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Some dynamical behavior of discrete Nagumo equation
Chaos, Solitons and Fractals 14 (2002) 1457–1464 www.elsevier.com/locate/chaos
Some dynamical behavior of discrete Nagumo equation q Yu-Rong Liu a b
a,*
, Zeng-Rong Liu
b
Department of Mathematics, Yangzhou University, Yangzhou 225002, China Department of Mathematics, Shanghai University, Shanghai 200436, China Accepted 26 March 2002 Communicated by W.M. Zheng
Abstract The spatiotemporal dynamics can be effectively studied by continuation from an anti-integrable limit. By using the anti-integrability method we consider the dynamical behavior of discrete Nagumo equation, prove the existence of breather and discuss the spatial disorder of the stationary solutions in the system. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently, the study of complicated dynamical systems is attracting considerable attention of many researchers and there are many results on it [1]. Because of complexity of the subject, however, the methods available are far from enough for studying the dynamics of the systems, and the numerical simulation is still a powerful weapon for understanding of complicated systems. Coupled map lattices (CMLs) [2], as the models of spatially extended systems, already have gained a lot of popularity among the researchers because of their convenience for computer simulations as well as for the rigorous mathematical analysis. On the other hand, CMLs yield interesting dynamical systems in their own right, which generally show a wide diversity of patterns including stationary structures, traveling waves, localized structures, etc., and also reveal various space–time phenomena such as spatiotemporal intermittency, spatiotemporal chaos, etc. [1–4]. Now CMLs serve as one of the most efficient tools to analyze space–time phenomena in extended systems. Here we consider the following CML: ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ uj ¼ uj þ k uj1 2uj þ ujþ1 þ af uj ; ð1Þ ðnÞ
where j 2 Z denotes a spatial coordinate and n 2 Z denotes time, uj 2 R; a; k > 0, and f ðuÞ ¼ uðu aÞð1 uÞ;
0 < a < 1:
This is a discrete analog of Nagumo equation ou o2 u ¼ D 2 þ f ðuÞ; ot ox
x 2 R1 ; t 2 Rþ ;
ð2Þ
where D is a positive constant. Eq. (2) is a well-known P.D.E., which is used as a model for the spread of genetic traits and for the propagation of nerve pulses in a nerve axon, neglecting recovery.
q
This work is supported by National Natural Science Foundation of China, 10171061. Corresponding author. E-mail address: [email protected] (Yu.-R. Liu).
*
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 8 2 - 6
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There are some results known for Eq. (1). In [5,6] authors discussed the existence of the spatial topological chaos and the traveling waves. It was shown that no traveling waves appear in Eq. (1) when coupling is weak, but there exist traveling waves if coupling is strong. In this paper, we consider breathers and the spatial disorder in Eq. (1) using anti-integrability method [7,8]. A breather of (1) is a time-periodic, spatially localized solution. ‘‘Spatially localized’’ means exponentially decaying with respect to spatial distance in the CML system. Breather was first found in the continuous sine–Gordon equation [9]. Recently MacKay and coworkers [10–14] have proved that a wide class of networks of weakly coupled oscillators, which are mathematically described by systems of coupled ordinary differential equations, possess breather solutions. Here we discuss the existence of discrete breathers in CML (1) with discrete-time and discrete-space. By the spatial disorder in (1) we mean the existence of a large number of stationary solutions randomly situated along spatial coordinates in the CML. By using the anti-integrability method instead of the complicated ‘‘horseshoe’’ technique, we show that system (1) exhibits the structure of the spatial disorder.
2. The existence of the discrete breathers ðnþ1Þ
First, Eq. (1) yields the system coordinates uðnþ1Þ ¼ fuj ðnÞ u ¼ fuj g 2 RZ at time n:
g 2 RZ at time n þ 1 as a function L of its coordinates
ðnÞ
uðnþ1Þ ¼ LðuðnÞ Þ: ðnÞ
Definition 1. A trajectory fuj g of Eq. (1) is said to be a breather (solution) if it is spatially localized and periodic in time, or equivalently if it is a time periodic solution which decays spatially to zero as j ! þ1. By setting k ¼ 1=a, Eq. (1) can be equivalently defined by the implicit equation ðnÞ ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ uj þ kk uj1 2uj þ ujþ1 ¼ 0: Fj ðu; kÞ ¼ uj uj a 1 uj k uj ðnÞ
ð3Þ
ðnÞ
Then a trajectory fuj g of Eq. (1) is the solution of Eq. (3), or the configuration u ¼ fuj g which fulfill for all n and j the equivalent implicit equation (3) with k ¼ 1=a. At the limit k ¼ 0, which we call anti-integrable limit, Eq. (3) reduces to ðnÞ ðnÞ ðnÞ ¼ 0; ð4Þ uj uj a 1 uj which has trivial solutions ðnÞ
ðnÞ
uj ¼ rj ;
ðnÞ
rj ¼ 0; 1 or
a:
Though at this limit, the solutions of this extended dynamical system do not correspond to trajectories of a deterministic dynamical system since uðnþ1Þ at time ðn þ 1Þ is not determined as a function of uðnÞ at the previous time n. However, we can prove that under some conditions, each of these solutions can be continued as trajectories of the deterministic dynamical system for jkj varying from zero up to some nonzero value d. Thus we can get understanding of the dynamics of (1) from (3). Now we give the main result of this section: ðnÞ
ðnÞ
Theorem 1. Consider any solution uð0Þ ¼ fuj;0 g of (3) at k ¼ 0 which, for some fixed j ¼ j0 , fuj0 ;0 g is s-periodic with ðnÞ respect to n (here s is an arbitrary fixed positive integer), and fuj0 ;0 g ¼ 0, for all n, j 6¼ j0 . Then this solution can be uniquely continued as the breather uðkÞ of (3) up to jkj < d. Here d ¼ fa2 ð1 aÞ2 g=f4ð1 þ 2kÞð23 þ 8aÞg. Before proving the theorem we make some preparations. Without loss of generality, we assume j0 ¼ 0 in the sequel. Let B ¼ fz ¼ fzðnÞ g j zðnÞ 2 RZ ; zðnþsÞ ¼ zðnÞ for all n 2 Zg with norm kzk ¼ supfjzðnÞ j; n 2 Zg: Definition 2. A 1D chain B is the product of an infinite sequence of Banach spaces Bj ¼ B; j 2 Z, with norm kuk ¼ supfjuj j; j 2 Zg. Here u ¼ fuj 2 B; j 2 Zg. ðnÞ
ðnÞ
Lemma 1. Let F ð; kÞ ¼ fFj ð; kÞg : B ! B here Fj ð; kÞ is as in (3), and let uð0Þ given in the above theorem. Then linear derivative operator ou F ðu; 0Þ is invertible for the trivial zero u ¼ uð0Þ of F ðu; 0Þ, and kou F 1 ðuð0Þ; 0Þk ¼ 1=fað1 aÞg.
Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464 ðnÞ
1459
ðnÞ
Proof. Let uð0Þ ¼ fuj;0 g 2 B and v ¼ fvj g 2 B. Then n o ou F ðu; kÞv ¼ ðou F ðu; kÞvÞðnÞ j o n ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ vj þ kk vjþ1 2vj þ vj1 : ¼ 3ðuj Þ2 þ 2ða þ 1Þuj a vj k vj
ð5Þ
Hence n o n o o n ðnÞ 2 ðnÞ ðnÞ ðnÞ ðnÞ ou F ðuð0Þ; 0Þv ¼ ðou F ðuð0Þ; 0ÞvÞðnÞ ¼ 3ðu ¼ b : Þ þ 2ða þ 1Þu a v v j j j j;0 j;0 j
ð6Þ
ðnÞ
Here bj ¼ a, j 6¼ 0, and 8 ðnÞ > if u0 ¼ 0; < a ðnÞ ðnÞ b0 ¼ a 1 if u0 ¼ 1; > : ðnÞ að1 aÞ if u0 ¼ a: We infer from (6) that ou F ðuð0Þ; 0Þ is invertible, and ( ) 1 ðnÞ 1 ; v ou F ðuð0Þ; 0Þv ¼ ðnÞ j bj kou F
1
ðuð0Þ; 0Þk ¼ sup kou F kvk 6 1
1
1 1 1 1 1 ; ; ¼ : ðuð0Þ; 0Þvk ¼ sup ðnÞ ¼ max a ð1 aÞ að1 aÞ að1 aÞ j;n bj
Lemma 2. Let X ; K; Y be Banach spaces, and F 2 C 1 ðX K; Y Þ with F ðx0 ; k0 Þ ¼ 0, Fx1 ðx0 ; k0 Þ 2 LðY ; X Þ, kFx1 ðx0 ; k0 Þk ¼ M. Suppose also that r; d are the constants satisfying the following conditions: (1) kFx0 ðx; kÞ Fx0 ðx0 ; kÞk 6 1=2M when kx x0 k 6 r; kk k0 k 6 d; (2) kF ðx0 ; kÞk < r=2M when kk k0 k < d. Then there exists exactly one function T : Bd ðk0 Þ ¼ fk 2 K j kk k0 k < dg ! Br ðx0 Þ ¼ fx 2 X j kx x0 k < rg; T ðkÞ ¼ xk such that xk0 ¼ x0 ; F ðxk ; kÞ ¼ 0. Proof. Lemma 2 can be inferred from the procedure of proof of the implicit function theorem (see, e.g. [15]).
Proof of Theorem 1. Note that kou F ðu; kÞ ou F ðuð0Þ; 0Þk ¼ sup kðou F ðu; kÞ ou F ðuð0Þ; 0ÞÞvk kvk 6 1
2 ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ¼ sup 3 uj þ 2ða þ 1Þuj a vj k vj vj j;n 2 ðnþ1Þ ðnÞ ðnþ1Þ ðnÞ ðnÞ ðnÞ þ kk vjþ1 2vj þ vj1 3 uj;0 þ 2ða þ 1Þuj;0 a vj 6 ð3ðkuk þ 1Þ þ 2ða þ 1ÞÞku uð0Þk þ 2k þ 4kk and
o n ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ kF ðuð0Þ; kÞk ¼ sup jkj uj;0 uj;0 þ kjkj ujþ1;0 uj;0 þ uj1;0 6 2jkjð1 þ 2kÞ: j;n
Therefore, when ku uð0Þk < r ¼
1 að1 aÞ ¼ 4ð3ð14 þ 1Þ þ 2ða þ 1ÞÞkou F 1 ðuð0Þ; 0Þk 23 þ 8a
and jkj 6
r a2 ð1 aÞ2 ¼ ¼d 4ð1 þ 2kÞkou F 1 ðuð0Þ; 0Þk 4ð1 þ 2kÞð23 þ 8aÞ
ð7Þ
ð8Þ
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we have from (7) and (8) that kou F ðu; kÞ ou F ðuð0Þ; 0Þk <
1 2kou F 1 ðuð0Þ; 0Þk
and kF ðuð0Þ; kÞk <
r 2kou
F 1 ðuð0Þ; 0Þk
:
By Lemma 2, we can infer that uð0Þ 2 B can be uniquely continued as the solution of (1) up to jkj < fa2 ð1 aÞ2 g=f4ð1 þ 2kÞð23 þ 8aÞg ¼ d. There only remains to prove that uðkÞ spatially exponentially decays. Note that Lu ¼ ffLugj2Z g ¼ L0 u þ Lþ u þ L u; where L0 u ¼ ffL0 ugj2Z g, o n ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ðnÞ uj 2kkuj ; n 2 Z ; fL0 ugj ¼ uj uj a 1 uj k uj ðnÞ
fLþ ug ¼ ffLþ ugj2Z g;
fLþ ugj ¼ fkkujþ1 ; n 2 Zg;
fL ug ¼ ffL ugj2Z g;
fL ugj ¼ fkkuj1 ; n 2 Zg:
ðnÞ
Hence the bounded linear operator L on 1D chain B has the block tridiagonal form with L0 consisting of blocks on the diagonal, Lþ consisting of blocks on the first super-diagonal and L consisting of blocks on the first sub-diagonal. By Theorems 2 and 3 in [10], we arrive at that uðkÞ decays exponentially as j ! 1. The proof of the theorem is complete.
