離散時間型競爭系統的全局一致性

國

立

交

通

大

學

應用數學系

碩

士

論

文

離散時間型競爭系統的全局一致性

Global Consensus for Discrete-time Competitive System

研 究 生：曾睿彬

指導教授：石至文 教授

離散時間型競爭系統的全局一致性

Global Consensus for Discrete-time Competitive System

研 究 生：曾睿彬 Student：Jui-Pin Tseng

指導教授：石至文 Advisor：Chih-Wen Shih

國 立 交 通 大 學

應用數學系

碩 士 論 文

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics January 2005

離散時間型競爭系統的全局一致性

學生:曾睿彬 指導教授

石至文教授

國立交通大學應用數學系(研究所)碩士班

摘

要

本論文討論一個離散時間型競爭系統，我們的興趣在於這個系統如何

達到全局一致性。不用 Lyapunov 函數，而使用分析的討論去推論出

隨著時間趨近於無限，每個解的軌跡收斂到某個定值 。

Global Consensus for Discrete-time

Competitive System

Student: Jui-Pin Tseng Advisors: Dr. Chih-Wen Shih

Department (Institute) of Applied Mathematics

National Chiao Tung University

ABSTRACT

A discrete-time competitive system is studied. We are interested in

how the dynamics of the system reach global consensus. Analytical

arguments are developed to conclude that every orbit converges to a point

as time tends to infinity, without knowing a Lyapunov function.

誌

謝

感謝我的指導老師石至文教授兩年半來辛苦的指導，讓我在交大

的求學過程中，獲益良多。並在完成論文的的過程中，給我方向，且

耐心的、細心地與我討論，方使這篇論文得已完成。老師除了在數學

專業上引領我進入動態系統的領域，另外，在日常生活上也讓我看到

了許多值得我學習、參考的生活態度。同時，感謝林文偉教授、莊重

教授、洪盟凱教授給予建議與指教。

感謝奕達學長在我初初踏入交大時，在課業及生活上的幫助。謝

謝光輝學弟在電腦方面給予支援。感謝昌源學長辛苦教我Latex。

最後要感謝我的家人，爸、媽、弟弟睿士、妹妹筱雯，給我精神

上、實資上的支持、幫助。尤其是我最心愛的老婆靜坤，有你的支持

與貼諒，我才能無後顧之憂地完成學業，並給我最大的力量與幸福。

當然，也不能忘了我那兩個最可愛的小孩，筵閎、筠閑，是你們讓老

爸疲憊的時候，仍能由衷感到喜悅。

目

錄

中文摘要 ………

i

英文摘要 ………

ii

誌謝

………

iii

目錄

………

iv

1

Introduction……… 1

2

Main Results………

3

3

Proofs of Lemmas………

8

4

A Comparison between Continuous-time and

Discrete-time Models………

19

References ………

23

Global Consensus for Discrete-time

Competitive System

Jui Pin Tseng

Department of Applied Mathematics

National Chiao Tung University

Hsinchu, Taiwan, R.O.C.

*

January 29, 2005

Abstract

A discrete-time competitive system is studied. We are interested in how the dynamics of the system reach global consensus. Analytical arguments are developed to conclude that every orbit converges to a point as time tends to infinity, without knowing a Lyapunov function.

1

Introduction

One of the commonest ways to guarantee convergence of dynamics is to find a Lyapunov function for the system, that is, a continuous real valued function V on state space, which is nonincreasing along trajectories of the system. One then applies the LaSalle’s invariance principle to conclude the convergence. For example, Cohen and Grossberg (1983) [1] proved one convergence theorem for neural network systems of the form

˙ xi = ai(x)[bi(xi) − n X j=1 ωijgj(xj)], i = 1, · · · , n, (1.1)

where ai ≥ 0, the matrix [ωij] of coupling weights is symmetric, and g

0

j ≥ 0 for all

j. There exists a Lyapunov function

V (x) = − n X i=1 Z _xi 0 bi(ξ)g 0 i(ξ)dξ + 1 2 n X i=1 n X j=1 ωijgi(xi)gj(xj).

They showed that if ai > 0 and g

0

i > 0 for every i, then V is a strict Liapunov

function and therefore the system is quasi-convergent, see also [9]. Forti et. al. (1995) [2] proved global stability of Hopfield-type neural network of the form

˙ xi = −dixi+ n X j=1 Tijgj(xj) + Ii, (1.2)

where di > 0, gj is nondecreasing function. Again, the results obtained therein

employed a Lyapunov function of the so-called generalized Lur’e-Postnikov type. However, it is not always easy to find a suitable Lyapunov function when considering convergent dynamics. Grossberg (1978) [3] proved a convergence theorem for a class of “competitive systems” for which no Lyapunov functions are known. He considered systems of the form

˙

xi = ai(x)[bi(xi) − c(x)], (1.3)

where ai ≥ 0, _∂x∂c_i ≥ 0, for i = 1, · · · , n. Herein, each bi is a function of only

one variable xi, and the function c does not depend on i. In this kind of system,

population xi at neuron i competes indirectly with other xj through a scalar c(x),

i.e., the interaction among neurons are through function c(x). Worth noticed, it is difficult to find a suitable Lyapunov function for (1.3). In fact, systems (1.1), (1.2) both can be written in the form

˙

xi = ai(x)[bi(xi) − ci(x)],

which has a crucial difference from (1.3).

The “competition” for (1.3) by Grossberg means ai ≥ 0,_∂x∂c_j ≥ 0, for all i, j

and therefore has a little different sense from the commonly used one. Usually , a system ˙xi = Gi(x1, x2, · · · , xn) is competitive if ∂G_∂xji ≤ 0, for i 6= j. The sense of

competition in Grossberg’s paper can be seen if we consider functions ai as positive

constants. The assumption on ai for the studied dynamics is more general though.

Let us give more details to Grossberg’s model. In (1.3), n is any integer greater than 1, x(t) = (x1(t), x2(t), · · · , · · · , xn(t)) ∈ Rn. Such a system can have

any number of competing populations, any interpopulation signal functions bi(xi),

any mean competition function, or adaptation level c(x), and any state-dependent amplifications ai(x) of the competitive balance. That work in [3] proved that any

initial value x(0) ≥ 0 (i.e. xi(0) ≥ 0, for any i) generates a limiting pattern x(∞) =

conditions on ai, bi, c. We shall summarize the main ideas of Grossberg’s work in

Section 4.

