Transiently chaotic neural networks with piecewise linear output functions

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Transiently chaotic neural networks with piecewise linear

output functions

Shyan-Shiou Chen

a

, Chih-Wen Shih

b,*,1

a_{Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC}

b_{Department of Applied Mathematics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu, Taiwan, ROC}

Accepted 2 January 2007

Communicated by Professor Ji-Huan He

Abstract

Admitting both transient chaotic phase and convergent phase, the transiently chaotic neural network (TCNN) pro-vides superior performance than the classical networks in solving combinatorial optimization problems. We derive con-crete parameter conditions for these two essential dynamic phases of the TCNN with piecewise linear output function. The confirmation for chaotic dynamics of the system results from a successful application of the Marotto theorem which was recently clarified. Numerical simulation on applying the TCNN with piecewise linear output function is car-ried out to find the optimal solution of a travelling salesman problem. It is demonstrated that the performance is even better than the previous TCNN model with logistic output function.

1. Introduction

Solving combinatorial optimization problems such as the travelling salesman problem (TSP) has been one of the main motifs for the development of neural networks[1]. While iterations in the classical Hopﬁeld model may be trapped at local minimum and fail to reach global minimum of the objective functions, the transiently chaotic neural network (TCNN) was developed to provide a global searching ability and thus achieved better performance in solving combi-natorial optimization problems[2–8]. The TCNN model employed in[2–4]is

xiðt þ 1Þ ¼ lxiðtÞ xiiðtÞ½yiðtÞ a0i þ a Xn j¼1;j–i xijyjþ vi " # ; ð1Þ jxiiðt þ 1Þj ¼ jð1 bÞxiiðtÞj ð2Þ

for i¼ 1; . . . ; n; t 2 N (positive integers), where xiis the internal state of neuron i; yiis the output of neuron i, which

corresponds to xithrough an output function; the output function adopted therein is the logistic function given by

*_{Corresponding author. Tel.: +886 3 5131209; fax: +886 3 5724679.}

E-mail address:[email protected](C.-W. Shih).

1

Partially supported by The National Science Council, and the MOEATU program, of ROC on Taiwan.

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y_iðtÞ ¼ 1=½1 þ expðxiðtÞ=eÞ; l is the damping factor of nerve membrane; xiiis the self-feedback connection weight; a0i

is the self-recurrent bias of neuron i; xijis the connection weight from neuron j to neuron i; viis the input bias of neuron

i; b with 0 < b < 1, is the damping factor for xii. Eq. (2)represents an exponential cooling schedule in the annealing

procedure.

There are chaotic phase and convergent phase for the TCNN with diﬀerent parameters. The purpose of this presen-tation is to provide in-depth understanding on the dynamics of the TCNN. In particular, we present detailed analysis and concrete parameter conditions for the chaotic and convergent phases of the TCNN with piecewise linear output functions: yiðtÞ ¼ geðxiðtÞÞ :¼ 2 þ xiðtÞ e þ 1 xiðtÞe 1 4; e >0: ð3Þ

Notably, piecewise linear output functions have been employed in various neural networks, cf.[9–13]. There are several advantages for the use of piecewise linear output function in the TCNN. In this presentation, it will be demonstrated that the mathematical description on the chaotic phase for the TCNN with output function(3)is more succinct than with the logistic one, cf.[14]. Subsequently, the parameter conditions derived are more concise. In addition, the model considered herein is a successful application of the Marotto’s theorem[15,16]in concluding chaotic dynamics for multi-dimensional and high multi-dimensional maps. Moreover, there also exists a Lyapunov function which is Lipschitz though not diﬀerentiable, for such a TCNN model. Theoretical conﬁrmations for convergence of iterations or evolutions of the network system to a steady state can then be established. Moreover, the parameters can be controlled so that the output component of the network tends to either exact one or zero. This consequence, for example, provides a lucid signal to further tune the parameters if the outcome is not feasible in the computation task of solving combinatorial optimization problems.

As an extension of the Li and York theorem on the one-dimensional maps, Marotto[15]established a significant theorem in confirming the chaotic dynamics for multi-dimensional systems. With presence of the so-called ‘‘snap-back repeller’’, the phase space possesses a topological structure which includes infinitely many periodic points and a scram-bled set. Very erratic behaviors of the system then occur, including the lack of global stability for solutions, and the existence of an uncountable collection of orbits which do not eventually approach any periodic points. The definition of snap-back repellers is further clarified for rigorousness recently in[16]. It will be demonstrated that the formulations and analysis in the considered model, i.e., the TCNN with piecewise linear output function, fit into the new definition of snap-back repellers in[16]pertinently.

We discuss the chaotic dynamics in Section2, and the convergent dynamics in Section3, for the TCNN. We will then make use of the TCNN properties to select suitable parameters for the computation of solving a travelling salesman problem. The task of solving the TSP via neural networks is to search for the minimizer of an objective function through iterations or evolutions of the network system. We arrange the setting for the application of the TCNN to the TSP in Section4. Some numerical simulations are performed in Section5.

2. Chaotic dynamics for the TCNN with piecewise linear output function

In this section, we plan to investigate the transiently chaotic behaviors for the TCNN with piecewise linear output functions. We consider the following system:

xiðt þ 1Þ ¼ lxiðtÞ þ x½yiðtÞ a0i þ

Xn j¼1

xijyjðtÞ þ vi; ð4Þ

where t2 N, and yiðtÞ ¼ geðxiðtÞÞ; i ¼ 1; . . . ; n, is the piecewise linear output function deﬁned in(3), and illustrated in

Fig. 1. Eqs.(1) and (2)are in fact a skew system of Eq.(4)over Eq.(2), cf.[17]. We denote by xðt þ 1Þ ¼ FðxðtÞÞ the iterations generated by Eq.(4)with x¼ ðx1; . . . ; xnÞ and F ¼ ðF1; . . . ; FnÞ. We shall analyze the chaotic dynamics of the

network from a geometrical observation on Eq. (4). These chaotic dynamics correspond to the transiently chaotic behaviors in the annealing process of the TCNN.

