Discrete Time Systems Part 18 doc

Multidimensional systems may also exhibit simple dynamics; for example, every orbitconverges to a fixed point, as time tends to infinity.. Typical mathematical tools for justifying suchdyn

The linearized of (70) has the matrices:

The analysis of the roots for the equation P2(λ) =0 is done for fixed values of the parameters.

The numerical simulation can be done for c1=0.1, c2=0.4, k=0.04, σ=0.4.

5 The discrete deterministic and stochastic Kaldor model

The discrete Kaldor model describes the business cycle for the state variables characterized by

the income (national income) Y n and the capital stock K n , where n∈ IN For the description

of the model’s equations we use the investment function I : IR+×IR+ → IR denoted by

I = I(Y, K) and the savings function S : IR+×IR+ → IR, denoted by S = S(Y, K) bothconsidered as being differentiable functions (Dobrescu & Opri¸s, 2009), (Dobrescu & Opri¸s,2009)

The discrete Kaldor model describes the income and capital stock variations using the

functions I and S and it is described by:

where p∈ (0, 1)is the propensity to save with the respect to the income

The investment function I is defined by taking into account a certain normal level of income

u and a normal level of capital stock pu

q , where u ∈ IR, u > 0 The coefficient q ∈ (0, 1)represents the capital depreciation

In what follows we admit Rodano’s hypothesis and consider the form of the investmentfunction as follows:

1+r

q

(1−r−q)k+f(y−u) +pu

The fixed points of the application (77) with respect to the model’s parameters s, q, p, r are the

solutions of the following system:

Taking into account that f satisfies f(0) =0, by analyzing (78) we have:

Proposition 5.1.(i) The point of the coordinates P

u, pu q

(iii) If f(x) = arctan x and p

Let(y0, k0) be a fixed point of the application (77) We use the following notations: ρ1 =

f(y0−u),ρ2= f (y0−u),ρ3= f (y0−u)and

a10=s(ρ1−p), a01= −rs, b10=ρ1, b01= −q−r.

Proposition 5.2.(i) The Jacobian matrix of (77) in the fixed point(y0, k0)is:

then equation (80) has the roots with their absolute values equal to 1

(iv) With respect to the change of variable:

s(β) = (( 1+β)2−1+q+r

1−q−r)(ρ1−p) +r

equation (80) becomes:

λ2−a1(β)λ+b1(β) =0 (81)where

a1(β) =2+(ρ1−p)((1+β)2−1+ρ+r)

(1−q−r)(ρ1−p) +r −q−r, b1(β) = (1+β)2.Equation (81) has the roots:

μ1,2(β) = (1+β)e ±iθ(β)

where

θ(β) =arccos a1(β)

2(1+β).

(v) The point s(0) =s0is a Neimark-Sacker bifurcation point

(vi) The eigenvector q∈ IR2, which corresponds to the eigenvalueμ(β) =μ and is a solution

of Aq=μq, has the components

q1=1, q2=μ−1−a10

The eigenvector p∈IR2, which corresponds to the eigenvalueμ and is a solution of A T p=μp,

has the components:

The proof follows by direct calculation using (77).

With respect to the translation y→y+y0, k→k+k0, the application (77) becomes:

y k

→

(1−sp)y−rsk+s f(y+y0−u) −f(y0−u)

2(μ2−μ)(μ−1) + |

s(β)p1+p2|2

1−μ ++ |s(β)p1+p2|2

2(μ2−μ)

ρ2+s(β)p1+p2

2 ρ3and l1(0) =Re(C1(0)e iθ(0)) If l1(0) <0 in the neighborhood of the fixed point(y0, k0)then

there is a stable limit cycle If l1(0) >0 there is an unstable limit cycle

(iii) The solution of (76) in the neighborhood of the fixed point(y0, k0)is:

The present chapter contains a part of the authors’ papers that have been published in journals

or proceedings, to which we have added the stochastic aspects

The methods used in this chapter allow us to study other models described by systems ofequations with discrete time and delay and their associated stochastic models

7 Acknowledgements

The research was done under the Grant with the title "The qualitative analysis andnumerical simulation for some economic models which contain evasion and corruption",CNCSIS-UEFISCU (grant No 1085/2008)

