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Turing instability for a two-dimensional Logistic coupled map lattice
Physics Letters A 374 (2010) 3447–3450
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Turing instability for a two-dimensional Logistic coupled map lattice L. Xu, G. Zhang ∗ , B. Han, L. Zhang, M.F. Li, Y.T. Han School of Science, Tianjin University of Commerce, Tianjin 300134, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 5 February 2010 Received in revised form 22 June 2010 Accepted 30 June 2010 Available online 9 July 2010 Communicated by A.R. Bishop
In this Letter, stability analysis is applied to a two-dimensional Logistic coupled map lattice with the periodic boundary conditions. The conditions of Turing instability are obtained, and various patterns can be exhibited by numerical simulations in the Turing instability region. For example, space–time periodic structures, periodic or quasiperiodic traveling wave solutions, stationary wave solutions, spiral waves, and spatiotemporal chaos, etc. have been observed. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The similar patterns have also been seen in relevant references. However, the present Letter owes to pattern formation via diffusion-driven instabilities because the system is stable in the absence of diffusion. © 2010 Elsevier B.V. All rights reserved.
Keywords: Logistic coupled map lattice Turing instability Diffusion-driven
1. Introduction Recently, Logistic Coupled Map Lattices (LCML) have been widely studied by a number of authors, for example, see [1–9] and the listed references. A one-dimensional CML with nearest neighbors coupling can be defined as
ε
ε
xti +1 = (1 − ε ) f xti +
f xti+1 + f xti−1 2
(1)
or
xti +1 = f xti +
2
f xti+1 − 2 f xti + f xti−1
,
(2)
where ε is the coupling parameter, the mapping function f (x) = λx(1 − x), and λ ∈ (0, 4]. In the two-dimensional lattices, each map is coupled to four of its nearest neighbors and given by
1 uti + = (1 − ε ) f uti j + j
ε
f uti+1, j + f uti, j +1 4 + f uti−1, j + f uti, j −1
or
(3)
1 uti + = f uti j + ε ∇ 2 f uti j , j
(4)
where
∇ 2 f uti j = f uti+1, j + f uti, j +1 + f uti−1, j + f uti, j −1 − 4 f uti j , and for the sake of convenience, we denote ε4 by
*
ε in (4).
Corresponding author. Tel.: +86 22 26675703; fax: +86 22 26686507. E-mail address: [email protected] (G. Zhang).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.06.065
For LCMLs (1) or (2), some exact periodic traveling wave solutions are obtained in [3], and the existence of quasiperiodic traveling wave solutions is considered in [5]. The analytical solutions are studied in [8], the control of spatiotemporal chaos is investigated in [6], and the symmetry breaking bifurcations are discussed in [7]. In particular, pattern dynamics of (1) or (3) have also attracted considerable attention [1,2,4,9]. It has been found that LCMLs exhibit a variety of space–time patterns: kink–antikink, space–time periodic structures or wavelike patterns, space–time intermittency and spatiotemporal chaos [1,2]. The numerical simulations and theory analysis of wavelike patterns are considered in [4]. The strong coupling cases are investigated in [9]. It is well known that Turing patterns are very important for a reaction–diffusion system. They have been proposed as mechanisms for biological pattern formation in embryological and ecological context [10]. All such works are based on the pioneering work of Turing [11]. In [12], the authors were the first to call attention to the fact that Turing’s idea would be applicable in ecological situation also. They conjectured that the nature of the equations which describe chemical interaction does not seem fundamentally different from the nature of those which describe ecological interaction among the species. Again, the idea that dispersal could give rise to instabilities and hence to spatial pattern was due to a number of authors [13]. A reaction–diffusion system exhibits diffusion-driven instability or Turing instability if the homogeneous steady state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present [10]. Unfortunately, a single reaction–diffusion partial differential equation cannot produce Turing instability. In fact, the systems (2) and (4) can been seen as discrete systems with diffusion terms. Then, can such systems produce Turing instability? Further, can Turing patterns be obtained? In fact, the frozen random, the pattern selection