2.1. Single neuron maps

Let us consider the following family of single neuron maps. For a ﬁxed h > 0 and a02 R, let f1:R! R be a function

deﬁned by

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whereh 6 1 6 h. We set hi¼Pn_j¼1jxijj þ jvij, for each i ¼ 1; . . . ; n, and deﬁne the upper map ^fiand the lower map fi as ^ fiðnÞ ¼ ln þ x½geðnÞ a0i þ hi; fiðnÞ ¼ ln þ x½geðnÞ a0i hi: ð6Þ Then for i¼ 1; . . . ; n, fiðxiÞ 6 FiðxÞ 6 ^fiðxiÞ; for all x¼ ðx1; . . . ; xnÞ 2 Rn: ð7Þ

In the following discussions, we shall propose a sequence of parameter conditions which guarantee the existence of repelling ﬁxed points and chaotic behaviors for each f1in Eq.(5). These conditions are based on geometrical

observa-tions on the conﬁguraobserva-tions of the maps f1. In our considerations, there are two sets of parameter conditions labelled as

(P1-) and (P2-), which correspond to two types of conﬁgurations for single neuron maps f1, as depicted inFigs. 2a and b

and3, respectively.

For a ﬁxed e > 0, we partition the real line into the left (‘), middle (m), right (r) parts; namely,

X‘:¼ ð1; eÞ; Xm:¼ ½e; e; Xr:¼ ðe; 1Þ: ð8Þ

The following parameter conditions are concerned with the existence of ﬁxed point for every member in the family of single neuron maps(5).

(P1-i) l < minf1 ½xa0þ h=e; 1 ½xð1 a0Þ þ h=eg.

(P2-i) (a) x> 0, 0 <1l_x < 1 2e, 1l x e h xþ a0>0, 1l x eþ h xþ a0<1, (b) x <0,1l_x <1 2e, 1l x eþ h xþ a0>0, 1l x e h xþ a0<1.

We remark thatFig. 2a and b correspond to the conﬁgurations of f1satisfying (P1-i), whileFig. 3corresponds to the

conﬁguration of f1satisfying (P2-i). The following proposition relies upon observations on the graphs of f1and ge. Its

proof is sketched inAppendix A.

Proposition 1. Consider the single neuron maps f1withh 6 1 6 h defined in Eq.(5). (i) If the parametersðl; x; e; a0; hÞ

satisfy (P1-i), then there exists a fixed point xm1 2 Xmfor f1. (ii) If the parametersðl; x; e; a0; hÞ satisfy (P2-i) (a) or (P2

-i) (b), then there exist three fixed points xr

12 Xr, xm1 2 Xmand x‘12 X‘for f1.

Let us recall the notion of snap-back repeller[15,16]. Consider a map x# FðxÞ where x 2 Rn_{, and F is C}1_or

piece-wise C1. Suppose x is a ﬁxed point of F with all eigenvalues of DFðxÞ exceeding 1 in magnitude, and there exists a point x0–x in a repelling neighborhood of x, such that Fmðx0Þ ¼ x and detðDFmðx0ÞÞ–0, for some positive integer m. Then x

is called a snap-back repeller of F. A scenario for such a snap-back repeller is depicted inFig. 4. The chaotic dynamics induced by the presence of a snap-back repeller have been established in Marotto’s theorem quoted inAppendix B.

The following parameter conditions are concerned with the existence of snap-back repellers for every member in the family of single neuron maps(5).

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(P1-ii) l > 0;1_lðe xa0þ hÞ < le xð1 a0Þ h.

(P1-iii) l > 0,1lðe xð1 a0Þ þ hÞ < le xa0þ h.

Notably, (P1-ii) (resp. (P1-iii)) is formulated for ﬁnding preimages of ﬁxed points from the left (resp. right) part of the

graph of f1;Fig. 2a (resp.Fig. 2b) corresponds to condition (P1-ii) (resp. (P1-iii)).

Proposition 2. If the parametersðl; x; e; a0; hÞ satisfy ðP1-iÞ and ðP1-iiÞ or ðP1-iiiÞ, then the fixed point xm1 2 Xmis a

snap-back repeller for f1withh 6 1 6 h.

Proof. We develop a scheme for constructing preimages of a repelling fixed point, and confirm that this fixed point is a snap-back repeller. We only explain the case satisfying (P1-i) and (P1-ii), with configuration illustrated inFig. 2a.

Nota-bly, the following quantities ‘‘bi’’ correspond to the vertical coordinates of points ‘‘Bi’’ inFig. 2a. Since (P1-i) holds,

there exists a ﬁxed point xm

1 lying in Xm, for each f1withh 6 1 6 h. In addition, (P1-i) also implies lþx2e<1;

sub-sequently, there exists a point xð1Þ

1 2 X‘such that f1ðxð1Þ1 Þ ¼ x m

1, if l > 0. Let b2¼ ðe þ xa0 hÞ=l with l > 0, then Fig. 2. The conﬁgurations for the graphs of fh, fhand the relative positions of certain points which correspond to the inequality in

condition (P1-i). (a) B1:ð0; le xa0 hÞ, B2:ð0; ðe þ xa0 hÞ=lÞ, B3:ð0; le þ xð1 a0Þ þ hÞ, B4:ð0; eÞ. (b)

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fhðb2Þ ¼ e. It follows from (P1-ii) that b2> b3¼ fhðeÞ ¼ le þ xð1 a0Þ þ h. Thus there exists xð2Þ1 2 Xm such that

f1ðxð2Þ1 Þ ¼ xð1Þ1 . Let b1¼ le xa0 h, then b1¼ fhðeÞ > b4¼ e, due to (P1-i). Hence, there exists a sequence of

pointsfxðiÞ

1 ji ¼ 3; 4; 5 . . .g lying in region Xm, which are successive preimages of xm1 under f1. Consequently, xm1 is a

snap-back repeller lying in XmandfxðiÞ1 ji ¼ 1; 2; . . .g is a homoclinic orbit for f1. h

There are also conditions for the ﬁxed points inProposition 1(ii) to be snap-back repellers; namely (P2-ii) (a) leþ xð1 a0Þ h >1_l½ð₂1x xa0þ hÞ=ð1 l _2exÞ x þ xa0 h,

(b) leþ xa0 h > 1l½ð 1

2x xa0 hÞ=ð1 l x2eÞ þ xa0þ h.