Recently, discrete-time systems have attracted much scientific interests, cf. [5], [6], [8]. In this study, we consider the following discrete-time version of Grossberg’s model

xi(k + 1) = xi(k) + βai(x(k))[bi(xi(k)) − c(x(k))], (1.4)

where i = 1, 2, · · · , n, k ∈ N0 := {0} S

N. Viewing from the δ-operator, (1.3) can be approximated by

xi((k + 1)δ) = xi(kδ) + δai(x(kδ))[bi(xi(kδ)) − c(x(kδ))]. (1.5)

One usually takes xi[k]δ := xi(kδ) as the k-th iteration of xi and

x[k]δ := (x1[k]δ, x2[k]δ, · · · , xn[k]δ) as the k-th iteration of x.

In this presentation, we mainly consider (1.4) with β = 1, i.e.

xi(k + 1) = xi(k) + ai(x(k))[bi(xi(k)) − c(x(k))]. (1.6)

We define ∆xi(k) := xi(k + 1) − xi(k), hence system (1.6) can be rewritten in the

form

∆xi(k) = ai(x(k))[bi(xi(k)) − c(x(k))]. (1.7)

The main purpose of this investigation is to find out under what conditions on functions ai, bi, and c, systems (1.4) or (1.6) possesses a global limiting pattern

x(∞) := (x1(∞), x2(∞), · · · , xn(∞)) with −∞ < xi(∞) := limt→∞xi(t) < ∞ for

every i, given any initial value x(0).

Below, in Section 2, we state the main results of this presentation. In Section 3, we prove three key lemmas for our main result Theorem 1. In Section 4, we summarize the work of Grossberg [3] and make a generalization. A comparison of the analysis in deriving the global consensus for the continuous-time and the discrete-time competitive systems is also made.

2

Main Results

Definition 2.1. (Global Consensus) A discrete-time competitive system is said

to achieve global consensus (or global pattern information) if, given any initial value

x(0) ∈ Rn_{, the limit x}

The main results require the following conditions : Condition (I): Each ai(x) is continuous, and

0 < ai(x) ≤ 1, for all x ∈ Rn, i = 1, · · · , n. (2.1)

Condition (I)0_{: Each a}

i(x) is continuous, and

0 < ai(x) ≤ A, for all x ∈ Rn, i = 1, · · · , n. (2.2)

Condition (II): c(x) is bounded and continuously differentiable with bounded deriva-tives; namely, there exist constants M1, M2, rj such that

M1 ≤ c(x) ≤ M2, (2.3) 0 ≤ ∂c

∂xj

(x) ≤ rj, (2.4)

for all x ∈ Rn_{, and j = 1, 2, · · · , n.}

Condition (III): bi(ξ) is continuously differentiable, strictly decreasing and there

exist di > 0, li ∈ R, ui ∈ R such that

−di ≤ b0i(ξ) < 0, for all ξ ∈ R, (2.5)

and

bi(ξ) > M2, for ξ ≤ li, (2.6)

bi(ξ) < M1, for ξ ≥ ui. (2.7)

Condition (IV): For i = 1, · · · , n,

0 < di ≤ 1 − n

X

j=1

rj < 1. (2.8)

Condition (IV)0_{: For i = 1, · · · , n,}

0 < di ≤ 1 β − n X j=1 rj < 1 β. (2.9)

Condition (IV)00_{: For i = 1, · · · , n,}

0 < di ≤ 1 Aβ − n X j=1 rj < 1 Aβ. (2.10) Set d := min{di : i = 1, 2, · · · , n}, (2.11) M := max{|M1|, |M2|}. (2.12)

Theorem 1. System (1.6) with functions ai,bi,and c satisfying Conditions (I), (II),

(III), and (IV) achieves global consensus.

The proof of Theorem 1 consists of three lemmas stated below. For system (1.4), the following corollary can be derived.

Corollary 2. System (1.4) with functions ai, bi, and c satisfying Conditions (I),(II),

(III), and (IV)0 _{achieves global consensus.}

In fact, we only need that function ai is continuous, positive and bounded

above by some real number, say A, for all i, instead of Condition (I). It is due to that (1.6) can be rewritten as

xi(k + 1) = xi(k) +

ai(x(k))

A [Abi(xi(k)) − Ac(x(k))].

We thus derive the following Corollary.

Corollary 3. System (1.4) whose functions ai, bi,and c satisfy Condition (I)0, (II),

(III), and (IV)00 _{achieves global consensus.}

Remark 2.1. From Corollary 3, we find that the smaller β in (1.4) (δ in (1.5)) is,

the weaker restrictions on functions a1, bi, c are. In other words, when we consider

(1.4) in stead of (1.3), and want to have the global consensus proposition, we must

choose sufficiently small β in (1.4), basically.

In order to state the key lemmas for our main result, Theorem 1, we introduce some notations and definition as follows:

Notation 2.2. gi(k) = bi(xi(k)) − c(x(k)), ∆gi(k) = gi(k + 1) − gi(k), ˆg(k) = max{gi(k) : i = 1, 2, · · · , n}, ˇg(k) = min{gi(k) : i = 1, 2, · · · , n}, I(k) = min{i : gi(k) = ˆg(k)}, J(k) = min{i : gi(k) = ˇg(k)}, ˆ x(k) = xI(k)(k), ˇ x(k) = xJ(k)(k), ˆb(k) = bI(k)(ˆx(k)), ˇb(k) = bJ(k)(ˇx(k)), ∆ˆb(k) = ˆb(k + 1) − ˆb(k), ∆ˇb(k) = ˇb(k + 1) − ˇb(k), ∆bi(xi(k)) = bi(xi(k + 1)) − bi(xi(k)).

Definition 2.3. (i) A jump of type-1 is said to occur from i to j at k-th iteration

if I(k) = i, I(k+1)=j, (ii) A jump of type-2 is said to occur from i to j at k-th iteration if J(k) = i, J(k + 1) = j.

Lemma 1. Consider system (1.6) with ai, bi, and c satisfying (2.1), (2.3), (2.5),

(2.6) and (2.7). Given any initial value x(0) ∈ Rn_{, {x(k)} will be attracted to some}

compact set contained in Rn_{. Hence sequence {x}

i(k) | k ∈ N0} are bounded above

and below for all i = 1, 2, · · · , n.