8 References

Dobrescu, L.; Opri¸s, D (2009) Neimark–Sacker bifurcation for the discrete-delay

Kaldor–Kalecki model, Chaos, Soliton and Fractals, Vol 39, Issue 5, 15 June 2009,

519–530, ISSN-0960-0779

Dobrescu, L.; Opri¸s, D (2009) Neimark–Sacker bifurcation for the discrete-delay Kaldor

model, Chaos, Soliton and Fractals, Vol 40, Issue 5, 15 June 2009, 2462–2468 Vol 39,

519–530, ISSN-0960-0779

Kloeden, P.E.; Platen, E (1995) Numerical Solution of Stochastic Differential Equations, Springer

Verlag, ISBN, Berlin, ISBN 3-540-54062-8

Kuznetsov, Y.A (1995) Elemets of Applied Bifurcations Theory, Springer Verlag, ISBN,

New-York, ISBN 0-387-21906-4

Lorenz, H.W (1993) Nonlinear Dynamical Economics and Chaotic Motion, Springer Verlag, ISBN,

Berlin, ISBN 3540568816

Mircea, G.; Neam¸tu, M.; Opri¸s, D (2004) Hopf bifurcation for dynamical systems with time delay

and applications, Mirton, ISBN, Timi¸soara, ISBN 973-661-379-8.

Mircea, G.; Neam¸tu, M., Cisma¸s, L., Opri¸s, D (2010) Kaldor-Kalecki stochastic model of

business cycles, Proceedings of 11th WSEAS International Conference on Mathematics and Computers in Business and Economics, pp 86-91, 978-960-474-194-6, Ia¸si, june 13-15,

2010, WSEAS Press

503Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications

Mircea, G.; Opri¸s, D (2009) Neimark-Sacker and flip bifurcations in a discrete-time dynamic

system for Internet congestion, Transaction on mathematics, Volume 8, Issue 2,

February 2009, 63–72, ISSN: 1109-2769

Neam¸tu, M (2010) The deterministic and stochastic economic games, Proceedings of

11th WSEAS International Conference on Mathematics and Computers in Business and Economics, pp 110-115, 978-960-474-194-6, Ia¸si, june 13-15, 2010, WSEAS Press

Kang-Ling Liao, Chih-Wen Shih and Jui-Pin Tseng

Department of Applied Mathematics, National Chiao Tung University

Hsinchu, Taiwan 300, R.O.C

1 Introduction

The most apparent look of a discrete-time dynamical system is that an orbit is composed

of a collection of points in phase space, in contrast to a trajectory curve for acontinuous-time system A basic and prominent theoretical difference between discrete-timeand continuous-time dynamical systems is that chaos occurs in one-dimensional discrete-timedynamical systems, but not for one-dimensional deterministic continuous-time dynamicalsystems; the logistic map and logistic equation are the most well-known example illustratingthis difference On the one hand, fundamental theories for discrete-time systems havealso been developed in a parallel manner as for continuous-time dynamical systems,such as stable manifold theorem, center manifold theorem and global attractor theoryetc On the other hand, analytical theory on chaotic dynamics has been developed morethoroughly for discrete-time systems (maps) than for continuous-time systems Li-Yorke’speriod-three-implies-chaos and Sarkovskii’s ordering on periodic orbits for one-dimensionalmaps are ones of the most celebrated theorems on chaotic dynamics

Regarding chaos theory for multidimensional maps, there are renowned Smale-Birkhoffhomoclinic theorem and Moser theorem for diffeomorphisms In addition, Marotto extendedLi-Yorke’s theorem from one-dimension to multi-dimension through introducing the notion

of snapback repeller in 1978 This theory applies to maps which are not one-to-one (notdiffeomorphism) But the existence of a repeller is a basic prerequisite for the theory Therehave been extensive applications of this theorem to various applied problems However,due to a technical flaw, Marotto fixed the definition of snapback repeller in 2005 WhileMarotto’s theorem is valid under the new definition, its condition becomes more difficult

to examine for practical applications Accessible and computable criteria for applying thistheorem hence remain to be developed In Section 4, we shall introduce our recent works andrelated developments in the application of Marotto’s theorem, which also provide an effectivenumerical computation method for justifying the condition of this theorem

