
F( t, x) = ( F 1( t, x), · · · , Fn( t, x))
where
n
Fi( t, x) = αxi + (1 − γ) tωii(0)( yi − a 0 i) + ∑ ωijyj + ai,
j= i
where yi = hi( xi), i = 1, · · · , n and hi is defined in (6). The orbits of TCNN are then given by the iterations x( t + 1) = F( t, x( t)) with components xi( t + 1) = Fi( t, x( t)). Note that y = H(x) = ( h 1( x 1), · · · , hn( xn)) is a diffeomorphism on R n. Let W 0 denote the n × n matrix obtained from the connection matrix W with its diagonal entries being replaced by zeros.
Restated, W 0 = W − diag[ W]. For given 0 < γ < 1, choose 0 < b < 1 such that | 1 −γ | < 1. We b
consider the following time-dependent energy-like function:
n
n
n
V( t, x) = − 1 ∑ ∑ w
∑ a
2
ijhi( xi) hj ( xj) −
ihi ( xi)
i=1 j= i
i=1
n
h
+(
i ( xi)
1 − α) ∑
h− 1( η) dη + bt.
(11)
i
i=1 0
Theorem (Chen & Shih, 2004b). Assume that W 0 is a cycle-symmetric matrix, and either one
of the following condition holds,
(i) 0 ≤ α ≤ 1 and W
3
0 + 4(1 − α) ε I is positive definite;
(ii) 1 ≤ α ≤ 1 and W
3
0 + 8 αε I is positive definite;
(iii) α ≥ 1 and W 0 + 8 εI is positive definite.
Then there exists an n 0 ∈ N so that V( t, x) defined by (11) is a Lyapunov function for the
TCNN (5)-(7) on N0 × R n.
3.2 Global consensus through a competing-component approach
Grossberg (1978) considered a class of competitive systems of the form
˙
xi = ai(x)[ bi( xi) − C( x 1, x 2, · · · , xn)], i = 1, 2, · · · , n, (12)
where ai ≥ 0, ∂C/ ∂xi ≥ 0, x = ( x 1, x 2, · · · , xn) ∈ R n. In such a system, n is the number of competing populations, ai(x) refers to competitive balance, bi( xi) represents interpopulation
signal functions, and C( x) stands for mean competition function, or adaptation level. System
(12) was proposed as a mathematical model for the resolution to a dilemma in science
for hundred of years: How do arbitrarily many individuals, populations, or states, each
obey unique and personal laws, succeed in harmoniously interacting with each other to
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Discrete Time Systems
form some sort of stable society, or collective mode of behavior. Systems of the form (12)
include the generalized Volterra-Lotka systems and an inhibitory network (Hirsch, 1989). A
suitable Lyapunov function for system (12) is not known, hence the Lyapunov method and
LaSalle invariant principle are invalid. The work in (Grossberg, 1978) employed a skillful
competing-component analysis to prove that for system (12), any initial value x(0) ≥ 0 (i.e.
xi(0) ≥ 0, for any i) evolves to a limiting pattern x(∞) = ( x 1(∞), x 2(∞), · · · , xn(∞)) with 0 ≤ xi(∞) := lim t→∞ xi( t) < ∞, under some conditions on ai, bi, C.
System (12) can be approximated, via Euler’s difference scheme or delta-operator circuit
implementation (Harrer & Nossek,1992), by
xi(( k + 1) δ) = xi( kδ) + δai(x( kδ))[ bi( xi( kδ)) − C(x( kδ))], where one takes xi( kδ) as the k-th iteration of xi. In this subsection, let us review the
competing-component analysis for convergent dynamics reported in (Shih & Tseng, 2009).
Consider the following discrete-time model,
xi( k + 1) = xi( k) + βai(x( k))[ bi( xi( k)) − C(x( k))], (13)
where i = 1, 2, · · · , n, k ∈ N0 := { 0 } N. We first consider the theory for (13) with β = 1, i.e.
xi( k + 1) = xi( k) + ai(x( k))[ bi( xi( k)) − C(x( k))].
(14)
The results can then be extended to (13). First, let us introduce the following definition for the
convergent property of discrete-time systems.
Definition. A discrete-time competitive system x( k + 1) = F(x( k)) is said to achieve global
consensus (or global pattern formation, global convergence) if, given any initial value x(0) ∈ R n,
the limit xi(∞) := lim k→∞ xi( k) exists, for all i = 1, 2, · · · , n.
The following conditions are needed for the main results.
Condition (A1): Each ai(x) is continuous, and
0 < ai(x) ≤ 1, for all x ∈ R n, i = 1, 2, · · · , n.
Condition (A2): C(x) is bounded and continuously differentiable with bounded derivatives;
namely, there exist constants M 1, M 2, rj such that for all x ∈ R n,
M 1 ≤ C(x) ≤ M 2,
0 ≤ ∂C (
∂
x) ≤ r
x
j, j = 1, 2, · · · , n.
j
Condition (A3): bi( ξ) is continuously differentiable, strictly decreasing and there exist di > 0,
li, ui ∈ R such that for all i = 1, 2, · · · n,
−di ≤ b ( ξ) <
i
0, for all ξ ∈ R,
bi( ξ) > M 2, for ξ ≤ li, and bi( ξ) < M 1, for ξ ≥ ui.
Multidimensional Dynamics: From Simple to Complicated
513
Condition (A4): For i = 1, 2, · · · , n,
n
0 < di ≤ 1 − ∑ rj < 1.
j=1
Theorem (Shih & Tseng, 2009). System (14) with ai, bi, and C satisfying conditions (A1)-(A4)
achieves global consensus.
The proof of this theorem consists of three lemmas which depict the properties for the
following terms:
gi( k) = bi( xi( k)) − C(x( k)), Δ gi( k) = gi( k + 1) − gi( k), ˆ g( k) = max {gi( k) : i = 1, 2, · · · , n}, ˇ g( k) = min {gi( k) : i = 1, 2, · · · , n}, I( k) = min {i : gi( k) = ˆ g( k) }, J( k) = min {i : gi( k) = ˇ