If Lemma 1 is valid, consider an arbitrary orbit {x(k)}. Then {|ai(x(k))| |

k ∈ N0} is bounded below by some positive number, say 0 < ρi ≤ |ai(x(k))| for all

k ∈ N0 and {b

0

i(xi(k)) | k ∈ N0} are bounded above by some negative number, say

b0_i(xi(k)) ≤ −²i < 0 for all k ∈ N0. We define

ρ := min{ρi : i = 1, 2, · · · , n}, (2.13)

² := min{²i : i = 1, 2, · · · , n}. (2.14)

Lemma 2. Consider system (1.6) with ai, bi, and c satisfying (2.1), (2.4), (2.5)

(I) for function ˆg, either case (ˆg-(i)) or case (ˆg-(ii)) holds, where

(ˆg-(i)): ˆg(k) < 0, for all k ∈ N0,

(ˆg-(ii)): ˆg(k) ≥ 0, for all k ≥ K1, for some K1 ∈ N0;

(II) for function ˇg, either case (ˇg-(i)) or case (ˇg-(ii)) holds, where

(ˇg-(i)): ˇg(k) > 0, for all k ∈ N0,

(ˇg-(ii)): ˇg(k) ≤ 0, for all k ≥ K2, for some K2 ∈ N0.

If Lemma 2 is valid, there are only four possibilities to consider.

case (i): Both (ˆg-(i)) and (ˇg-(i)) hold. This case is impossible from our definition of ˆg and ˇg.

case (ii): Both (ˆg-(i)) and (ˇg-(ii)) hold, then sequence {xi(k)} will always decrease

as k increases, for all i = 1, 2, · · · , n. By Lemma 1, {xi(k)} are bounded below for

every i, hence the limit xi(∞) exists, for every i = 1, 2, · · · , n.

case (iii): Both (ˆg-(ii)) and (ˇg-(i)) hold, then sequence {xi(k)} will always increase

as k increases, for all i = 1, 2, · · · , n. By Lemma 1, {xi(k)} are bounded above for

every i, hence the limit xi(∞) exists, for every i = 1, 2, · · · , n.

case (iv): Both (ˆg-(ii)) and (ˇg-(ii)) hold.

Accordingly, we are left with the case case (iv) only, for the conclusion of global consensus for (1.6). We thus assume that ˆg(0) ≥ 0, ˇg(0) ≤ 0, without loss of generality.

Lemma 3. Consider system (1.6) with ai, bi, and c satisfying Conditions (I), (II),

(III), and (IV) then,

(i) limk→∞ˆb(k) exists, denoted by ˆB, and limk→∞c(x(k)) = ˆB,

(ii) limk→∞ˇb(k) exists, denoted by ˇB, and limk→∞c(x(k)) = ˇB.

If Lemma 3 holds, we find that lim

k→∞ˆb(k) = limk→∞ˇb(k) =: ¯B, (2.15)

since limk→∞c(x(k))= ˆB= ˇB. For any i = 1, 2, · · · , n, ˇg(k) ≤ gi(k) ≤ ˆg(k), for all

k ∈ N0. Equivalently,

ˇb(k) − c(x(k)) ≤ bi(xi(k)) − c(x(k)) ≤ ˆb(k) − c(x(k)),

for all k ∈ N0. Thus, ˇb(k) ≤ bi(xi(k)) ≤ ˆb(k), for all k ∈ N0. Therefore lim

We obtain

lim

k→∞ˆb(k) = limk→∞bi(xi(k)) = limk→∞ˇb(k) = ¯B,

by (2.15). Therefore we conclude that lim

k→∞bi(xi(k)) = ¯B, for all i = 1, 2, · · · , n. (2.16)

Moreover, limk→∞xi(k) exists, for every i = 1, 2, · · · , n, by (2.5) and (2.16). Hence,

global consensus of system (1.6) is achieved, if functions ai, bi, and c satisfy

Condi-tions (I), (II), (III), (IV).

3

Proofs of Lemmas

Proof of Lemma 1 : For any initial vale x(0), we consider the iteration sequence

{xi(k)} and their components xi(k). We divide the proof into several steps.

(i) By (2.3) and(2.7),

bi(xi) − c(x) < 0, (3.1)

for all xi ≥ ui. Therefore

∆xi(k) = ai(x(k))[bi(xi(k)) − c(x(k))] < 0, (3.2)

if xi(k) ≥ ui. Similarly, By (2.3) and (2.6),

bi(xi) − c(x) > 0, (3.3)

for all xi ≤ li. Therefore

∆xi(k) = ai(x(k))[bi(xi(k)) − c(x(k))] > 0, (3.4)

if xi(k) ≤ li. We claim that for all k ∈ N0,

|bi(xi(k))| ≤ di|xi(k)| + |bi(0)|. (3.5)

This follows from

bi(xi(k)) − bi(0) = b

0

i(·)xi(k),

where “·” means some real number between xi(k) and 0. Thus, by (2.5),

|bi(xi(k))| = |bi(0) + b 0 i(·)xi(k)| ≤ |bi(0)| + |b 0 i(·)xi(k)| ≤ |bi(0)| + di|xi(k)|.

(ii) Next, we show that for fixed constant Li, there exist some constants u 0 i and d 0 i, where u0_i > 0, 0 < di < d 0 i < 1 such that di|xi| + Li < d 0 i|xi|, if |xi| ≥ u 0 i. (3.6)

Let us verify this. Notably,

di|xi| + Li

|xi|

= di+ Li

|xi|

→ di < 1,

as |xi| → ∞. Therefore, there exist some u

0 i and d 0 i, where u 0 i > 0, 0 < di < d 0 i < 1 such that (di|xi| + Li)/|xi| < d0i, if |xi| ≥ u 0 i. (iii) |∆xi(k)| = |ai(x(k))[bi(xi(k)) − c(x(k))]| ≤ |bi(xi(k)) − c(x(k))| (by (2.1)) ≤ |bi(xi(k))| + |c(x(k))| ≤ di|xi(k)| + |bi(0)| + |c(x(k))| (by (3.5)) ≤ di|xi(k)| + |bi(0)| + M (by (2.3), (2.12)).