Multidimensional systems may also exhibit simple dynamics; for example, every orbitconverges to a fixed point, as time tends to infinity Such a scenario is referred to asconvergence of dynamics or complete stability Typical mathematical tools for justifying suchdynamics include Lyapunov method and LaSalle invariant principle, a discrete-time version

Multidimensional Dynamics: From Simple to Complicated

27

However, it is not always possible to construct a Lyapunov function to apply this principle,especially for multidimensional nonlinear systems We shall illustrate other technique thatwas recently formulated for certain systems in Section 3.

As neural network models are presented in both continuous-time and discrete-time forms,and can exhibit both simple dynamics and complicated dynamics, we shall introduce somerepresentative neural network models in Section 2

2 Neural network models

In the past few decades, neural networks have received considerable attention and weresuccessfully applied to many areas such as combinatorial optimization, signal processingand pattern recognition (Arik, 2000, Chua 1998) Discrete-time neural networks have beenconsidered more important than their continuous-time counterparts in the implementations(Liu, 2008) The research interests in discrete-time neural networks include chaotic behaviors(Chen & Aihara, 1997; Chen & Shih, 2002), stability of fixed points (Forti & Tesi, 1995; Liang &Cao, 2004; Mak et al., 2007), and their applications (Chen & Aihara, 1999; Chen & Shih, 2008)

We shall introduce some typical discrete-time neural networks in this section

Cellular neural network (CNN) is a large aggregation of analogue circuits It was firstproposed by Chua and Yang in 1988 The assembly consists of arrays of identical elementaryprocessing units called cells The cells are only connected to their nearest neighbors Thislocal connectivity makes CNNs very suitable for VLSI implementation The equations fortwo-dimension layout of CNNs are given by

where u k , x ij , h(x ij) are the controlling input, state and output voltage of the specified

CNN cell, respectively CNNs are characterized by the bias I and the template set A and

B which consist of a ij,k and b ij,k , respectively a ij,k represents the linear feedback, and b ij,k the linear control The standard output h is a piecewise-linear function defined by h(ξ) =1

2(| ξ+1| − | ξ −1|) C is the linear capacitor and R is the linear resistor For completeness

of the model, boundary conditions need to be imposed for the cells on the boundary of theassembly, cf (Shih, 2000) The discrete-time cellular neural network (DT-CNN) counterpartcan be described by the following difference equation

δx[k] = x[k+1] − x[k]

T

is an approximation of the derivative of x(t) Indeed, limT→0 δx[k] = ˙x(t )| t =kT In this case,

μ = 1− T

τ , where T is the sampling period, and τ = RC The parameters a ij,k, b ij,kin (2)

correspond to a ij,k , b ij,kin (1) under sampling, cf (Hänggi et al., 1999) If (2) is considered in

conjunction with (1), then T is required to satisfy τ ≥ T to avoid aliasing effects Under this

situation, 0≤ μ ≤1 Thus CT-CNN is the limiting case of delta operator based CNNs with

T →0 If the delta operator based CNNs is considered by itself, then there is no restriction

on T, and thus no restrictions on μ in (2) On the other hand, a sampled-data based CNN

has been introduced in (Harrer & Nossek, 1992) Such a network corresponds to the limiting

case of delta operator based CNNs as T → 1 For an account of unifying results on theabove-mentioned models, see (Hänggi et al., 1999) and the references therein In addition,Euler’s difference scheme for (1) takes the form

Note that CNN of any dimension can be reformulated into a one-dimensional setting, cf (Shih

& Weng, 2002) We rewrite (2) into a one-dimensional form as

x i(t+1) =μx i(t) +w ii(t)[y i(t ) − a 0i] +Σn

k=i w ik y k(t) +a i (5)

where i = 1,· · · , n, t ∈ N (positive integers), ε, γ are fixed numbers with ε > 0, 0 <