Proposition 3. If the parametersðl; x; e; a0; hÞ satisfy either ðP2-iÞðaÞ and ðP2-iiÞðaÞ or ðP2-iÞðbÞ and ðP2-iiÞðbÞ, then the

fixed points inProposition 1(ii) are snap-back repellers for the single neuron maps f1withh 6 1 6 h.

Boundedness of iterations are necessary in the applications of TCNN. For the case 0 < l < 1, i.e., the situations of

Fig. 2a and b, interval [M, M] is a trapping region for map(5), provided M > 0 is large enough. For the case l <1, the following condition is needed:

Ω

Fig. 3. The conﬁgurations for the graphs of fh, fhand the relative positions of certain points which correspond to the inequality in

condition (P2-i). B1:ðe þ ðx 2hÞ=l; 0Þ, B2:ðle xa0i h; 0Þ, B3:ðle xa0iþ h; 0Þ, B4:ððð12x xa0i hÞ=

ð1 l x 2eÞ þ xa0iþ hÞ=l; 0Þ, B5:ðð12x xa0iþ hÞ=ð1 l 2exÞ; ð 1 2x xa0iþ hÞ=ð1 l 2exÞÞ, B6:ðð12x xa0i hÞ=ð1 l x2eÞ; ð1 2x xa0i hÞ=ð1 l x2eÞÞ, B7:ðð12x xa0iþ hÞ=ð1 l 2exÞ xð1 a0iÞ hÞ=l; 0Þ, B8:ðle þ xð1 a0iÞ h; 0Þ, B9: ðle þ xð1 a0iÞ þ h; 0Þ, B10:ðe ðx 2hÞ=l; 0Þ.

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(P2-iii) eþ_l1ðx 2hÞ < minfle xa0 h; le xð1 a0Þ hg.

Proposition 4. Assume that the parameters ðl; x; e; a0; hÞ satisfy l < 1, ðP2-iÞðaÞ and ðP2-iiiÞ, then interval

½e þx

l2hl;e xlþ2hl is a trapping region for the single neuron maps f1withh 6 1 6 h.

2.2. Multi-dimensional chaotic neural networks

We shall apply Propositions2 and 3for single neuron maps to establish the chaotic behaviors for the n-dimensional neural networks(4). Note that Rn _{can be written as the direct sum of the following subsets:}

Xq1qn ¼ fðx1; . . . ; xnÞ 2 R n_jx

i2 Xqi; qi¼ \r"; \m"; \‘"; i ¼ 1; . . . ; ng; ð9Þ

as illustrated inFig. 5for n = 2. Xmmis called the interior region; each Xq1qn, with qi¼ \‘"; \r", for all i, is called a

saturated region; each Xq1qn, with qi¼ \‘", or ‘‘r’’, for some i, and qj¼ \m" for some j, is called a mixed region.

We derive the following theorem on the existence of ﬁxed point for system(4). Recall hi¼P n

j¼1jxijj þ jvij.

Theorem 1. If the parametersðl; x; e; a0i; hiÞ, i ¼ 1; . . . ; n, satisfy ðP1-iÞ (resp. ðP2-iÞðaÞ), then there exists one fixed point

(resp. 3n_{fixed points) in X}

mm (resp. the 3nregions Xq₁qn defined in(9)) for system(4).

Proof. We prove the case that (P2-i)(a) holds for parametersðl; x; e; a0i; hiÞ, i ¼ 1; . . . ; n. For a given ð~n1; . . . ; ~nnÞ 2 Rn,

by Proposition1(ii), there exist points n‘ i 2 X‘, n m i 2 Xm, n r i 2 Xr such that n_i ¼ lni þ x½geðn iÞ a0i þ Xn j¼1 xijgeð~njÞ þ vi; ð10Þ

where ¼ \‘"; \m"; \r" for each i. Restated, each n

i is a ﬁxed point of the one-dimensional map ni# lniþ

x½geðniÞ a0i þP n

j¼1xijgeð~njÞ þ vi. In fact, each of these ðn1; . . . ;n

n) lies in a compact proper subset X q1qn of

Xq1qn, according to our formation in Section 2.1. Consider a ﬁxed region X q1qn for certain qi¼ \‘"; \m"; \r" (i¼ 1; . . . ; n). Let H : X_q 1qn# X q1qnbe deﬁned by Hð~n1; . . . ; ~nnÞ ¼ ðn q1 1; . . . ;n qn

nÞ. We shall show that there exists a ﬁxed

point for H. Deﬁne G : X_q

1qn X q1qn! X q1qn, G¼ ðG1; . . . ; GnÞ, by Gið~x; xÞ ¼ xi lxi x½geðxiÞ a0i Xn j¼1 xijgeð~xjÞ vi;

where ~x¼ ð~x1; . . . ; ~xnÞ, i ¼ 1; . . . ; n. Notably, Gðex; Hð~xÞÞ ¼ 0 for every ~x 2 Xq1qn, by Eq.(10). Now,

oG

ox¼ diag½v1; . . . ;vn;