Hence, by (3.6), we choose |bi(0)| + M = Li, there exist some constants u

0 i and d 0 i where u0_i > 0, 0 < di < d 0 i < 1 such that |∆xi(k)| < d 0 i|xi(k)| < |xi(k)|, if |xi(k)| ≥ u 0 i. (3.7)

(iv) Set, for each i,

q_i0 := max{|ui|, |li|, u

0

i}. (3.8)

Let Q0 _{:= [−q}0

1, q10] × · · · × [−q0n, qn0]. Q0 is a compact set, hence |ai(x)[bi(xi) − c(x)]|

is bounded on Q0_{, say}

|ai(x)[bi(xi) − c(x)]| ≤ K, (3.9)

for all x ∈ Q0, for all i. Set

qi := q

0

i+ K, (3.10)

Q := [−q1, q1] × · · · × [−qn, qn]. (3.11)

We shall utilize (3.2), (3.4), (3.7), (3.8), (3.9), (3.10) in the following discus-sions.

case (a): If xi(0) ∈ [−qi, −q 0 i], then ∆xi(0) > 0, due to xi(0) ≤ −q 0 i ≤ li, and |∆xi(0)| < |xi(0)|, due to xi(0) ≤ −u 0

i, hence xi(1) still stays in (−qi, −q

0

i], or moves

into (−q_i0, q_i0). If the former case occurs, we consider xi(1) as case (a) again. If the

latter case occurs, we consider xi(1) as in the following case (b).

case (b): If xi(0) ∈ (−q

0

i, q

0

i), then |∆xi(0)| < K, by (3.9), hence xi(1) will stay

in [−qi, −q 0 i] or (−q 0 i, q 0 i) or [q 0

i, qi]. Then we can still consider xi(1) as in case (a),

case (b), and case (c), respectively. case (c): If xi(0) ∈ [q 0 i, qi], then ∆xi(0) < 0, by xi(0) ≥ q 0 i ≥ ui, and |∆xi(0)| < |xi(0)|, by xi(0) ≥ u 0

i, hence xi(1) still stays in [q

0 i, qi), or moves into (−q 0 i, q 0 i). If

the former case occurs, we consider xi(1) as in case (c) again. If the latter case

occurs, we consider xi(1) as in case (b). From the above arguments, we find that if

−qi ≤ xi(0) ≤ qi, then −qi < xi(1) < qi, and we can prove that −qi < xi(k) < qi,

for all k ≥ 2, by induction. (vi): If xi(0) < −qi, then

case (d): {xi(k)} either increases as k increases and remains bounded above

by −qi, or

case (e): {xi(k)} enter [−qi, qi] at some iteration, and never leaves [−qi, qi]

again.

(vii) if xi(0) > qi, then

case (f): {xi(k)} either decreases as k increases and remains bounded below

by qi, or

case (g): {xi(k)} enters [−qi, qi] at some iteration, and never leaves [−qi, qi]

again.

We find that no matter which case above occurs, {xi(k)} are bounded above

and below for all i. Therefore, {|ai(x(k))|} are bounded below by some positive

number, say 0 < ρ0_i ≤ |ai(x(k))|, and {b

0

i(xi(k))} are bounded above by some

nega-tive number, say b0_i(xi(k)) ≤ −²

0

i < 0. In fact, it is impossible for the above case (d)

and case (f) to occur. This is due to that if case (d) occurs, then

bi(xi(k)) − c(x(k)) = bi(xi(k)) − bi(li) + bi(li) − c(x(k))

> bi(xi(k)) − bi(li)

= b0_i(·)[xi(k) − li]

≥ ²0_iK,

for all xi(k) ≤ −qi ≤ li − K, by (2.5), (3.3), where “·” means some real number

between xi(k) and li. Therefore ∆xi(k) = ai(x(k))[bi(xi(k)) − c(x(k))] > ²

0

iKρ

0

Hence {xi(k)} will increase unboundedly, and this yields a contradiction. Therefore

case (d) never occurs. Similarly, case (f) never occurs, either. By the arguments above, we can find that given any initial value x(0), {x(k)} will be attracted by Q. Proof of Lemma 2:

For function ˆg, if ˆg(k) ≥ 0 for some k, say I(k) = i, then gj(k) ≤ gi(k), for all

j 6= i. Consider two possibilities |∆gi(k)| ≤ gi(k), and |∆gi(k)| > gi(k).

case (i) |∆gi(k)| ≤ gi(k): It follows that

ˆg(k + 1) ≥ gi(k + 1) = gi(k) + ∆gi(k) ≥ 0.

case (ii) |∆gi(k)| > gi(k): Let us elaborate.

∆gi(k) = gi(k + 1) − gi(k) = bi(xi(k + 1)) − c(x(k + 1)) − [bi(xi(k)) − c(x(k))] = bi(xi(k + 1)) − bi(xi(k)) − [c(x(k + 1)) − c(x(k))] = b0_i(·)[xi(k + 1) − xi(k)] − n X j=1 ∂c ∂xj (•)[xj(k + 1) − xj(k)],

where “·” means some real number between xi(k + 1) and xi(k), “•” means some

vector between x(k + 1) and x(k). Thus, ∆gi(k) = b 0 i(·)ai(x(k))gi(k) − n X j=1 ∂c ∂xj (•)aj(x(k))gj(k) ≥ −diai(x(k))gi(k) − n X j=1 rjaj(x(k))gi(k) (by (2.3), (2.5) and gj(k) ≤ gi(k) ≥ 0) ≥ −digi(k) − n X j=1 rjgi(k) (by (2.1)) = (−di− n X j=1 rj)gi(k) ≥ −gi(k) (by (2.8)).

Hence ∆gi(k) > 0, since |∆gi(k)| > gi(k) and ∆gi(k) ≥ −gi(k). Therefore, ˆg(k+1) ≥

gi(k + 1) = gi(k) + ∆gi(k) > 0.