3. The spatial disorder of the stationary solutions of discrete Nagumo equation In this section, we consider the spatial pattern of the stationary solutions of Eq. (1), and prove the system (1) exhibits the structure of spatial disorder. The stationary solutions u ¼ fuj gj2Z of Eq. (1) satisfy the equation kðuj1 2uj þ ujþ1 Þ þ af ðuj Þ ¼ 0;
ð9Þ
Z
where u 2 R . Set l ¼ k=a, we rewrite Eq. (9) as lðuj1 2uj þ ujþ1 Þ þ f ðuj Þ ¼ 0:
ð10Þ
For the sake of simplicity, we assume that a ¼ 12, consequently f ðuj Þ ¼ uj ðuj 12Þð1 uj Þ. The goal of this section is to discuss the disorder of the stationary solutions under some topology. Let B ¼ fu j u ¼ fuj g; j 2 Z; uj 2 R1 g, and let S : B ! B be the translational map defined by ðSðuÞÞj ¼ ujþ1 . Now define the map Gð; lÞ : l1 ! l1 by ðGðu; lÞÞj ¼ lðuj1 2uj þ ujþ1 Þ þ f ðuj Þ:
ð11Þ
Then the stationary solutions of Eq. (1), namely the solutions of Eq. (10), are in one-to-one correspondence with the zeros of the map G. Obviously, the solutions of Eq. (10) at l ¼ 0 are the set X ¼ f uj u ¼ f uj g; uj 2 f0; 12 ; 1gg l1 , and moreover, such a solution in X has a unique extension to l 6¼ 0 sufficiently small. 1 Theorem 2. There exists a positive constant d ¼ 1280 such that for every l; 0 < jlj < d, each solution u ¼ f uj g of Eq. (10) at l ¼ 0 has the unique extension uðlÞ with kuðlÞ ukl1 < r, where r ¼ 801 .
Proof. Let map G be as above. Then 0. . ... B . B l 2l 3u2j þ 3uj 12 DGðu; lÞ ¼ B B l @
1 C C C C A
l 2l 3u2jþ1 þ 3ujþ1 12 .. .
l .. .
..
.
ð12Þ
Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464
and for any u 2 X 0
1461
1
..
B . B B B B B B DGðu; 0Þ ¼ B B B B B B @
..
C C C C C C C C; C C C C C A
. bjj bjþ1;jþ1 ..
. ..
ð13Þ
.
where 8 1 > < 2 if uj ¼ 0; bjj ¼ 12 if uj ¼ 1; > :1 if uj ¼ 12 : 4 j < r, we have Hence DG1 ðu; 0Þ exists with M ¼ kDG1 ðu; 0Þk ¼ 4. When jlj < d, and ju u 0 1 . .. ... ... B C B C B C B C l wjj l B C kDGðu; lÞ DGðu; 0Þkl1 ¼ B C B C l w l B jþ1;jþ1 C B C @ A .. .. .. . l1 . . 1 1 u2j þ 3 6 where wjj ¼ 2l 3u2j þ 3uj 3 uj 2 2 ukl1 ku þ ukl1 6 4jlj þ 3ku ukl1 þ ku 6
1 1 ¼ 8 2M
and kGðu; lÞkl1 ¼ sup jlðuj1 2uj þ ujþ1 Þj 6 2jlj < j
1=80 r ¼ : 8 2M
Hence for any l; 0 < jlj < d, each u 2 X , by Lemma 2, has a unique extension. The proof of the theorem is complete. According to the above theorem, to every u 2 X , there corresponds a unique solution uðlÞ of Eq. (10) for any l; 0 < jlj < d. We denote the correspondence by the map hl , i.e., hl ð uÞ ¼ uðlÞ, and set X ðlÞ ¼ hl ðX Þ. Proposition 1. Let S be the translational map defined as above, then (1) 8u 2 X , we have hl SðuÞ ¼ S hl ðuÞ, (2) SðX ðlÞÞ ¼ X ðlÞ. Proof uÞ. It is not (1) 8u 2 X , by Theorem 2 there exists uðlÞ 2 X ðlÞ such that uðlÞ is the solution of Eq. (10), i.e., uðlÞ ¼ hl ð hard to verify that SðuðlÞÞ is also the solution of Eq. (10). Also noticing that kS hl ðuÞ SðuÞkl1 ¼ khl ðuÞ ukl1 < r; we can infer from the uniqueness of the solution of Eq. (10) that hl Sð uÞ ¼ S hl ð uÞ, proving Part (1).
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(2) We can easily get the desired result from Part (1). Now we return to Eq. (10). Let EðlÞ ðl 6¼ 0Þ be the set of all solutions of Eq. (10). Obviously X ðlÞ EðlÞ; SðEðlÞÞ ¼ EðlÞ. From Eq. (10), we have 1 ujþ1 ¼ 2uj þ f ðuj Þ uj1 ; l
ð14Þ
1 uj1 ¼ 2uj þ f ðuj Þ ujþ1 : l
ð15Þ
Hence an element of EðlÞ ðl 6¼ 0Þ is uniquely determined by its any two neighboring components. For any u; v 2 EðlÞ, u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ, v ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ, we define dðu; vÞ ¼ ju vjd ¼ maxfju1 v1 j; ju0 v0 jg:
ð16Þ
It is easy to verify that d is a distance on EðlÞ. With d; EðlÞ becomes a metric space, which we denote by ðEðlÞ; dÞ. Now we state and prove the main theorem of this section. Theorem 3. For metric space ðEðlÞ; dÞ with 0 < jlj < d, there exists a subset of EðlÞ on which the translational dynamical system S is topologically conjugate to the shift map with three symbols. Hence the translational map S on ðEðlÞ; dÞ is chaotic. Hence system (1) admits a structure of spatial disorder. Before proving the theorem, we prove a lemma. Let uðlÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X ðlÞ, then by Theorem 2 there exists a unique u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X such that hl ðuÞ ¼ uðlÞ and ku uðlÞkl1 < r. Let R3 be the well-known space of bi-infinite sequences with three symbols under standard topology. Define the map g : X ! R3 as follows: 8u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ; ui 2 f0; 1; 12g;
gð uÞ ¼ t ¼ ð. . . ; t2 ; t1 ; t0 ; t1 ; t2 ; . . .Þ;
ð17Þ
where 8 < 0 if ui ¼ 0; ti ¼ 1 if ui ¼ 1; : 2 if ui ¼ 12 : Denote fl ¼ g h1 l : X ðlÞ ! R3 . Now we give Lemma 3 8u; v 2 X ðlÞ; 0 < l < d;
u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ;
v ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ
with ju vjd < 2r. Then (1) If ju vjd ¼ ju1 v1 j, then jSðuÞ SðvÞjd > 316ju vjd . In this case, jSðuÞ SðvÞjd ¼ jðSðuÞÞ1 ðSðvÞÞ1 j, hence jS 2 ðuÞ S 2 ðvÞjd > 316jSðuÞ SðvÞjd > 3162 ju vjd provided that jSðuÞ SðvÞjd < 2r. Generally if ju vjd ¼ ju1 v1 j, then we have jS k ðuÞ S k ðvÞjd > 316k ju vjd ; provided that jS i ðuÞ S i ðvÞjd < 2r ði ¼ 1; 2; . . . ; k 1Þ; (2) If ju vjd ¼ ju0 v0 j, then jS 1 ðuÞ S 1 ðvÞjd > 316ju vjd . In this case, jS 1 ðuÞ S 1 ðvÞjd ¼ jðS 1 ðuÞÞ0 ðS 1 ðvÞÞ0 j, hence jS 2 ðuÞ S 2 ðvÞjd > 316jS 1 ðuÞ S 1 ðvÞjd > 3162 ju vjd provided that jS 1 ðuÞ S 1 ðvÞjd < 2r. Generally if ju vjd ¼ ju0 v0 j, then we have jS k ðuÞ S k ðvÞjd > 316k ju vjd , provided that jS i ðuÞ S i ðvÞjd < 2r ði ¼ 1; 2; . . . ; k 1Þ; Proof ju vjd ¼ ju1 v1 j ! ju1 v1 j P ju0 v0 j:
ð18Þ
1 ; u 0 2 f0; 1; 12g such that u1 ; v1 2 ð Since u; v 2 X ðlÞ and ju vjd < 2r, we can infer that there exist u u1 r; u1 þ rÞ; u0 ; v0 2 ðu0 r; u0 þ rÞ. Noticing that SðuÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ; SðvÞ ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ, and
Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464
1463
jðSðuÞÞ1 ðSðvÞÞ1 j ¼ ju2 v2 j 1 1 ¼ 2u1 þ f ðu1 Þ u0 2v1 þ f ðv1 Þ v0 l l P
1 jf ðu1 Þ f ðv1 Þj 2ju1 v1 j ju0 v0 j jlj
P ðOwing to the mean value theorem and ð18ÞÞ 1 0 jf ðnÞjju1 v1 j 3ju1 v1 j jlj 1 0 jf ðnÞj > 319; n 2 ½ r; r [ ½1 r; 1 þ r [ ½12 r; 12 þ r P jlj
P
> 316ju1 v1 j:
ð19Þ
Hence we have jðSðuÞÞ1 ðSðvÞÞ1 j > 316jðSðuÞÞ0 ðSðvÞÞ0 j; which implies that jSðuÞ SðvÞjd ¼ jðSðuÞÞ1 ðSðvÞÞ1 j > 316ju vjd : Resuming the same procedure as above we can prove the rest of Part (1). The similar method can be applied to the proof of Part (2). Now we return to Theorem 3: Proof of Theorem 3. We know that ðX ðlÞ; dÞ ðEðlÞ; dÞ is an invariant set for S. It is sufficient to prove that on ðX ðlÞ; dÞ the translational map S is topologically conjugate to the shift map r with three symbols. Let fl ¼ g h1 l defined as above. We will prove that fl is a homeomorphism. First we prove that fl is continuous. Given u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X ðlÞ. Let u ¼ h1 u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ; t ¼ fl ðuÞ ¼ ð. . . ; t2 ; t1 ; l ðuÞ ¼ ð. . . ; t0 ; t1 ; t2 ; . . .Þ 8e > 0, there exists a positive integer n0 such that 1=2n0 < e. According to Eqs. (14) and (15) we can find a d1 > 0 such that for any v ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ 2 X ðlÞ satisfying jv ujd < d1 one has jui vi j < 2r; jij 6 n0 . This implies that there exists ui 2 f0; 1; 12g; jij 6 n0 such that ui ; vi 2 ðui r; ui þ rÞ;
jij 6 n0 :
Therefore ðfl ðvÞÞi ¼ ðfl ðuÞÞi ; jij 6 n0 , and jfl ðvÞ fl ðuÞjR3 6
1 < e: 2n0
This proves the continuity of fl . Finally we prove the continuity of fl1 . Given t ¼ ð. . . ; t2 ; t1 ; t0 ; t1 ; t2 ; . . .Þ 2 R3 . Let u ¼ fl1 ðtÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X ðlÞ and h1 l ðuÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ:8e > 0, there exists a positive integer n1 such that 2r=316n1 < e. Let d2 ¼ 1=2n1 þ1 , then for any y 2 R3 satisfying jy tjR3 < d2 with fl1 ðyÞ ¼ v, we have yi ¼ t i ;
jij 6 n1 þ 1:
ð20Þ
Hence ui ; vi 2 ðui r; ui þ rÞ;
jij 6 n1 þ 1:
According to Lemma 3, either jS n1 ðuÞ S n1 ðvÞjd > 316n1 ju vjd
ð21Þ
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or jS n1 ðuÞ S n1 ðvÞjd > 316n1 ju vjd
ð22Þ
holds. In the first case, we have 2r > jun1 þ1 vn1 þ1 j > 316n1 ju vjd ; In the second case, we have 2r > jun1 vn1 j > 316n1 ju vjd : Therefore in any case we obtain that jfl1 ðyÞ fl1 ðtÞjd ¼ ju vjd <
2r < e: 316n1
This implies that fl1 is also continuous map, and we complete the proof of the theorem.
ð23Þ
4. Discussion In this presentation, we have proved the existence of breathers and the structure of spatial disorder in Eq. (1). Similarly we can also prove the existence of multi-breathers and spatiotemporal solutions which are periodic with respect to both time n and space j. Discussion of other dynamical properties such as kink and anti-kink, spatiotemporal chaos and the stability of the solutions in the system is very meaningful, which is left for the future studies.
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Cross MC, Hohenberg PC. Pattern formation: outside of equilibrium. Rev Mod Phys 1993;65(3):851–1113. Kaneko K. Theory and applications of coupled map lattices. New York: Wiley; 1993. Kaneko K. Chaotic travelling waves in a CML. Physica D 1993;68:299–317. Ginzburg S, Sbitnev V. Spatiotemporal chaos in 2D coupled map lattice with pinning-like force. Physica D 1999;132:87–99. Keener JP. Propagation and its failure in couled systems of discrete excitable cells. SIAM J Appl Math 1987;47:556–72. Chow S-N, Shen W. Dynamics in a discrete Nagumo equation: spatial topological chaos. SIAM J Appl Math 1995;55:1764–81. Aubry S, Abramovici G. Chaotic trajectories in the standard map: The concept of anti-integrability. Physica D 1990;43:199–219. Aubry S. Anti-integrability in dynamical and variational problems. Physica D 1995;86:284–96. Ablowitz MJ, Kanp DJ, Newll AC, Segur H. The inverse scattering transform-Fourier analysis for nonlinear problem. Stud Appl Math 1974;53:249–315. Mackay RS, Aubry S. Proof of existence of breathers for time-reversiber or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 1994;7:1623–43. Marin JL, Aubry S. Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit. Nonlinearity 1996;9:1501–28. Aubry S. Breathers in nonlinear lattices: Existence, linear stability and quantization. Physica D 1997;103:201–50. Sepulchre JA, Mackay RS. Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators. Nonlinearity 1997;10:679–713. Sepulchre JA, Mackay RS. Discrete brethers in diordered media. Physica D 1998;113:342–5. Deimling K. Nonlinear functional analysis. Berlin: Springer; 1985.