3448
L. Xu et al. / Physics Letters A 374 (2010) 3447–3450
and the fully developed spatiotemporal chaos via spatiotemporal intermittency have been observed in [5] and [6]. The generalized Turing patterns emerging from global synchronized limit cycles or chaotic states have been studied in [14,15]. However, there has been little attempt to study Turing patterns for CMLs. In this Letter, we will answer the above problems for the system (4). To this end, we need to add the periodic boundary conditions
uts,0 = uts,m ,
uti,1 = uti,m+1
(5)
and t ut0,k = um ,k ,
t ut1,k = um +1,k
(6) Z + , where m is a positive
for k, s ∈ {1, 2, . . . , m} = [1, m] and t ∈ integer. The Letter is organized as follows. After a brief presentation of the logistic model, the Turing instability theory analysis will be given for the two-dimensional Logistic coupled map lattices. Turing instability conditions can then be deduced combining linearization method and inner product technique in Section 2. Based on the results of Section 2, a series of numerical simulations are performed and different patterns such as space–time periodic structures, periodic or quasiperiodic traveling wave, wave, spiral waves, and spatiotemporal chaos, etc. can be selectively realized in Section 3. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The final section is the conclusion. 2. Turing instability
(7)
has already been investigated in detail. With the increase of nonlinear parameter λ, the attractor of the isolated logistic map shows the bifurcation from the fixed point x∗ = (λ − 1)/λ to period 2, and then the period-doubling cascade to the onset of chaos. For discussing Turing instability of the problem (4)–(6), we will assume that the nonlinear parameter λ satisfies the condition 1 < λ < 3. At this time, the positive fixed point of (7) exists and is asymptotically stable. We also need to consider eigenvalues of the following equation
∇ 2 X i j + λ X i j = 0,
(8)
with the periodic boundary conditions
X i ,1 = X i ,m+1
X i ,0 = X i ,m ,
(9)
and
X 1 , j = X m +1 , j .
X 0, j = X m , j ,
(10)
In view of [16], the eigenvalue problem (8)–(10) has the eigenvalues
(l − 1)π (s − 1)π = kls2 λl,s = 4 sin2 + sin2 m
m
for l, s ∈ [1, m].
(11)
The linearization equation of (4) can be written by
1 uti + = (2 − λ) uti j + ε ∇ 2 uti j . j
(12)
Then taking the inner product of (12) with the corresponding ij eigenfunction X ls of the eigenvalue λl,s , we see that m
i , j =1
ij 1 X ls uti + j
= (2 − λ)
m
i , j =1
ij X ls uti j
+ε
m
i , j =1
ij X ls ∇ 2 uti j
.
(13)
ij
2 U t +1 = (2 − λ) 1 − εkls Ut.
(14) ij
If U t is a solution of (14), then uti j = U t Xls is also clearly a solution of (12) with the periodic boundary conditions (5) and (6). Thus, the unstable system (14) will imply that the problem (4)–(6) is unstable. By the above discussion, we easily see that the conditions of Turing instability for the problem (4)–(6) are: there exist l, s ∈ [1, m] and ε > 0 such that
ε>
λ−1 (l−1)π 4(λ − 2)(sin2 m
+ sin2
( s −1 ) π
+ sin2
( s −1 ) π
m
)
for 2 < λ < 3
(15)
for 1 < λ < 2.
(16)
or
ε>
3−λ 4(2 − λ)(sin2
(l−1)π m
m
)
3. Numerical simulation and discussion For the problem (4)–(6), we employed the periodic Neumann boundary conditions with a system size of 200 × 200 space units. Here, we performed a series of two-dimensional simulations, in each, the initial condition was always a small amplitude random perturbation 1% around the steady state. As a numerical example, we firstly consider the distribution of time evolution. Let λ = 2.9, then (2.9 − 1)/2.9 ≈ 0.6552 and
In this section, we will consider Turing instability for the problem (4)–(6). It is well known that the logistic model of the form 1 xti + = λxti j 1 − xti j j
m
Let U t = i , j =1 Xls uti j and use the periodic boundary conditions (5) and (6), then we have
max
l,s∈[1,200]
sin2
(l − 1)π 200
+ sin2
(s − 1)π 200
= 2.