For function ˇg, if ˇg(k) ≤ 0 for some k, say J(k) = i. Then gj(k) ≥ gi(k), for

case (i) |∆gi(k)| ≤ −gi(k): It follows that ˇg(k + 1) ≤ gi(k + 1) = gi(k) + ∆gi(k) ≤ 0. case(ii) |∆gi(k)| > −gi(k): ∆gi(k) = gi(k + 1) − gi(k) = bi(xi(k + 1)) − c(x(k + 1)) − [bi(xi(k)) − c(x(k))] = bi(xi(k + 1)) − bi(xi(k)) − [c(x(k + 1)) − c(x(k))] = b0_i(·)[xi(k + 1) − xi(k)] − n X j=1 ∂c ∂xj (•)[xj(k + 1) − xj(k)],

where “·” means some real number between xi(k + 1) and xi(k), “•” means some

vector between x(k + 1) and x(k). Thus

|∆gi(k)| = b 0 i(·)ai(x(k))gi(k) − n X j=1 ∂c ∂xj (•)aj(x(k))gj(k) ≤ −diai(x(k))gi(k) − n X j=1 rjaj(x(k))gi(k) (by (2.3), (2.5) and gj(k) ≥ gi(k) ≤ 0) ≤ −digi(k) − n X j=1 rjgi(k) (by (2.1)) = (di+ n X j=1 rj)(−gi(k)) ≤ −gi(k) (by (2.8)).

Hence ∆gi(k) < 0, since |∆gi(k)| > −gi(k) and ∆gi(k) ≤ −gi(k). So, ˇg(k + 1) ≤

gi(k + 1) = gi(k) + ∆gi(k) < 0.

From the above arguments, we find that function ˆg may keep negative at all iterations. But once it becomes nonnegative at some iteration, it will always remain nonnegative after this iteration. Similarly, ˇg may keep positive at all iterations. But once it get nonpositive at some iteration, it will always be nonpositive after this iteration. This completes the proof of Lemma 2. With Lemma 2, we assume that ˆg(0) ≥ 0, ˇg(0) ≤ 0, without loss of generality.

Proof for Lemma 3:

We assert that limk→∞ˆb(k) exists, and denote it by ˆB; moreover, limk→∞c(x(k))= ˆB.

Case (i): There exist finitely many jumps of type-1.

In this case, there exist some K3 ∈ N, some i, say 1, such that ˆg(k) = g1(k) ≥ 0, for all k ≥ K3. Hence {x1(k)} will be non-decreasing as k increases. By Lemma 1,

{x1(k)} are bounded above. Therefore, limk→∞x1(k) exists, hence limk→∞b1(x1(k)) exists, denoted by ˆB. Restated, limk→∞ˆb(k) = ˆB.

Next, we justify that limk→∞c(x(k))= ˆB. Assume otherwise, limk→∞c(x(k)) 6=

ˆ

B. It follows from ˆg(k) = g1(k) ≥ 0, for all k ≥ K3, that b1(x1(k)) ≥ c(x(k)), for all k ≥ K3. There exists some ε > 0, and subsequence {kl}∞l=1 of positive

integer numbers with k1 > K3 such that |c(x(kl)) − ˆB| > ε, for all l ∈ N. Because

limk→∞b1(x1(k)) = ˆB, for such ε, there exists K4 ∈ N, such that |b1(x1(k))− ˆB| ≤ ε₂, for all k ≥ K4. Therefore g1(kl) = b1(x1(kl)) − c(x(kl)) > ε₂, for all kl ≥ K4. We find that {x1(k)} is always increasing after K4-th iteration. In fact,

∆x1(kl) = a1(x(kl))[b1(x1(kl) − c(x(kl))] > ρ

ε

2,

if kl ≥ K4. Hence {x1(k)} will increase unboundedly, and yields a contradiction to Lemma 1.

Case (ii): There exist infinitely many jumps of type-1.

We shall justify that {ˆb(k)} decreases as {k} ↑ ∞. Consider a fixed k ∈ N0. Subcase (ii-a): no jump of type-1 occurs at k-th iteration.

Suppose I(k) = I(k + 1) = i, then gi(k) ≥ 0, gi(k + 1) ≥ 0. In addition,

ˆb(k + 1) = bi(xi(k + 1))

≤ bi(xi(k))

= ˆb(k),

thank to (2.5), and ∆xi(k) = ai(x(k))gi(k) ≥ 0. Thus {ˆb(k)} decreases as k

in-creases.

Subcase (ii-b): jump of type-1 occurs at k-th iteration and gi(k) ≥ 0, gj(k) ≥ 0,

where I(k) = i 6= I(k + 1) = j. It follows that

ˆb(k + 1) = bj(xj(k + 1))

≤ bj(xj(k))

≤ bi(xi(k))

= ˆb(k),

due to (2.5), ∆xj((k)) = aj(x(k))gj(k) ≥ 0, and by I(k) = i 6= j.

Subcase (ii-c): jump of type-1 occurs at k-th iteration and gi(k) ≥ 0, gj(k) < 0,

Notably, we still have gj(k + 1) ≥ 0. We claim that bj(xj(k + 1)) − bj(xj(k)) ≤ bi(xi(k)) − bj(xj(k)). (3.12) Indeed, LHS = b0_j(·)∆xj(k) = b0_j(·)aj(x(k))gj(k) ≤ b0_j(·)gj(k) (by (2.1)) ≤ −djgj(k) (by (2.5), and gj(k) < 0)) ≤ gi(k) − gj(k) (by (1 − dj)gj(k) < 0 ≤ gi(k)) = bi(xi(k)) − bj(xj(k)) = RHS.

Herein, “ · ” is defined as before. Hence, ˆb(k + 1) = bj(xj(k + 1)) ≤ bi(xi(k)) = ˆb(k).

All these cases indicate that {ˆb(k)} decreases as {k} increases. By Lemma 1, {x(k)} are attracted into some compact set Q contained in Rn_{. Therefore, {b}

i(xi(k))} are

bounded below, and so are {ˆb(k)}. Hence {ˆb(k)} decreases and converges to some number ˆB as k tends to infinity (denoted by {ˆb(k)} ↓ ˆB).