Some dynamical behavior of discrete Nagumo equation q Yu-Rong Liu a b
a,*
, Zeng-Rong Liu
b
Department of Mathematics, Yangzhou University, Yangzhou 225002, China Department of Mathematics, Shanghai University, Shanghai 200436, China Accepted 26 March 2002 Communicated by W.M. Zheng
Abstract The spatiotemporal dynamics can be effectively studied by continuation from an anti-integrable limit. By using the anti-integrability method we consider the dynamical behavior of discrete Nagumo equation, prove the existence of breather and discuss the spatial disorder of the stationary solutions in the system. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently, the study of complicated dynamical systems is attracting considerable attention of many researchers and there are many results on it [1]. Because of complexity of the subject, however, the methods available are far from enough for studying the dynamics of the systems, and the numerical simulation is still a powerful weapon for understanding of complicated systems. Coupled map lattices (CMLs) [2], as the models of spatially extended systems, already have gained a lot of popularity among the researchers because of their convenience for computer simulations as well as for the rigorous mathematical analysis. On the other hand, CMLs yield interesting dynamical systems in their own right, which generally show a wide diversity of patterns including stationary structures, traveling waves, localized structures, etc., and also reveal various space–time phenomena such as spatiotemporal intermittency, spatiotemporal chaos, etc. [1–4]. Now CMLs serve as one of the most efficient tools to analyze space–time phenomena in extended systems. Here we consider the following CML: ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ uj ¼ uj þ k uj1 2uj þ ujþ1 þ af uj ; ð1Þ ðnÞ
where j 2 Z denotes a spatial coordinate and n 2 Z denotes time, uj 2 R; a; k > 0, and f ðuÞ ¼ uðu aÞð1 uÞ;
0 < a < 1:
This is a discrete analog of Nagumo equation ou o2 u ¼ D 2 þ f ðuÞ; ot ox
x 2 R1 ; t 2 Rþ ;
ð2Þ
where D is a positive constant. Eq. (2) is a well-known P.D.E., which is used as a model for the spread of genetic traits and for the propagation of nerve pulses in a nerve axon, neglecting recovery.
q
This work is supported by National Natural Science Foundation of China, 10171061. Corresponding author. E-mail address: [email protected] (Yu.-R. Liu).
*
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 8 2 - 6
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There are some results known for Eq. (1). In [5,6] authors discussed the existence of the spatial topological chaos and the traveling waves. It was shown that no traveling waves appear in Eq. (1) when coupling is weak, but there exist traveling waves if coupling is strong. In this paper, we consider breathers and the spatial disorder in Eq. (1) using anti-integrability method [7,8]. A breather of (1) is a time-periodic, spatially localized solution. ‘‘Spatially localized’’ means exponentially decaying with respect to spatial distance in the CML system. Breather was first found in the continuous sine–Gordon equation [9]. Recently MacKay and coworkers [10–14] have proved that a wide class of networks of weakly coupled oscillators, which are mathematically described by systems of coupled ordinary differential equations, possess breather solutions. Here we discuss the existence of discrete breathers in CML (1) with discrete-time and discrete-space. By the spatial disorder in (1) we mean the existence of a large number of stationary solutions randomly situated along spatial coordinates in the CML. By using the anti-integrability method instead of the complicated ‘‘horseshoe’’ technique, we show that system (1) exhibits the structure of the spatial disorder.
2. The existence of the discrete breathers ðnþ1Þ
First, Eq. (1) yields the system coordinates uðnþ1Þ ¼ fuj ðnÞ u ¼ fuj g 2 RZ at time n:
g 2 RZ at time n þ 1 as a function L of its coordinates
ðnÞ
uðnþ1Þ ¼ LðuðnÞ Þ: ðnÞ
Definition 1. A trajectory fuj g of Eq. (1) is said to be a breather (solution) if it is spatially localized and periodic in time, or equivalently if it is a time periodic solution which decays spatially to zero as j ! þ1. By setting k ¼ 1=a, Eq. (1) can be equivalently defined by the implicit equation ðnÞ ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ uj þ kk uj1 2uj þ ujþ1 ¼ 0: Fj ðu; kÞ ¼ uj uj a 1 uj k uj ðnÞ
ð3Þ
ðnÞ
Then a trajectory fuj g of Eq. (1) is the solution of Eq. (3), or the configuration u ¼ fuj g which fulfill for all n and j the equivalent implicit equation (3) with k ¼ 1=a. At the limit k ¼ 0, which we call anti-integrable limit, Eq. (3) reduces to ðnÞ ðnÞ ðnÞ ¼ 0; ð4Þ uj uj a 1 uj which has trivial solutions ðnÞ
ðnÞ
uj ¼ rj ;
ðnÞ
rj ¼ 0; 1 or
a:
Though at this limit, the solutions of this extended dynamical system do not correspond to trajectories of a deterministic dynamical system since uðnþ1Þ at time ðn þ 1Þ is not determined as a function of uðnÞ at the previous time n. However, we can prove that under some conditions, each of these solutions can be continued as trajectories of the deterministic dynamical system for jkj varying from zero up to some nonzero value d. Thus we can get understanding of the dynamics of (1) from (3). Now we give the main result of this section: ðnÞ
ðnÞ
Theorem 1. Consider any solution uð0Þ ¼ fuj;0 g of (3) at k ¼ 0 which, for some fixed j ¼ j0 , fuj0 ;0 g is s-periodic with ðnÞ respect to n (here s is an arbitrary fixed positive integer), and fuj0 ;0 g ¼ 0, for all n, j 6¼ j0 . Then this solution can be uniquely continued as the breather uðkÞ of (3) up to jkj < d. Here d ¼ fa2 ð1 aÞ2 g=f4ð1 þ 2kÞð23 þ 8aÞg. Before proving the theorem we make some preparations. Without loss of generality, we assume j0 ¼ 0 in the sequel. Let B ¼ fz ¼ fzðnÞ g j zðnÞ 2 RZ ; zðnþsÞ ¼ zðnÞ for all n 2 Zg with norm kzk ¼ supfjzðnÞ j; n 2 Zg: Definition 2. A 1D chain B is the product of an infinite sequence of Banach spaces Bj ¼ B; j 2 Z, with norm kuk ¼ supfjuj j; j 2 Zg. Here u ¼ fuj 2 B; j 2 Zg. ðnÞ
ðnÞ
Lemma 1. Let F ð; kÞ ¼ fFj ð; kÞg : B ! B here Fj ð; kÞ is as in (3), and let uð0Þ given in the above theorem. Then linear derivative operator ou F ðu; 0Þ is invertible for the trivial zero u ¼ uð0Þ of F ðu; 0Þ, and kou F 1 ðuð0Þ; 0Þk ¼ 1=fað1 aÞg.
Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464 ðnÞ
1459
ðnÞ
Proof. Let uð0Þ ¼ fuj;0 g 2 B and v ¼ fvj g 2 B. Then n o ou F ðu; kÞv ¼ ðou F ðu; kÞvÞðnÞ j o n ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ vj þ kk vjþ1 2vj þ vj1 : ¼ 3ðuj Þ2 þ 2ða þ 1Þuj a vj k vj
ð5Þ
Hence n o n o o n ðnÞ 2 ðnÞ ðnÞ ðnÞ ðnÞ ou F ðuð0Þ; 0Þv ¼ ðou F ðuð0Þ; 0ÞvÞðnÞ ¼ 3ðu ¼ b : Þ þ 2ða þ 1Þu a v v j j j j;0 j;0 j
ð6Þ
ðnÞ
Here bj ¼ a, j 6¼ 0, and 8 ðnÞ > if u0 ¼ 0; < a ðnÞ ðnÞ b0 ¼ a 1 if u0 ¼ 1; > : ðnÞ að1 aÞ if u0 ¼ a: We infer from (6) that ou F ðuð0Þ; 0Þ is invertible, and ( ) 1 ðnÞ 1 ; v ou F ðuð0Þ; 0Þv ¼ ðnÞ j bj kou F
1
ðuð0Þ; 0Þk ¼ sup kou F kvk 6 1
1
1 1 1 1 1 ; ; ¼ : ðuð0Þ; 0Þvk ¼ sup ðnÞ ¼ max a ð1 aÞ að1 aÞ að1 aÞ j;n bj
Lemma 2. Let X ; K; Y be Banach spaces, and F 2 C 1 ðX K; Y Þ with F ðx0 ; k0 Þ ¼ 0, Fx1 ðx0 ; k0 Þ 2 LðY ; X Þ, kFx1 ðx0 ; k0 Þk ¼ M. Suppose also that r; d are the constants satisfying the following conditions: (1) kFx0 ðx; kÞ Fx0 ðx0 ; kÞk 6 1=2M when kx x0 k 6 r; kk k0 k 6 d; (2) kF ðx0 ; kÞk < r=2M when kk k0 k < d. Then there exists exactly one function T : Bd ðk0 Þ ¼ fk 2 K j kk k0 k < dg ! Br ðx0 Þ ¼ fx 2 X j kx x0 k < rg; T ðkÞ ¼ xk such that xk0 ¼ x0 ; F ðxk ; kÞ ¼ 0. Proof. Lemma 2 can be inferred from the procedure of proof of the implicit function theorem (see, e.g. [15]).
Proof of Theorem 1. Note that kou F ðu; kÞ ou F ðuð0Þ; 0Þk ¼ sup kðou F ðu; kÞ ou F ðuð0Þ; 0ÞÞvk kvk 6 1
2 ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ¼ sup 3 uj þ 2ða þ 1Þuj a vj k vj vj j;n 2 ðnþ1Þ ðnÞ ðnþ1Þ ðnÞ ðnÞ ðnÞ þ kk vjþ1 2vj þ vj1 3 uj;0 þ 2ða þ 1Þuj;0 a vj 6 ð3ðkuk þ 1Þ þ 2ða þ 1ÞÞku uð0Þk þ 2k þ 4kk and
o n ðnþ1Þ ðnÞ ðnÞ ðnÞ ðnÞ kF ðuð0Þ; kÞk ¼ sup jkj uj;0 uj;0 þ kjkj ujþ1;0 uj;0 þ uj1;0 6 2jkjð1 þ 2kÞ: j;n
Therefore, when ku uð0Þk < r ¼
1 að1 aÞ ¼ 4ð3ð14 þ 1Þ þ 2ða þ 1ÞÞkou F 1 ðuð0Þ; 0Þk 23 þ 8a
and jkj 6
r a2 ð1 aÞ2 ¼ ¼d 4ð1 þ 2kÞkou F 1 ðuð0Þ; 0Þk 4ð1 þ 2kÞð23 þ 8aÞ
ð7Þ
ð8Þ
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we have from (7) and (8) that kou F ðu; kÞ ou F ðuð0Þ; 0Þk <
1 2kou F 1 ðuð0Þ; 0Þk
and kF ðuð0Þ; kÞk <
r 2kou
F 1 ðuð0Þ; 0Þk
:
By Lemma 2, we can infer that uð0Þ 2 B can be uniquely continued as the solution of (1) up to jkj < fa2 ð1 aÞ2 g=f4ð1 þ 2kÞð23 þ 8aÞg ¼ d. There only remains to prove that uðkÞ spatially exponentially decays. Note that Lu ¼ ffLugj2Z g ¼ L0 u þ Lþ u þ L u; where L0 u ¼ ffL0 ugj2Z g, o n ðnÞ ðnÞ ðnÞ ðnþ1Þ ðnÞ ðnÞ uj 2kkuj ; n 2 Z ; fL0 ugj ¼ uj uj a 1 uj k uj ðnÞ
fLþ ug ¼ ffLþ ugj2Z g;
fLþ ugj ¼ fkkujþ1 ; n 2 Zg;
fL ug ¼ ffL ugj2Z g;
fL ugj ¼ fkkuj1 ; n 2 Zg:
ðnÞ
Hence the bounded linear operator L on 1D chain B has the block tridiagonal form with L0 consisting of blocks on the diagonal, Lþ consisting of blocks on the first super-diagonal and L consisting of blocks on the first sub-diagonal. By Theorems 2 and 3 in [10], we arrive at that uðkÞ decays exponentially as j ! 1. The proof of the theorem is complete.