γ < 1 The main feature of TCNN contains chaotic dynamics temporarily generated forglobal searching and self-organizing As certain variables (corresponding to temperature

in the annealing process) decrease, the network gradually approaches a dynamical structurewhich is similar to classical neural networks The system then settles at stationary states andprovides a solution to the optimization problem Equations (5)-(6) with constant self-feedback

connection weights, that is, w ii(t) = w ii =constant, has been studied in (Chen & Aihara,

1995, 1997); therein, it was shown that snapback repellers exist if| w ii |are large enough Theresult hence implicates certain chaotic dynamics for the system More complete analyticalarguments by applying Marotto’s theorem through the formulation of upper and lowerdynamics to conclude the chaotic dynamics have been performed in (Chen & Shih, 2002, 2008,

2009) As the system evolves, w iidecreases, and the chaotic behavior vanishes In (Chen &Shih, 2004), they derived sufficient conditions under which evolutions for the system converge

to fixed points of the system Moreover, attracting sets and uniqueness of fixed point for thesystem were also addressed

Time delays are unavoidable in a neural network because of the finite signals switching andtransmission speeds The implementation of artificial neural networks incorporating delays

507Multidimensional Dynamics: From Simple to Complicated

has been an important focus in neural systems studies (Buric & Todorovic, 2003; Campbell,2006; Roska & Chua, 1992; Wu, 2001) Time delays can cause oscillations or alter the stability

of a stationary solution of a system For certain discrete-time neural networks with delays,the stability of stationary solution has been intensively studied in (Chen et al., 2006; Wu et al.,2009; Yua et al., 2010), and the convergence of dynamics has been analyzed in (Wang, 2008;Yuan, 2009) Among these studies, a typically investigated model is the one of Hopfield-type:

u i(t+1) =a i(t)u i(t) +∑m

j=1b ij(t)g j(u j(t − r ij(t))) +J i , i=1, 2,· · · , m. (8)

Notably, system (8) represents an autonomous system if a i(t ) ≡ a i , and b ij(t ) ≡ b ij(Chen etal., 2006), otherwise, a non-autonomous system (Yuan, 2009)

The class of Z-matrices consists of those matrices whose off-diagonal entries are less than

or equal to zero A M-matrix is a Z-matrix satisfying that all eigenvalues have positive real parts For instance, one characterization of a nonsingular square matrix P to be a M-matrix is that P has non-positive off-diagonal entries, positive diagonal entries, and non-negative row sums There exist several equivalent conditions for a Z-matrix P to be M-matrix, such as the one where there exists a positive diagonal matrix D such that PD is a diagonally dominant matrix, or all principal minors of P are positive (Plemmons, 1977) A common approach

to conclude the stability of an equilibrium for a discrete-time neural network is throughconstructing Lyapunov-Krasovskii function/functional for the system In (Chen, 2006), based

on M-matrix theory, they constructed a Lyapunov function to derive the delay-independent

and delay-dependent exponential stability results

Synchronization is a common and elementary phenomenon in many biological and physical

systems Although the real network architecture can be extremely complicated, richdynamics arising from the interaction of simple network motifs are believed to providesimilar sources of activities as in real-life systems Coupled map networks introduced byKaneko (Kaneko, 1984) have become one of the standard models in synchronization studies.Synchronization in diffusively coupled map networks without delays is well understood,and the synchronizability of the network depends on the underlying network topologyand the dynamical behaviour of the individual units (Jost & Joy, 2001; Lu & Chen, 2004).The synchronization in discrete-time networks with non-diffusively and delayed coupling isinvestigated in a series of works of Bauer and coworkers (Bauer et al., 2009; Bauer et al., 2010)

3 Simple dynamics

Orbits of discrete-time dynamical system can jump around wildly However, there aresituations that the dynamics are organized in a simple manner; for example, every solutionconverges to a fixed point as time tends to infinity Such a notion is referred to as

convergence of dynamics or complete stability Moreover, the simplest situation is that all orbits

converge to a unique fixed point We shall review some theories and results addressingsuch simple dynamics In Subsection 3.1, we introduce LaSalle invariant principle andillustrate its application in discrete-time neural networks In Subsection 3.2, we review thecomponent-competing technique and its application in concluding global consensus for adiscrete-time competing system