Next, we verify that limk→∞c(x(k))= ˆB. Assume otherwise: limk→∞c(x(k)) 6=

ˆ

B. There exist some positive µ, subsequence {kl}∞l=1 of positive integers, such that

|c(x(kl)) − ˆB| >

µ

²ρ, (3.13)

Where ², ρ are defined in (2.13) and (2.14). Because {ˆb(k)} ↓ ˆB, for µ0 _{:= min{}µ ²ρ, µ} >

0, there exists L ∈ N such that ˆ

B ≤ bI(k)(xI(k)(k)) ≤ ˆB + µ0, (3.14)

for all k ≥ L. Moreover

ˆg(`) = bI(`)(xI(`)(`) − c(x(`)) ≥ 0, (3.15)

for all ` ∈ N. Consider the kL-th iteration. Notably, kL> L. By (3.13), (3.14), and

(3.15), we have

ˆg(kL) = b1(x1(kL)) − c(x(kL)) > µ

where, for convenience, we set I(kL)=1 without loss of generality. There are two

possibilities at kL-th iteration, either jump of type-1 occurs or not. If it does not

occur, then |∆ˆb(kL)| = |ˆb(kL+ 1) − ˆb(kL)| = |b1(x1(kL+ 1)) − b1(x1(kL))| = |b0₁(·)||x1(kL+ 1) − x1(kL)| = |b0₁(·)||a1(x(kL))||g1(kL)| = |b0₁(·)||a1(x(kL))||ˆg(kL)| > ²ρµ ²ρ = µ. But it is impossible, because of (3.14).

If jump of type-1 occurs at kL-th iteration. Assume that I(kL+ 1)=2. Below

we consider three different cases for b2(x2(kL)):

Case (a): ˆB ≤ b2(x2(kL)) < b1(x1(kL)). Then g2(kL) > _²ρµ, and |∆b2(x2(kL))| =

|b0₂(·)||a2(x(kL))||g2(kL)| > ²ρ_²ρµ = µ. It is impossible, due to (3.14).

Case (b): ˆB > b2(x2(kL)) ≥ c(x(kL)). Then g2(kL) ≥ 0, and x2(kL+ 1) ≥

x2(kL). Thus,

ˆb(kL+ 1) = b2(x2(kL+ 1))

≤ b2(x2(kL))

< B.ˆ

It is impossible, since {ˆb(k)} ↓ ˆB.

Case (c): b2(x2(kL)) < c(x(kL)). Then g2(kL) < 0, and

∆b2(x2(kL)) = b2(x2(kL+ 1)) − b2(x2(kL))

= b0₂(·)a2(x(kL))g2(kL)

≤ −d2g2(kL)

< −g2(kL).

Thus, b2(x2(kL+1)) = b2(x2(kL)) + ∆b2(x2(kL)) < b2(x2(kL)) − g2(kL) = c(x(kL)).

From the above discussions, we conclude that limk→∞c(x(k)) = ˆB.

The second part of the lemma asserts that limk→∞ˇb(k) exists, denoted by ˇB,

and limk→∞c(x(k))= ˇB. The proof for the assertion resembles the first part. Let us

elaborate.

Case (i): There exist finitely many jumps of type-2.

In this case, there exists some K5 ∈ N, some i, say 1, such that ˇg(k) = g1(k) ≤ 0, for all k ≥ K5. Hence {x1(k)} will be non-increasing as k increases. By Lemma 1,

{x1(k)} are bounded below. Therefore, limk→∞x1(k) exists, hence limk→∞b1(x1(k)) exists, denoted by ˇB. Restated, limk→∞ˇb(k) = ˇB.

Next,we justify that limk→∞c(x(k))= ˇB. Assume otherwise, limk→∞c(x(k)) 6=

ˇ

B. It follows from ˇg(k) = g1(k) ≤ 0, for all k ≥ K5, b1(x1(k)) ≤ c(x(k)), for all k ≥

K5. There exists some ε > 0, and subsequence {kl}∞l=1 of positive integer numbers

with k1 > K5 such that |c(x(kl))− ˇB| > ε, for all l ∈ N. Because limk→∞b1(x1(k)) = ˇ

B, for such ε, there exists K6 ∈ N, such that |b1(x1(k)) − ˇB| ≤ ε₂, for all k ≥ K6. Therefore g1(kl) = b1(x1(kl)) − c(x(kl)) < −ε₂, for all kl ≥ K6. We find that {x1(k)} are always decreasing after K6− th iteration. In fact,

∆x1(kl) = a1(x(kl))[b1(x1(kl) − c(x(kl))] < −ρε

2,

if kl≥ K6. Hence, {x1(k)} will decrease unboundedly, and yields a contradiction to Lemma 1.

Case (ii): There exist infinitely many jumps of type-2.

We shall justify that {ˇb(k)} increases as {k} ↑ ∞. Consider a fixed k ∈ N0,

Subcase (ii-a): no jump of type-2 occurs at k-th iteration. Suppose J(k) =

J(k + 1) = i, then gi(k) ≤ 0, gi(k + 1) ≤ 0. In addition,

ˇb(k + 1) = bi(xi(k + 1))

≥ bi(xi(k))

= ˇb(k)

thank to (2.5), and ∆xi((k)) = ai(x(k))gi(k) ≤ 0. Thus {ˇb(k)} increases as {k}

increases.

Subcase (ii-b): jump of type-2 occurs at k-th iteration and gi(k) ≤ 0, gj(k) ≤ 0,

It follows that ˇb(k + 1) = bj(xj(k + 1)) ≥ bj(xj(k)) ≥ bi(xi(k)) = ˇb(k) due to (2.5), ∆xj((k)) = aj(x(k))gj(k) ≤ 0 and J(k) = i 6= j.

Subcase (ii-c): jump of type-2 occurs at k-th iteration and gi(k) ≤ 0, gj(k) > 0,

where J(k) = i 6= J(k + 1) = j.

Notably, we still have gj(k + 1) ≤ 0. We claim that

bj(xj(k + 1)) − bj(xj(k)) ≥ bi(xi(k)) − bj(xj(k)). (3.16) Indeed, LHS = b0_j(·)∆xj(k) = b0_j(·)aj(x(k))gj(k) ≥ b0_j(·)gj(k) (by (2.1)) ≥ −djgj(k) (by (2.5), and gj(k) > 0)) ≥ gi(k) − gj(k) (by (1 − dj)gj(k) > 0 ≥ gi(k)) = bi(xi(k)) − bj(xj(k)) = RHS.

Herein, “ · ” is defined as before. Hence, ˇb(k + 1) = bj(xj(k + 1)) ≥ bi(xi(k)) = ˇb(k).