3. The spatial disorder of the stationary solutions of discrete Nagumo equation In this section, we consider the spatial pattern of the stationary solutions of Eq. (1), and prove the system (1) exhibits the structure of spatial disorder. The stationary solutions u ¼ fuj gj2Z of Eq. (1) satisfy the equation kðuj1 2uj þ ujþ1 Þ þ af ðuj Þ ¼ 0;
ð9Þ
Z
where u 2 R . Set l ¼ k=a, we rewrite Eq. (9) as lðuj1 2uj þ ujþ1 Þ þ f ðuj Þ ¼ 0:
ð10Þ
For the sake of simplicity, we assume that a ¼ 12, consequently f ðuj Þ ¼ uj ðuj 12Þð1 uj Þ. The goal of this section is to discuss the disorder of the stationary solutions under some topology. Let B ¼ fu j u ¼ fuj g; j 2 Z; uj 2 R1 g, and let S : B ! B be the translational map defined by ðSðuÞÞj ¼ ujþ1 . Now define the map Gð; lÞ : l1 ! l1 by ðGðu; lÞÞj ¼ lðuj1 2uj þ ujþ1 Þ þ f ðuj Þ:
ð11Þ
Then the stationary solutions of Eq. (1), namely the solutions of Eq. (10), are in one-to-one correspondence with the zeros of the map G. Obviously, the solutions of Eq. (10) at l ¼ 0 are the set X ¼ f uj u ¼ f uj g; uj 2 f0; 12 ; 1gg l1 , and moreover, such a solution in X has a unique extension to l 6¼ 0 sufficiently small. 1 Theorem 2. There exists a positive constant d ¼ 1280 such that for every l; 0 < jlj < d, each solution u ¼ f uj g of Eq. (10) at l ¼ 0 has the unique extension uðlÞ with kuðlÞ ukl1 < r, where r ¼ 801 .
Proof. Let map G be as above. Then 0. . ... B . B l 2l 3u2j þ 3uj 12 DGðu; lÞ ¼ B B l @
1 C C C C A
l 2l 3u2jþ1 þ 3ujþ1 12 .. .
l .. .
..
.
ð12Þ
Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464
and for any u 2 X 0
1461
1
..
B . B B B B B B DGðu; 0Þ ¼ B B B B B B @
..
C C C C C C C C; C C C C C A
. bjj bjþ1;jþ1 ..
. ..
ð13Þ
.
where 8 1 > < 2 if uj ¼ 0; bjj ¼ 12 if uj ¼ 1; > :1 if uj ¼ 12 : 4 j < r, we have Hence DG1 ðu; 0Þ exists with M ¼ kDG1 ðu; 0Þk ¼ 4. When jlj < d, and ju u 0 1 . .. ... ... B C B C B C B C l wjj l B C kDGðu; lÞ DGðu; 0Þkl1 ¼ B C B C l w l B jþ1;jþ1 C B C @ A .. .. .. . l1 . . 1 1 u2j þ 3 6 where wjj ¼ 2l 3u2j þ 3uj 3 uj 2 2 ukl1 ku þ ukl1 6 4jlj þ 3ku ukl1 þ ku 6
1 1 ¼ 8 2M
and kGðu; lÞkl1 ¼ sup jlðuj1 2uj þ ujþ1 Þj 6 2jlj < j
1=80 r ¼ : 8 2M
Hence for any l; 0 < jlj < d, each u 2 X , by Lemma 2, has a unique extension. The proof of the theorem is complete. According to the above theorem, to every u 2 X , there corresponds a unique solution uðlÞ of Eq. (10) for any l; 0 < jlj < d. We denote the correspondence by the map hl , i.e., hl ð uÞ ¼ uðlÞ, and set X ðlÞ ¼ hl ðX Þ. Proposition 1. Let S be the translational map defined as above, then (1) 8u 2 X , we have hl SðuÞ ¼ S hl ðuÞ, (2) SðX ðlÞÞ ¼ X ðlÞ. Proof uÞ. It is not (1) 8u 2 X , by Theorem 2 there exists uðlÞ 2 X ðlÞ such that uðlÞ is the solution of Eq. (10), i.e., uðlÞ ¼ hl ð hard to verify that SðuðlÞÞ is also the solution of Eq. (10). Also noticing that kS hl ðuÞ SðuÞkl1 ¼ khl ðuÞ ukl1 < r; we can infer from the uniqueness of the solution of Eq. (10) that hl Sð uÞ ¼ S hl ð uÞ, proving Part (1).
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(2) We can easily get the desired result from Part (1). Now we return to Eq. (10). Let EðlÞ ðl 6¼ 0Þ be the set of all solutions of Eq. (10). Obviously X ðlÞ EðlÞ; SðEðlÞÞ ¼ EðlÞ. From Eq. (10), we have 1 ujþ1 ¼ 2uj þ f ðuj Þ uj1 ; l
ð14Þ
1 uj1 ¼ 2uj þ f ðuj Þ ujþ1 : l
ð15Þ
Hence an element of EðlÞ ðl 6¼ 0Þ is uniquely determined by its any two neighboring components. For any u; v 2 EðlÞ, u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ, v ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ, we define dðu; vÞ ¼ ju vjd ¼ maxfju1 v1 j; ju0 v0 jg:
ð16Þ
It is easy to verify that d is a distance on EðlÞ. With d; EðlÞ becomes a metric space, which we denote by ðEðlÞ; dÞ. Now we state and prove the main theorem of this section. Theorem 3. For metric space ðEðlÞ; dÞ with 0 < jlj < d, there exists a subset of EðlÞ on which the translational dynamical system S is topologically conjugate to the shift map with three symbols. Hence the translational map S on ðEðlÞ; dÞ is chaotic. Hence system (1) admits a structure of spatial disorder. Before proving the theorem, we prove a lemma. Let uðlÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X ðlÞ, then by Theorem 2 there exists a unique u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X such that hl ðuÞ ¼ uðlÞ and ku uðlÞkl1 < r. Let R3 be the well-known space of bi-infinite sequences with three symbols under standard topology. Define the map g : X ! R3 as follows: 8u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ; ui 2 f0; 1; 12g;