Fig. 5. Illustration of Xq1q2 in R

2_{, where q}

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where vi¼ 1 l xg0eðxiÞ, i ¼ 1; . . . ; n. Note that ge0 ¼ 0 in regions Xr, X‘, and g0e¼ 1

2ein the interior region Xm. For

x¼ ðx1; . . . ; xnÞ 2 Xq1qn, we have vi¼ 1 l x

2e or 1 l for each i, which are nonzero due to

0 <ð1 lÞ=x < 1=ð2eÞ. It follows that H is a C1

function on region X_q

1qn, by the implicit function theorem. Thus,

there exists one ﬁxed point x of H in X_q

1qn, which is also a ﬁxed point of system(4), by the Brouwer’s ﬁxed point

the-orem. Consequently, there are 3n_{ﬁxed points of system}₍₄₎_{in R}n_. _h

Theorem 2. If the parametersðl; x; e; a0i; hiÞ, i ¼ 1; . . . ; n, satisfy ðP1-iÞ, and ðP1-iiÞ or ðP1-iiiÞ, then there exists a

snap-back repeller in the interior region Xmm for system(4).

Proof. By Theorem1, there exists a ﬁxed point x¼ ðx1; . . . ; xnÞ in Xmmsuch that FiðxÞ ¼ xi, i¼ 1; . . . ; n. To ﬁnd

pre-images of x under F for the system, we need to solve FiðxÞ ¼ xi; i¼ 1; . . . ; n. To achieve this, recall Eq. (7) that

fiðxiÞ 6 FiðxÞ 6 ^fiðxiÞ; i ¼ 1; . . . ; n: We observe that for a given ð~x1; . . . ; ~xnÞ 2 X‘‘, one can always ﬁnd

ðx1; . . . ; xnÞ 2 X‘‘satisfying

xi¼ lxiþ x½geðxiÞ a0i þ

Xn j¼1

xijgeð~xjÞ þ vi; ð11Þ

if the parametersðl; x; e; a0i; hiÞ; i ¼ 1; . . . ; n, satisfy conditions (P1-i), and (P1-ii), according to our previous

formula-tions and Proposition2. Deﬁne a map G : X‘‘ X‘‘! Rnby

Gið~x; xÞ ¼ xi lxi x½geðxiÞ a0i

Xn j¼1

xijgeð~xjÞ vi;

where G¼ ðG1; . . . ; GnÞ. Then ðoG=oxÞð~x; xÞ ¼ diag½v1; . . . ;vn, where vi¼ l xg0eðxiÞ, i ¼ 1; . . . ; n. Thus,

detðoG=oxÞð~x; xÞ–0: By similar arguments as the ones in the proof of Theorem1, there exists a point x2 X‘‘such

that Gðx; xÞ ¼ 0. Denoting xð1Þ_{¼ x, it follows that Fð}_xð1Þ_{Þ ¼}_{x. Successively, by similar arguments, we also obtain}

xð2Þ in Xmm with Fiðxð2ÞÞ ¼ x

ð1Þ

i ; i¼ 1; . . . ; n, and then a sequence fxðkÞjk P 3; k 2 Ng in Xmm with

Fiðxðkþ1ÞÞ ¼ xðkÞi ; i¼ 1; . . . ; n. Furthermore, the entries for the Jacobian matrix of F in Xmmare

½DF_ij¼ l þxþ xii 2e ; if i¼ j; ¼ xij 2e; if i–j: Notably, lþx 2eþ 1 2emaxi¼1;...;nðP n

j¼1jxijjÞ < 1, due to conditions (P1-i), and (P1-ii). Thus, the absolute values of all

eigenvalues of DF are greater than one, by the Gerschgorin’s theorem. Therefore, F is expanding in Xmm. Hence,

the sequence fxðkÞ_{jk ¼ 1; 2; . . .g approaches the ﬁxed point in X}

mm and thus the ﬁxed point x is a snap-back

repeller. h

Theorem 3. If the parametersðl; x; e; a0i; hiÞ, i ¼ 1; . . . ; n, satisfy ðP2-iÞðaÞ, and ðP2-iiÞðaÞ or ðP2-iiÞðbÞ, then there exists a

snap-back repeller for the multi-dimensional neural network(4).

The proof employs Proposition3and resembles the one of Theorem2.

3. Stability analysis for the TCNN

We plan to study the asymptotic behaviors for the TCNN in this section. Analogous to the TCNN with logistic out-put function, there also exists a time-dependent Lyapunov function for the TCNN with piecewise linear outout-put function

(3). We shall derive this Lyapunov function in Section 3.1and employ the non-autonomous discrete-time LaSalle’s invariant principle to analyze the convergence of the TCNN. In Section3.2, we analyze the instability of the ﬁxed points in the interior region and the mixed regions so that the evolutions of TCNN, with suitably chosen parameters, settle at saturated regions from almost all initial points.

3.1. Lyapunov function for the TCNN

We consider the TCNN with more general cooling schedule, namely xiðt þ 1Þ ¼ lxiðtÞ þ ð1 bÞ qðtÞ x½geðxiðtÞÞ a0i þ Xn j¼1 xijgeðxjðtÞÞ þ vi; ð12Þ

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where i¼ 1; . . . ; n, 0 < b < 1; ge is deﬁned in (3); qðtÞ satisﬁes the condition that there exists an n12 N such that

qðtÞ t P 0 for all t > n1. The standard cooling process simply takes q(t) = t. Correspondingly, we deﬁne