3.1 Lyapunov method and LaSalle invariant principle

Let us recall LaSalle invariant principle for difference equations We consider the differenceequation

where F : Rn −→ Rn is a continuous function Let U be a subset ofRn For a function

V : U −→ R, define ˙V(x) =V(F(x)) − V(x) V is said to be a Lyapunov function of (9) on U if (i) V is continuous, and (ii) ˙ V(x) ≤0 for all x∈ U Set

S0:= {x∈ U | V˙(x) =0}

LaSalle Invariant Principle(LaSalle, 1976) Let F be a continuous mapping onRn , and let V

be a Lyapunov function for F on a set U ⊆Rn If orbitγ : = { F n(x)| n ∈N}is contained in a

compact set in U, then its ω-limit set ω(γ ) ⊂ S0

V −1(c)for some c=c(x).This principle has been applied to the discrete-time cellular neural network (4) in (Chen &Shih, 2004a), where the Lyapunov function is constructed as

Proposition(Chen & Shih, 2004a) Let W be a positive-definite symmetric matrix and 0 ≤ μ ≤

1 Then V is a Lyapunov function for (4) onRn

Consider the condition

in (Chen & Shih, 2004b) The alternative conditions derived therein is considered moreapplicable and has been applied to study the convergence of the TCNN

LetN be the set of positive integers For a given continuous function F : N×Rn −→Rn, weconsider the non-autonomous difference equation

A sequence of points{x(t )}∞

1 inRnis a solution of (10) if x(t+1) =F(t, x(t)), for all t ∈N.

LetOx = {x(t ) | t ∈ N, x(1) = x}, be the orbit of x We say that p is aω-limit point of

Ox if there exists a sequence of positive integers{ t k } with t k → ∞ as k → ∞, such that

p=limk→∞x(t k) Denote byω(x)the set of allω-limit points of Ox

509Multidimensional Dynamics: From Simple to Complicated

LetNi represent the set of all positive integers larger than n i , for some positive integer n i Let

G be any set inRn 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, n1≥ n0, and

(iii) there exists a continuous function Q0 : G → R such that ˙V(t, x ) ≤ − Q0(x) ≤0

for all x∈ G and for all t ∈N2, n2≥ n1,

or

(iii) there exist a continuous function Q0: G →R and an equi-continuous family of

functions Q :N2× G →R such that limt→∞| Q(t, x ) − Q0(x)| =0 for all x∈ G and

˙

V(t, x ) ≤ − Q(t, x ) ≤0 for all(t, x ) ∈N2× G, n2≥ n1

Define

S0 = {x∈ G : Q0(x) =0}

Theorem(Chen & Shih, 2004a) Let V :N0× G →R be a Lyapunov function for (10) and let

Oxbe an orbit of (10) lying in G for all t ∈N0 Then limt→∞ Q(t, x(t)) =0, andω(x) ⊂ S0.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 { t k }∞

1, t k → ∞ as k → ∞ and x(t k ) → p as k → ∞ Since

V(t k, x(t k))is non-increasing and bounded below, V(t k, x(t k))approaches a real number as

k → ∞ Moreover, V(t k+1, x(t k+1)) − V(t1, x(t1)) ≤ −∑t k+1−1

t =t1 Q(t, x(t)), by (iii) Thus,

∑∞t =t1Q(t, x(t )) < ∞ Hence, Q(t, x(t )) → 0 as t → ∞, since Q(t, x(t )) ≥ 0 Notably,

Q(t k, x(t k )) → Q0(x(t k))as k →∞ This can be justified by observing that

| Q(t k, x(t k )) − Q0(x(t k ))|

≤ | Q(t k, x(t k)) +Q(t k, p) − Q(t k, p) +Q0(p) − Q0(p) − Q0(x(t k ))|

In addition, | Q0(x(t ))| ≤ | Q(t, x(t ))| + | Q(t, x(t )) − Q0(x(t ))| It follows from (iii) that

Q0(x(t k )) → 0 as k → ∞ Therefore, Q0(p) =0, since Q0is continuous Thus, p∈ S0

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) ismore achievable

We are interested in knowing whether if an orbit of the system (10) approaches an equilibriumstate or fixed point as time tends to infinity The structure ofω-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 theω-limit set ω(x)is contained in the set of fixed points of T, and the closure of the orbit