All these cases indicate that {ˇb(k)} increase as {k} increases. By Lemma 1, {x(k)} are attracted into some compact set Q contained in Rn_{. Therefore, {b}

i(xi(k))} are

bounded above, and so are {ˇb(k)}. Hence {ˇb(k)} increase and converge to some number, say ˇB as {k} tend to infinity (denoted by ˇb(k)} ↑ ˇB).

Next, we verify that limk→∞c(x(k))= ˇB. Assume otherwise: limk→∞c(x(k)) 6=

ˇ

B. There exist some positive µ, subsequence {kl}∞l=1 of positive integers, such that

|c(x(kl)) − ˇB| >

µ

²ρ. (3.17)

Where ², ρ are defined in (2.13) and (2.14). Because ˇb(k)} ↑ ˇB, for µ0 := min{_²ρµ, µ} >

0, there exists L ∈ N, such that ˇ

B ≥ bJ(k)(xJ(k)(k)) ≥ ˇB − µ

0

for all k ≥ L. Moreover

ˇg(`) = bJ(`)(xJ(`)(`) − c(x(`)) ≤ 0, (3.19)

for all ` ∈ N. Consider the kL-th iteration. Notably, kL > L. By (3.17), (3.18),and

(3.19), we have

ˇg(kL) = b1(x1(kL)) − c(x(kL)) < −

µ ²ρ,

where, for convenience, we set J(kL)=1 without loss of generality. There are two

possibilities at kL− th iteration, either jump of type-2 occurs or not. If it dose not

occur, then |∆ˇb(kL)| = |ˇb(kL+ 1) − ˇb(kL)| = |b1(x1(kL+ 1)) − b1(x1(kL))| = |b0₁(·)||x1(kL+ 1)) − (x1(kL))| = |b0₁(·)||a1(x(kL))||g1(kL)| = |b0₁(·)||a1(x(kL))||ˇg(kL)| > ²ρµ ²ρ = µ. But it is impossible, because (3.18).

If jump of type-2 occurs at kL− th iteration. Assume that J(kL+ 1)=2. Below

we consider three different cases for b2(x2(kL)):

Case (a): ˇB ≥ b2(x2(kL)) > b1(x1(kL)). Then g2(kL) < −_²ρµ, and |∆b2(x2(kL))| =

|b0₂(·)||a2(x(kL))||g2(kL)| > ²ρ_²ρµ = µ. It is impossible, due to (3.18).

Case (b): ˇB < b2(x2(kL)) ≤ c(x(kL)). Then g2(kL) ≤ 0, and x2(kL+ 1) ≤

x2(kL). Thus

ˇb(kL+ 1) = b2(x2(kL+ 1))

≥ b2(x2(kL))

> B.ˇ

Case (c): b2(x2(kL)) > c(x(kL)). Then g2(kL) > 0, and ∆b2(x2(kL)) = b2(x2(kL+ 1)) − b2(x2(kL)) = b0₂(·)a2(x(kL))g2(kL) ≥ −d2g2(kL) > −g2(kL). Thus, b2(x2(kL+1)) = b2(x2(kL)) + ∆b2(x2(kL)) > b2(x2(kL)) − g2(kL) = c(x(kL)).

Hence ˇb(kL+ 1) = b2(x2(kL+ 1)) > c(x(kL)) > ˇB. It is impossible, since {ˇb(k)} ↑ ˇB.

From the above discussions, we conclude that limk→∞c(x(k)) = ˇB.

4

A Comparison between Continuous-time and

Discrete-time Models

We first introduce the results for (1.3) stated in Grossberg’s paper [4].

Definition 4.1. A competitive system is said to achieve weak global consensus (or

weak global pattern formation), if given any initial value x(0) ≥ 0, all the limits bi(xi(∞)) := limt→∞bi(xi(t)) exist, for all i = 1, 2, · · · , n.

Definition 4.2. A competitive system is said to achieve strong global consensus (or

strong global pattern formation) if, given any initial value x(0) ≥ 0, all the limits xi(∞) := limt→∞xi(t) exist, for all i = 1, 2, · · · , n.

The following conditions are needed for the main results in Grossberg’s paper [4].

Condition (G1):

(a): ai(x) is continuous for x ≥ 0,

(b): bi(xi) is either continuous with piecewise derivative for xi ≥ 0, or is continuous

with piecewise derivative for xi > 0 and bi(0) = ∞,

(c): c(x) is continuous with piecewise derivative for x ≥ 0. Condition (G2):

ai(x) > 0 if xi > 0 and xj ≥ 0, j 6= i,and ai(x) = 0 if xi = 0 and xj ≥ 0,

j 6= i. Moreover, there exist a function ¯ai(xi) such that, for sufficiently small λ > 0,

¯ ai(xi) ≥ ai(xi) if x ∈ [0, λ]n and Z _λ 0 dω ¯ ai(ω) = ∞ (4.1)

Condition (G3): lim sup_ω→∞bi(ω) < c(0, 0, · · · , ∞, · · · , 0), where ”∞” occurs in the

ith entry, i = 1, 2, · · · , n.

Condition (G4): ∂c

∂xj ≥ 0 , j = 1, 2, · · · , n

Theorem 4 (Grossberg). Any system of the form (1.3) satisfying Conditions

(G1), (G2), (G3) and(G4) achieves weak global consensus. Moreover, bi(xi(∞)) =

c(x(∞)), for every i.

Similar to proof of Theorem 1, the one of Theorem 4 consists of three main parts which we describe as follows:

First, the theorem will be proved for the case that bi ≡ b, then this proof can

be adapted to the case of i-dependent bi.