Fiðt; xÞ ¼ lxiþ ð1 bÞqðtÞx½geðxiÞ a0i þ

Xn j¼1

xijgeðxjÞ þ vi: ð13Þ

Let us recall the notion of Lyapunov function for discrete-time non-autonomous system, cf.[18]. Let N be the set of positive integers. For a given continuous function F : N Rn_{! R}n_{, we consider the non-autonomous dynamical}

equation

xðt þ 1Þ ¼ Fðt; xðtÞÞ: ð14Þ

A sequence of pointsfxðtÞjt ¼ 1; 2; . . .g in Rn_{is a solution of}₍₁₄₎_{if xðt þ 1Þ ¼ Fðt; xðtÞÞ, for all t 2 N. Let O}

x¼ fxðtÞjt 2

N; xð1Þ ¼ xg be the orbit of x. p is an x-limit point of Oxif there exists a sequence of positive integersftkg with tk! 1

as k! 1, such that p ¼ limk!1xðtkÞ. Denote by xðxÞ the set of all x-limit points of Ox. Let nibe a positive integer, and

let Ni denote the set of all positive integers larger than ni. Denote by X the closure of X # Rn. For a function

V :N0 X ! R, we deﬁne _V ðt; xÞ ¼ V ðt þ 1; Fðt; xÞÞ V ðt; xÞ so that if fxðtÞg is a solution of Eq. (14), then

_

Vðt; xðtÞÞ ¼ V ðt þ 1; xðt þ 1ÞÞ V ðt; xðtÞÞ. V is said to be a Lyapunov function for(14)if (i) each Vðt; Þ is continuous, and (ii) for each p2 X, there exists a neighborhood U of p such that V ðt; xÞ is bounded below for x 2 U \ X and t 2 N1,

n1P n0, and (iii) there exists a non-degenerate continuous function Q0:X! R such that _V ðt; xÞ 6 Q0ðxÞ 6 0 for all

x2 X and for all t 2 N2, n2P n1, or (iii)0 there exist a non-degenerate continuous function Q0:X! R and an

equi-continuous family of functions Qðt; Þ : X ! R such that limt!1jQðt; xÞ Q0ðxÞj ¼ 0 for all x 2 X and _V ðt; xÞ 6

Qðt; xÞ 6 0 for all ðt; xÞ 2 N2 X, n2P n1. Deﬁne

S0¼ fx 2 X : Q0ðxÞ ¼ 0g: ð15Þ

We recall the following theorem for the asymptotic behaviors of solutions to Eq.(14).

Theorem 4 [19]. Let n02 N, V : N0 X ! R be a Lyapunov function for Eq.(14)and Oxbe an orbit of Eq.(14)lying in

X for all t2 N1. Then limt!1Qðt; xðtÞÞ ¼ 0, and xðxÞ S0.

Further properties for xðxÞ can be derived if there exists a limiting map FðxÞ for Fðt; xÞ, i.e., there exists a continuous map F : Rn_{! R}n_{such that lim}

t!1jFðt; xÞ FðxÞj ¼ 0, for all x 2 Rn. The following theorem generalizes Theorem 2.2 in

[19].

Theorem 5. Assume that FðxÞ is the limiting map of Fðt; xÞ, and the orbit Oxis bounded, then xðxÞ is positively invariant

under F; moreover, if the x-limit set xðxÞ of Oxis contained in the set of fixed points of F, then xðxÞ is connected. Under

this circumstances, if F has only finitely many fixed points, then the orbit Oxapproaches some single fixed point of F, as time

tends to infinity.

Let us deﬁne the limiting mapping F¼ ðF1; . . . ; FnÞ of F in(13)by

FiðxÞ ¼ lxiþ

Xn j¼1

xijgeðxjÞ þ vi; i¼ 1; . . . ; n: ð16Þ

It follows thatjFðxÞ Fðt; xÞj converges uniformly to zero in x 2 Rn _{as t}_{! 1, since j1 bj < 1, jFðxÞ Fðt; xÞj}2

¼ Pn i¼1jð1 bÞ qðtÞ_xðy i a0iÞj 2

andPn_i¼1jxðyi a0iÞj 2

is bounded. Let us consider the function

Vðt; xÞ ¼ 1 2 Xn i¼1 Xn j¼1 xijyiyj Xn i¼1 viyi ðl 1Þe Xn i¼1 ðy2 i yiÞ þ c t_; _ð17Þ

where y_i¼ geðxiÞ, for i ¼ 1; . . . ; n, 0 < c < 1. Note that V is globally Lipschitz, but not C1. Let W ¼ ½xijnn.

Theorem 6. If 0 6 l 6 1, e > 0, j1b_c j < 1 and the matrix W þ 2eð1 lÞI is positive-definite, then there exists n22 N; n2> n1so that Vðt þ 1; xðt þ 1ÞÞ 6 V ðt; xðtÞÞ for t P n2and V is a Lyapunov function for system(12)on N2 Rn.

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Proof. Vðt þ 1; xðt þ 1ÞÞ V ðt; xðtÞÞ ¼ 1 2 Xn i¼1 Xn j¼1 xijDtyiDtyj ðl 1Þe Xn i¼1 ½y2 iðt þ 1Þ y 2 iðtÞ yiðt þ 1Þ þ yiðtÞ þX n i¼1 ½lxiðtÞ xiðt þ 1ÞDtyiþ Xn i¼1 ð1 bÞqðtÞx½yiðtÞ a0iDtxiþ ctþ1 ct ¼ 1 2 Xn i¼1 Xn j¼1 xijDtyiDtyj ðl 1Þe Xn i¼1 ½y2 iðt þ 1Þ y 2 iðtÞ yiðt þ 1Þ þ yiðtÞ þ lX n i¼1 ½xiðtÞ xiðt þ 1ÞDtyiþ ðl 1Þ Xn i¼1 xiðt þ 1ÞDtyiþ Xn i¼1 ð1 bÞqðtÞx½yiðtÞ a0iDtyiþ ctþ1 c t_¼1 2 Xn i¼1 Xn j¼1 xijDtyiDtyj l Xn i¼1 ½xiðt þ 1Þ xiðtÞDtyi þ ðl 1ÞX n i¼1 fxiðt þ 1ÞDtyi e½y 2 iðt þ 1Þ y 2 iðtÞ þ eDtyig þ Xn i¼1 ð1 bÞqðtÞx½yiðtÞ a0iDtyiþ ctþ1 c t : ð18Þ

where Dty¼ ðDty1; . . . ;DtynÞ with Dtyi¼ geðxiðt þ 1ÞÞ geðxiðtÞÞ. Let us compute the term in the bracket of(18). Set