Oxis compact Thenω(x)is connected

This proposition can be extended to non-autonomous systems for which there exist limitingmaps Namely,

(A)There exists a continuous map F : Rn →Rnsuch that limt→∞ F(t, x ) −F(x) =0, for all

x∈Rn

Theorem (Chen & Shih, 2004b) Assume that (10) satisfies(A), the orbitOx is bounded,andω(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

Oxapproaches 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) = (F1(t, x),· · · , F n(t, x))where

F i(t, x) =αx i+ (1− γ)t ω ii(0)(y i − a 0i) +∑n

j=i ω ij y j+a i,

where y i = h i(x i), i =1,· · · , n and h i is defined in (6) The orbits of TCNN are then given

by the iterations x(t+1) = F(t, x(t))with components x i(t+1) = F i(t, x(t)) Note that

y=H(x) = (h1(x1),· · · , h n(x n))is a diffeomorphism on Rn Let W0denote the n × n matrix obtained from the connection matrix W with its diagonal entries being replaced by zeros Restated, W0=W −diag[W] For given 0< γ <1, choose 0< b <1 such that|1−γ

b | <1 Weconsider the following time-dependent energy-like function:

i=1

h i (x i)

0 h −1 i (η)dη+b t (11)

Theorem(Chen & Shih, 2004b) Assume that W0is a cycle-symmetric matrix, and either one

of the following condition holds,

(i) 0≤ α ≤ 1

3and W0+4(1− α)εI is positive definite;

(ii)13 ≤ α ≤ 1 and W0+8αεI is positive definite;

(iii)α ≥ 1 and W0+8εI is positive definite

Then there exists an n0 ∈ N so that V(t, x)defined by (11) is a Lyapunov function for theTCNN (5)-(7) onN0×Rn

3.2 Global consensus through a competing-component approach

Grossberg (1978) considered a class of competitive systems of the form

511Multidimensional Dynamics: From Simple to Complicated

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) Asuitable Lyapunov function for system (12) is not known, hence the Lyapunov method andLaSalle 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

x i(0) ≥ 0, for any i) evolves to a limiting pattern x(∞) = (x1(∞), x2(∞),· · · , x n(∞))with

0≤ x i(∞):=limt→∞ x i(t ) < ∞, under some conditions on a i , b i , C.

System (12) can be approximated, via Euler’s difference scheme or delta-operator circuitimplementation (Harrer & Nossek,1992), by

x i((k+1)δ) =x i(kδ) +δa i(x(kδ))[b i(x i(kδ )) − C(x(kδ))],

where one takes x i(kδ) as the k-th iteration of x i In this subsection, let us review thecompeting-component analysis for convergent dynamics reported in (Shih & Tseng, 2009).Consider the following discrete-time model,

x i(k+1) =x i(k) +βa i(x(k))[b i(x i(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

x i(k+1) =x i(k) +a i(x(k))[b i(x i(k )) − C(x(k))] (14)The results can then be extended to (13) First, let us introduce the following definition for theconvergent 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) ∈ Rn,

the limit x i(∞):=limk→∞ x i(k)exists, for all i=1, 2,· · · , n.

The following conditions are needed for the main results

Condition (A1): Each a i(x)is continuous, and

0< a i(x) ≤1, for all x∈Rn , i=1, 2,· · · , n.

Condition (A2): C(x)is bounded and continuously differentiable with bounded derivatives;

namely, there exist constants M1, M2, r jsuch that for all x∈Rn,

M1≤ C(x) ≤ M2,

0≤ ∂x ∂C

j(x) ≤ r j , j=1, 2,· · · , n.

Condition (A3): b i(ξ)is continuously differentiable, strictly decreasing and there exist d i >0,

l i , u i ∈ R such that for all i=1, 2,· · · n,

− d i ≤ b i (ξ ) <0, for allξ ∈R,

b i(ξ ) > M2, for ξ ≤ l i , and b i(ξ ) < M1, for ξ ≥ u i