gð uÞ ¼ t ¼ ð. . . ; t2 ; t1 ; t0 ; t1 ; t2 ; . . .Þ;
ð17Þ
where 8 < 0 if ui ¼ 0; ti ¼ 1 if ui ¼ 1; : 2 if ui ¼ 12 : Denote fl ¼ g h1 l : X ðlÞ ! R3 . Now we give Lemma 3 8u; v 2 X ðlÞ; 0 < l < d;
u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ;
v ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ
with ju vjd < 2r. Then (1) If ju vjd ¼ ju1 v1 j, then jSðuÞ SðvÞjd > 316ju vjd . In this case, jSðuÞ SðvÞjd ¼ jðSðuÞÞ1 ðSðvÞÞ1 j, hence jS 2 ðuÞ S 2 ðvÞjd > 316jSðuÞ SðvÞjd > 3162 ju vjd provided that jSðuÞ SðvÞjd < 2r. Generally if ju vjd ¼ ju1 v1 j, then we have jS k ðuÞ S k ðvÞjd > 316k ju vjd ; provided that jS i ðuÞ S i ðvÞjd < 2r ði ¼ 1; 2; . . . ; k 1Þ; (2) If ju vjd ¼ ju0 v0 j, then jS 1 ðuÞ S 1 ðvÞjd > 316ju vjd . In this case, jS 1 ðuÞ S 1 ðvÞjd ¼ jðS 1 ðuÞÞ0 ðS 1 ðvÞÞ0 j, hence jS 2 ðuÞ S 2 ðvÞjd > 316jS 1 ðuÞ S 1 ðvÞjd > 3162 ju vjd provided that jS 1 ðuÞ S 1 ðvÞjd < 2r. Generally if ju vjd ¼ ju0 v0 j, then we have jS k ðuÞ S k ðvÞjd > 316k ju vjd , provided that jS i ðuÞ S i ðvÞjd < 2r ði ¼ 1; 2; . . . ; k 1Þ; Proof ju vjd ¼ ju1 v1 j ! ju1 v1 j P ju0 v0 j:
ð18Þ
1 ; u 0 2 f0; 1; 12g such that u1 ; v1 2 ð Since u; v 2 X ðlÞ and ju vjd < 2r, we can infer that there exist u u1 r; u1 þ rÞ; u0 ; v0 2 ðu0 r; u0 þ rÞ. Noticing that SðuÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ; SðvÞ ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ, and
Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464
1463
jðSðuÞÞ1 ðSðvÞÞ1 j ¼ ju2 v2 j 1 1 ¼ 2u1 þ f ðu1 Þ u0 2v1 þ f ðv1 Þ v0 l l P
1 jf ðu1 Þ f ðv1 Þj 2ju1 v1 j ju0 v0 j jlj
P ðOwing to the mean value theorem and ð18ÞÞ 1 0 jf ðnÞjju1 v1 j 3ju1 v1 j jlj 1 0 jf ðnÞj > 319; n 2 ½ r; r [ ½1 r; 1 þ r [ ½12 r; 12 þ r P jlj
P
> 316ju1 v1 j:
ð19Þ
Hence we have jðSðuÞÞ1 ðSðvÞÞ1 j > 316jðSðuÞÞ0 ðSðvÞÞ0 j; which implies that jSðuÞ SðvÞjd ¼ jðSðuÞÞ1 ðSðvÞÞ1 j > 316ju vjd : Resuming the same procedure as above we can prove the rest of Part (1). The similar method can be applied to the proof of Part (2). Now we return to Theorem 3: Proof of Theorem 3. We know that ðX ðlÞ; dÞ ðEðlÞ; dÞ is an invariant set for S. It is sufficient to prove that on ðX ðlÞ; dÞ the translational map S is topologically conjugate to the shift map r with three symbols. Let fl ¼ g h1 l defined as above. We will prove that fl is a homeomorphism. First we prove that fl is continuous. Given u ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X ðlÞ. Let u ¼ h1 u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ; t ¼ fl ðuÞ ¼ ð. . . ; t2 ; t1 ; l ðuÞ ¼ ð. . . ; t0 ; t1 ; t2 ; . . .Þ 8e > 0, there exists a positive integer n0 such that 1=2n0 < e. According to Eqs. (14) and (15) we can find a d1 > 0 such that for any v ¼ ð. . . ; v2 ; v1 ; v0 ; v1 ; v2 ; . . .Þ 2 X ðlÞ satisfying jv ujd < d1 one has jui vi j < 2r; jij 6 n0 . This implies that there exists ui 2 f0; 1; 12g; jij 6 n0 such that ui ; vi 2 ðui r; ui þ rÞ;
jij 6 n0 :
Therefore ðfl ðvÞÞi ¼ ðfl ðuÞÞi ; jij 6 n0 , and jfl ðvÞ fl ðuÞjR3 6
1 < e: 2n0
This proves the continuity of fl . Finally we prove the continuity of fl1 . Given t ¼ ð. . . ; t2 ; t1 ; t0 ; t1 ; t2 ; . . .Þ 2 R3 . Let u ¼ fl1 ðtÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ 2 X ðlÞ and h1 l ðuÞ ¼ ð. . . ; u2 ; u1 ; u0 ; u1 ; u2 ; . . .Þ:8e > 0, there exists a positive integer n1 such that 2r=316n1 < e. Let d2 ¼ 1=2n1 þ1 , then for any y 2 R3 satisfying jy tjR3 < d2 with fl1 ðyÞ ¼ v, we have yi ¼ t i ;
jij 6 n1 þ 1:
ð20Þ
Hence ui ; vi 2 ðui r; ui þ rÞ;
jij 6 n1 þ 1:
According to Lemma 3, either jS n1 ðuÞ S n1 ðvÞjd > 316n1 ju vjd
ð21Þ
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Yu.-R. Liu, Z.-R. Liu / Chaos, Solitons and Fractals 14 (2002) 1457–1464
or jS n1 ðuÞ S n1 ðvÞjd > 316n1 ju vjd
ð22Þ
holds. In the first case, we have 2r > jun1 þ1 vn1 þ1 j > 316n1 ju vjd ; In the second case, we have 2r > jun1 vn1 j > 316n1 ju vjd : Therefore in any case we obtain that jfl1 ðyÞ fl1 ðtÞjd ¼ ju vjd <
2r < e: 316n1
This implies that fl1 is also continuous map, and we complete the proof of the theorem.
ð23Þ
4. Discussion In this presentation, we have proved the existence of breathers and the structure of spatial disorder in Eq. (1). Similarly we can also prove the existence of multi-breathers and spatiotemporal solutions which are periodic with respect to both time n and space j. Discussion of other dynamical properties such as kink and anti-kink, spatiotemporal chaos and the stability of the solutions in the system is very meaningful, which is left for the future studies.
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