Ci¼ CiðxiðtÞ; xiðt þ 1ÞÞ :¼ xiðt þ 1ÞDtyi e½y 2 iðt þ 1Þ y 2 iðtÞ þ eDtyi ¼ FiðxðtÞÞDtyi eDtyi½geðFiðxðtÞÞÞ þ geðxiðtÞÞ 1:

We shall justify that CiP eðDtyiÞ 2

inAppendix C. On the other hand,½xiðt þ 1Þ xiðtÞ½geðxiðt þ 1ÞÞ geðxiðtÞÞ P 0,

due to that ge is non-decreasing for e > 0. Let eW ¼ W þ 2eð1 lÞI. Then,

Vðt þ 1; xðt þ 1ÞÞ V ðt; xðtÞÞ 6 1 2ðDtyÞ T _e WDtyþ Mð1 bÞqðtÞþ ctþ1 ct; where M¼1 nmax n

i¼1fsupt2N;y2Rnjxðy_i a0iÞDtyijg: Since j 1b

c j < 1 and qðtÞ t P 0 for all t > n1, there exists n22 N such

that Mð1 bÞqðtÞþ ctþ1_ct _{¼ c}t_{½Mð1 bÞ}qðtÞt_ð1b c Þ t þ c 1 < ct_½Mð1b c Þ t þ c 1 < 0 if t P n2. Hence, Vðt þ 1; xðt þ 1ÞÞ V ðt; xðtÞÞ 6 1 2ðDtyÞ T _e WDty 6 0: Deﬁne Qðt; xÞ ¼ 1 2ðDtyÞ T _e WDty; Q0ðxÞ ¼ 1 2ðDyÞ T _e WDy; where Dy¼ ðDy1; . . . ;DynÞ, Dyi¼ geðFiðxÞÞ geðxiÞ. Notice that

lim

t!1jQðt; xÞ Q0ðxÞj ¼ 0

for all x2 Rn_{. Thus, V}_{ðt þ 1; xðt þ 1ÞÞ 6 V ðt; xðtÞÞ for t P n}

2. h

Accordingly, under the assumption ofTheorem 6, xðxÞ S0¼ fz : geðFiðzÞÞ geðziÞ ¼ 0; i ¼ 1; . . . ; ng

for any x2 Rn_{, due to}_{Theorem 4}_{. We observe that the only points which are positively invariant under F are ﬁxed}

points of F, in respecting the definition of the limiting map F defined in Eq.(16). Thus, xðxÞ is connected and contained in the set of fixed points of F. If, furthermore, F has only finitely many fixed points, then every orbit Oxtends to a single

ﬁxed point of F.

3.2. Instability for ﬁxed points in the interior and mixed regions

For the previous TCNN models in the literatures, there is a situation that the output component tends to a value which is neither close to one nor to zero. For example, a numerical simulation for the TCNN with logistic output func-tion exhibiting such a phenomenon is illustrated inFig. 6.

In this section, we discuss the parameter conditions so that almost every trajectory generated by F tends to a stable ﬁxed point of F, which lies in a saturated region. Restated, starting from almost all initial points, every entry of the

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output matrix½yijðtÞnntends to 0 or 1 as t! 1. This convergence then corresponds to a possible route as the TCNN is

applied to solve the TSP.

For a mixed region Xq1...qn, we deﬁne the index sets

J0¼ fi 2 f1; . . . ; ngjjxij 6 eg; J1¼ fi 2 f1; . . . ; ngjjxij > eg;

and denote byjJ0j, jJ1j their cardinalities. Let W0be thejJ0j jJ0j submatrix of W, which is obtained by deleting the jth

row and the jth column of W, for all j2 J1. The linear part DF of F restricted to a mixed region Xq1qn, after

rearrang-ing the indices, can be represented in the matrix form: DF¼ B

0 D

; ð19Þ

where * is an jJ1j jJ0j matrix, B ¼ lI1and D¼ lI0þ2e1W0with I1(resp. I0) being the identity matrix of sizejJ1j (resp.

jJ0j). Thus, the eigenvalues of matrix(19)are the eigenvalues of B and D. Note that W0is symmetric. If W0has a

non-zero eigenvalue, which holds generically, then there exists an eigenvalue of DF with moduli greater than one, provided e is suﬃciently small. By similar reasoning, it can be seen that the ﬁxed point in the interior region Xmmis unstable, if e is

suﬃciently small. On the other hand, a ﬁxed point in a saturated region is always stable if 0 < l < 1, due to that the linear part of F restricted to a saturated region is lInn.

Moreover, it is possible to impose a condition so that the ﬁxed points of F in certain mixed regions do not exist. For example, let us consider a mixed region Xq1qnwithjJ0j ¼ 1. Without loss of generality, we suppose that qk¼ \m". It is

computed that if x¼ ðx1; . . . ; xnÞ is a ﬁxed point of F in this region, then

xk¼ Pn j¼1xkjyjþ vk 1 l ; where yj¼ 1 or 0. Hence, if j Pn

j¼1xkjyjþ vkj=j1 lj > e, which is more likely to hold for smaller e, there does not exist

any ﬁxed point of F in this region. Our discussions above have demonstrated that almost every trajectory generated by F converges to a ﬁxed point of F in a saturated region, under conditions described in Section3.1and that e > 0 is small enough.