Part (I): (This part works as Lemma 1)

By Conditions (G1) and (G2), if xi(0) > 0, then xi(t) > 0 for t ≥ 0. If xi(0) = 0, then

component xican be deleted from the network without loss of generality [4]. By (4.1)

and Condition (G3), there exist a B such that xi(t) ∈ [0, B] for all i = 1, 2, · · · , n,

t ≥ 0. Hence our attention is restricted to positive initial values. It is then derived

that x(t) stays in some compact subset in Rn_{, for all time t ≥ 0.}

Part (II): (This part works as Lemma 2) Define

gi(t) = b(xi(t)) − c(x(t)) (4.2)

ˆg(t) = max{gj(t) : j = 1, 2, · · · , n} (4.3)

Then either ˆg(t) < 0, for all t ≥ 0, or there exists t = T such that ˆg(T ) ≥ 0 implies ˆg(t) ≥ 0, for t ≥ T . This is due to Condition (G4). If at any time t = s, ˆg(s) = 0, say ˆg(s) = gi(s), then lim t→0+ ˆg(s + t) − ˆg(s) t ≥ ˙gi(s) = b 0 (xi(s)) ˙xi(s) − n X j=1 ∂c ∂xj (x(s)) ˙xj(s) ≥ 0. (4.4)

If ˆg(t) < 0 for all t ≥ 0, it is a trivial case. Hence ˆg(t) ≥ 0 are assumed below without loss of generality

Part (III): (This part works as Lemma 3)

Definition 4.3. (i): A jump is said to occur from i to j at time t = T , if there

exists time s and u such that ˆg(t) = gi(t), for s ≤ t ≤ T , and ˆg(t) = gj(t), for

T ≤ t ≤ u. (ii): I(t)=min{i : ˆg(t) = gi(t)}. (iii): ˆb(t)=b(xI(t)(t)).

If Part (I)and (II) are valid, then we have the following three conclusions. (i): ˆb(t) is monotone at all large time, hence limt→∞ˆb(t) exists , and denoted by

ˆ

B, (ii): limt→∞c(x(t))= ˆB, (iii): limt→∞b(xi(t))= ˆB, for all i = 1, 2, · · · , n. Hence,

weak global consensus is achieved.

Corollary 5 (Grossberg). Any system of the form (1.3) whose functions satisfy

Condition(G1)-(G4) and whose bi possess finitely many local maxima, or intervals

of local maxima ,within the range of xi, achieves strong global consensus.

From the process of the proof above, we can find that Part (I) and (II) play very dominant roles for Theorem 4. From this view point, we can extend Theorem 4 to Theorem 6, i.e. the phase space for (1.3) can be extended to Rn_.

We need some conditions for the theorem.

Condition (A): (a): ai(x) is continuous and positive for all x ∈ Rn,

(b): bi(ξ) is continuously differentiable for ξ ∈ R,

(c): c(x) is continuously differentiable for x ∈ Rn_.

Condition (B): Given any initial value x(0), x(t) will be attracted by some compact set contained in Rn_.

Condition (B)0: (a): limξ→∞bi(ξ) = −∞ , limξ→−∞bi(ξ) = ∞ ,i = 1, 2, · · · , n, (b):

c(x) is bounded below.

Condition (C):

∂c

∂xk ≥ 0 , k = 1, 2, · · · , n

Theorem 6. Any system of form (1.3) whose functions satisfy Condition (A), (B),

(C) achieves weak global consensus (herein, I mean that given any initial value

x(0) ∈ Rn_{, all the limits b}

i(xi(∞)) := limt→∞bi(xi(t)) exist , for all i = 1, 2, · · · , n.).

Moreover, each bi(xi(∞)) = c(x(∞)).

The proof of the Theorem 6 is mainly because that Condition (B) works as Part (I) in the proof of Theorem 4, Condition (C) works as Part (III) in the proof of Theorem 4 (because of (4.4), mainly). Therefore the work as Part (III) in the proof of Theorem 4 will also be done, and weak global consensus will be achieved. By the

arguments above, for the purpose of comparing the difference of convergence theo-rems for continuous-time and discrete-time competitive network (details in Section 4), we can see that the proof of Theorem 6 can also be completed by the parallel three parts, just as in Theorem 4.

Corollary 7. Any system of form (1.3) whose functions satisfies Condition (A),

(B), (C) and whose bi possess finitely many local maxima, or intervals of local

max-ima, within the range of xi, achieves strong global consensus.

Remark 4.1. In Theorem 6, Condition (B) is a more abstract condition, and it can

be achieved by the more concrete one as Condition (B0).

Proof. By Condition (B)0 and (C). For each i, there exist pi, qi ∈ R, such that

˙

xi(t) = ai(x(t))[bi(xi(t)) − c(x(t))] < 0, if xi(t) ≥ qi, and ˙xi(t) = ai(x(t))[bi(xi(t)) −

c(x(t))] > 0, if xi(t) ≤ pi. Hence, given initial valve x(0), x(t) will be bounded,

for all t. Then both |ai(x(t))|, |[bi(xi(t)) − c(x(t))]| are bounded below from some

positive number, and so is | ˙xi(t)|. Therefore if xi(t) > qi at some time, say Si, then

xi(t) must be decreasing until xi(t) enters [p1, qi] and never leave it again, as time

goes by after Si. If xi(t) < pi at some time, say Ti, then xi(t) must be increasing

until xi(t) enters [pi, qi] and never leave it again, as time goes by after Ti. Hence

x(t) will be attracted by [p1, q1] × · · · × [pn, qn].

Below, let us compare the difference of convergence theorems for continuous-time and discrete-continuous-time competitive network via Theorem 1 and Theorem 6 (with Condition (B)0).

In the process of proving Theorem 1, we can find that different from (1.3) with “continuous solution”, the behavior of solution{x(k)} is much unpredictable. Hence we have to control ∆xi(k) at each iteration. Details are shown as follows:

(I): Different from Part (I) for Theorem 6 (just as Remark 4.1 for Theorem 6), we need more conditions as those function ai, c ,b0i must be bounded; namely, “0 <

(x) ≤ 1”, “|b0

i(ξ)| ≤ di ” and “M1 ≤ c(x) ≤ M2” to achieve Lemma 1 ,

(II): Different from Part (II) for Theorem 6 with continuous ˆg(t), {ˇg(k)}, {ˇg(k)} in Lemma 2 are sequences. We must control functions bi, c in addition to make

the same work as Part (II). Hence we need more conditions ,“b0

i(ξ) ≥ −di and

0 < di ≤ 1 −

P_n

i=1ri < 1” to achieve Lemma 2,

(III): Different from Part (III) for Theorem 6, Lemma 1 and Lemma 2 are “not sufficient” for Theorem 1. To achieve the “monotonicity” of {ˆb(k)} and {ˇb(k)}, we

need function bi to be decreasing; namely 0 ≥ b0i(ξ) ≥ −di > −1. For the purpose

“limk→∞c(x(k)) = ˆB = ˇB”, we demand function bito be strictly decreasing; namely

0 > b0

i(ξ) ≥ −di > −1 .

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