4. Application to the TSP

We will make use of the properties derived in previous sections to choose suitable parameters for the TCNN to solv-ing the TSP. The task of solvsolv-ing the TSP via neural networks is to search for the minimizer of an objective function through evolutions of the network system. The objective function proposed by Hopﬁeld and Tank [1] consists of two parts which are both quadratic functions, namely,

E¼ E1þ Ed; ð20Þ where E1¼ c1 2 Xn i¼1 Xn j¼1 y_ij 1 !2 þX n j¼1 Xn i¼1 y_ij 1 !2 2 4 3 5; Ed¼ c2 2 Xn i¼1 Xn k¼1 Xn j¼1 ðy_kðjþ1Þþ y_kðj1ÞÞyijdik;

Fig. 6. An example that the TCNN with the logistic output function has an infeasible solution, i.e., there exists an output entry yi

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y_ij2 ½0; 1; i; j ¼ 1; . . . ; n, is the probability for the ith city to be visited the jth time, and dikis the distance between city i

and city k. We call Y ¼ ½yijnn the output matrix. Note that E1attains its minimal value, i.e., zero, at a permutation

matrix. Edis related to the tour distance of the TSP. Therefore, 2Ed=c2gives the tour length when Y is a permutation

matrix. In our setting of solving the TSP through the TCNN computations, the meaningful asymptotic output matrix should be a permutation matrix. Notably, if the global minimum of E is attained at a permutation matrix Y, then it is attained at all permutations to Y, which give the same value to E.

To apply the TCNN to the TSP, we change the setting of the TSP with two-dimensional indices into the one-dimen-sional form. Restated, by letting sði; jÞ ¼ j þ ði 1Þn, where n is the number of cities for the TSP, Eq.(20)becomes

EðyÞ ¼ 1 2yW y

T_2c

1In2_n2yþ c₁n: ð21Þ

Herein, In2_n2 is the identity matrix of size n2 n2, y¼ ðy₁; . . . ; y_sði;jÞ; . . . ; y_n2Þ and

W ¼ c1½Inn 1nnþ 1nn Inn c2D B; ð22Þ

1nnis the matrix whose entries are all one, D¼ ½dij T

and B¼ ½bij with bi;j¼ 0 except that bi;iþ1¼bi;i1¼b1;n¼bn;1¼ 1;

the n2_n2_{block matrix A}_{B is deﬁned by the formula ½A B}

ij¼ ½aijB, where A = [aij] and B = [bij]. We consider the

TCNN

xiðt þ 1Þ ¼ lxiðtÞ þ ð1 bÞqðtÞx½yiðtÞ a0i þ

Xn2 j¼1

WijyjðtÞ þ 2c1; ð23Þ

where W ¼ ½Wij :¼ W diag½W =2, i ¼ 1; . . . ; n2. According to previous discussions, there is a Lyapunov function for

Eq.(23): Vðt; yÞ ¼ 1 2 Xn2 i¼1 Xn2 j¼1 Wijyiyj Xn2 i¼1 2c1yi ðl 1Þe Xn2 i¼1 ðy2 i yiÞ þ ct; ð24Þ

where y¼ ðy1; . . . ; yn2Þ. The conditions for chaotic and convergent dynamical phases of the TCNN are all computable.

The range of the parameters satisfying these conditions can also be depicted numerically. We shall describe how we choose parameters in the numerical simulations for solving a TSP via the TCNN in the next section.

5. Numerical simulations

We present numerical simulations for solving the TSP via the evolutions of the TCNN(23). We employ the ten-city TSP problem by Hopﬁeld and Tank to illustrate the performance. The ten-city data is illustrated inFig. 7.

Let us describe how to choose parameters in the numerical simulations. We ﬁrst take an l with 0 < l < 1 for bound-edness of iterations for the TCNN. Set x = 0, and take e; a0i; hi; i¼ 1; . . . ; n2, so that the TCNN with these parameters is

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in convergent phase, where hi¼Pn 2

k¼1fjWikj þ 2jc1jg. We then let jxj increase from 0 to see whether if parameters

ðl; e; x; a0i; hiÞ enter the chaotic regime. These computations can be assisted by a computer programming. In our

illus-tration, the parameters are set as l = 0.9, b = 0.005, e = 0.01, a0i= 0.65, x =0.08, c1¼ 0:015, c2= 0.015 and q(t) = t.

The iterations are demonstrated inFig. 8, with the synchronously updating mode, in accordance with our theory. We also employ the cyclic updating iterations in Fig. 8b. Our simulation indicates that the best route for this TSP is 2! 3 ! 1 ! 4 ! 5 ! 6 ! 7 ! 8 ! 9 ! 10 ! 2. The other best route such as 8 ! 7 ! 6 ! 5 ! 4 ! 1 ! 3! 2 ! 10 ! 9 ! 8 and 9 ! 10 ! 2 ! 3 ! 1 ! 4 ! 5 ! 6 ! 7 ! 8 ! 9 also have the same value E ¼ 2:7257. In fact, all of them represent the same loop. Under these parameters, 100% of 3000 initial conditions lead to the optimal route with E¼ 2:7257 in the evolutions of system(23).

6. Conclusion

In this presentation, we have analyzed the dynamics for the chaotic phase and the convergent phase for the TCNN with piecewise linear output function. The analysis provides concrete and computable parameter conditions for the respective dynamic phase. The investigation makes use of the structure of piecewise linear output function. The param-eter conditions thus derived are much simpler than the ones for logistic output functions. We have also observed that the TCNN with piecewise linear output function has even better performance than with the logistic output function in the applications. For example, with suitably chosen parameters, the output component of the TCNN with piecewise linear output function tends to either one or zero, with the objective function decreasing, as the system evolves. If the output matrix does not tend to a permutation matrix, one can enlarge slightly the parameter c1in Eq.(22). The

investigation demonstrates a skillful employ of the Marotto’s theorem for the n-dimensional TCNN map and is expected to contribute toward applications of the TCNN as an annealing machine in solving various combinatorial optimization problems.

Appendix A. Proof ofProposition 1

nis a fixed point of the single neuron map f1,h 6 1 6 h if and only if n¼ lnþ x½geðnÞ a0 þ 1. To find a fixed

point in Xmfor the single neuron map f1, we seek for nwithe 6 n 6 esatisfying½ð1 lÞn 1=x þ a0¼ geðnÞ.

Equiv-alently, we look for an intersection of equations g¼ ½ð1 lÞn 1=x þ a0 and g¼ geðnÞ in Xm. Notably, (P1-i):

l <minf1 ðxa0þ hÞ=e; 1 ½xð1 a0Þ h=eg is equivalent to fhðeÞ > e and fhðeÞ < e. Thus, under condition

(P1-i), there exists a ﬁxed point of f1in Xm, with reference ofFig. 2a. Let x > 0 and L1ðxÞ :¼ ½ð1 lÞx 1=x þ a0. Then

Lh(resp. Lh) is the upper (resp. lower) line inFig. 9. Note that LhðxÞ 6 L1ðxÞ 6 LhðxÞ with h 6 1 6 h for all x 2 R. If Fig. 8. The iterations of components x1;1; . . . ; x1;10, and E for the TCNN in solving a 10-city TSP. The iteration is synchronous

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LhðeÞ 6 1 and LhðeÞ P 0, then it is obvious that there exists a ﬁxed point n2 Xm for L1withh 6 1 6 h. Actually,

under this circumstances, there exist three ﬁxed points of f1forh 6 1 6 h. h

Appendix B. Marotto’s Theorem

Consider a dynamical system: x# FðxÞ, x 2 Rn_{and F is in C}1_ðRn_;Rn_{Þ or piecewise C}1_{. Suppose that}

x is a ﬁxed point of F with all eigenvalues of DFðxÞ exceeding 1 in magnitude, and suppose there exists a point x0–x in a repelling

neigh-borhood of x, such that Fmðx0Þ ¼ x and detðDFmðx0ÞÞ–0, for some 1 < m 2 N. Then x is called a snap-back repeller[16]

of F, as depicted inFig. 4. If F has a snap-back repeller, then the system of F is chaotic in the following sense: (i) There exists a positive integer m0such that F has p-periodic points for each integer p P m0. (ii) There exists a scrambled set,

that is, an uncountable set L containing no periodic points so that the following pertains: (a) FðLÞ L; (b) for every y2 L and any periodic point x of F,

lim sup

m!1

kFm_{ðyÞ F}m_{ðxÞk > 0;}

(c) for every x, y2 L with x–y, lim sup

m!1

kFm_{ðyÞ F}m_{ðxÞk > 0;}

(iii) There exists an uncountable subset L0of L such that for every x, y2 L0,

lim inf

m!1 kF

m_{ðyÞ F}m_{ðxÞk ¼ 0:}

Marotto ﬁrst reported this theorem in 1978. The theorem has been applied to justify chaotic behaviors for several dynamical systems[20]. The original deﬁnition of snap-back repeller employed the notion that there exists a real number r >0 and x02 Bðx; rÞ with x0–x, called snap-back point, such that all eigenvalues of DFðxÞ exceed unity in norm for all

x2 Bðx; rÞ. This condition does not imply that x0lies in an expanding neighborhood of x. Such a deﬁnition then leads

to an insuﬃciency in the proof of the theorem. The deﬁnition of snap-back repeller is then revised recently[16]. There were other related discussions in[20,21].

Appendix C. Supplemental proof forTheorem 6

When no confusion arises, we will omit the variable t in Ci. We arrange the justiﬁcation into three cases: FiðxÞ P e,

jFiðxÞj < e and FiðxÞ 6 e. Case I: FiðxÞ P e, Ci¼ FiðxÞDtyi eDtyi geðxiÞ ¼ Dtyi½FiðxÞ egeðxiÞ ¼ ½1 geðxiÞ½FiðxÞ egeðxiÞ P ½1 geðxiÞ½e egeðxiÞ ¼ e½1 geðxiÞ 2 ¼ eðDtyiÞ 2 :

Fig. 9. The conﬁguration explains condition (P1-i)(a). The quantities 1l_xeþh_xþ a0 and 1l_x ðeÞ _xhþ a0 correspond to the

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Case II: FiðxÞ 6 e, Ci¼ FiðxÞgeðxiÞ þ egeðxiÞ½geðxiÞ 1 ¼ geðxiÞ½FiðxÞ e þ egeðxiÞ P eðgeðxiÞÞ2¼ eðDtyiÞ 2 : Case III: jFiðxÞj < e, Ci¼ FiðxÞDtyi eDtyi 1 2eFiðxÞ þ 1 2þ geðxiÞ 1 ¼1 2FiðxÞDtyiþ e 2Dtyi eDtyi geðxiÞ ¼1 2½FiðxÞ þ eDtyi eDtyi geðxiÞ ¼ 1 2eFiðxÞ þ 1 2 geðxiÞ 1 2ðFiðxÞ þ eÞ egeðxiÞ ¼: :

For case III, we consider three subcases: xiP e,jxij < e and xi6e.

Subcase III(a): xiP e, ¼ 1 2eFiðxÞ þ 1 2 1 1 2ðFiðxÞ þ eÞ e ¼ 1 4e½FiðxÞ e 2 ¼ eðDtyiÞ 2 : Subcase III(b): xi6e,

¼ 1 2eFiðxÞ þ 1 2 1 2ðFiðxÞ þ eÞ ¼ 1 4e½FiðxÞ þ e 2 ¼ eðDtyiÞ 2 : Subcase III(c):jxij < e, ¼ 1 2eFiðxÞ þ 1 2 1 2exi 1 2 ₁ 2ðFiðxÞ þ eÞ e 1 2exiþ 1 2 ¼ 1 4e½FiðxÞ xi 2 ¼ eðDtyiÞ 2 : References

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