Thus, when ε > 0.2639, the problem (4)–(6) is Turing instable. Fig. 1(a)–(f) exhibits in detail the distribution of time evolution. In Fig. 1(a), the symmetry breaking around the fixed point can be observed. Fig. 1(b)–(d) shows the self-organization process of the system, space–time periodic characteristics begin to appear, and then steady periodic structures emerge in both Fig. 1(e) and Fig. 1(f). To explore clearly the dynamical behavior of 2-dimensional coupled map lattices, we investigate the effect of coupling parameter by keeping the nonlinearity parameter λ of the system fixed. Following the arrows we can realize different wavelength patterns by adjusting the coupled strength increasingly, as depicted in Fig. 2. A stable pattern of rhombi shapes for ε = 0.32 is given in Fig. 2(a). Fig. 2(b)–(c) shows the stable traveling wave pattern for ε = 0.33 and ε = 0.34, in particular, clear stripe pattern appears in Fig. 2(b), similar patterns can also be seen in [15,17]. Comparing Fig. 2(b) with Fig. 2(c), we can obtain the similar quasiperiodic traveling wave but different spatial behaviors. In Fig. 2(b), the direction of spatial propagation of the waves is the vertical axis, but diagonal direction in Fig. 2(c). An interesting situation is depicted in Fig. 3 where an ordered circular wave pattern [18] first is seen. Then with the evolution time proceeding, a stable pattern of square shapes, namely, stationary wave is observed. If we let ε = 0.35, spiral structure will be obtained in Fig. 4(a). With the further increase of ε , the system will gradually lack longtime, large-distance coherence in spite of an organized regular behavior at the local scale. Spatiotemporal chaos which absolutely arises from spatial diffusion will emerge. Fig. 4(b)–(c) just puts to the proof. The above simulations show that the dynamics of a lattice site changes from the fixed point, with the increasing of ε in the Turing instability region. We have seen:
L. Xu et al. / Physics Letters A 374 (2010) 3447–3450
Fig. 1. Snapshots of contour pictures of the time evolution of CML system at some instants when 199 900. (f) t = 200 000.
Fig. 2. Pattern selection with the increase of
3449
ε = 0.335. (a) t = 100. (b) t = 3100. (c) t = 3200. (d) t = 21 300. (e) t =
ε in the Turing instability region. (a) t = 50 000, ε = 0.32. (b) t = 50 000, ε = 0.33. (c) t = 100 000, ε = 0.34.
Fig. 3. Snapshots of contour pictures of the time evolution of CML system at some instants when (e) t = 32 670. (f) t = 32 870.
ε = 0.345. (a) t = 50. (b) t = 9700. (c) t = 26 450. (d) t = 32 130.
3450
L. Xu et al. / Physics Letters A 374 (2010) 3447–3450
Fig. 4. Pattern selection with the increase of
ε in the Turing instability region. (a) t = 200 000, ε = 0.35. (b) t = 200 000, ε = 0.36. (c) t = 200 000, ε = 0.37.
Fig. 5. Selective patterns resulting from random initial values when
(i) (ii) (iii) (iv) (v)
space–time periodic structures; quasiperiodic traveling wave; stationary wave; spiral wave; spatiotemporal chaos.
If the coupling parameter ε is further increased, transient pattern is observed. A domain is unstable and its boundary moves in time till a single domain covers the space. For kinds of purposes a more realistic hypothesis is obviously to allow for parameter fluctuations along heterogeneous lattice. The main point here is to report that many patterns may be easily produced by varying the parameters controlling local diffusion. The results of extensive numerical simulations also show that the type of the system dynamics is determined by the values of λ and ε . If λ has been selected, we run the simulations until they reach a stationary state or until they show a behavior that does not seem to change its characteristics anymore. For different sets of parameters ε , the features of the spatial patterns become essentially different. Of course, similar kinetics behaviors can be exhibited with the variation of λ in the parameter space. But the more patterns can be found if the parameter λ is closer to 3. It is plausible to argue that patterns observed in natural phenomena may be modeled as a combination of two complementary procedures, (i) fluctuations of system parameters; (ii) variation of the initial conditions. Although the affection of random initial values is not the object of the present writing, but it deserves attention and the mathematical mechanism should be explained in a future work. Here, an example has been shown for ε = 0.34 and λ = 2.9 in Fig. 5. Even if all parameters are fixed, different patterns such as stripe pattern (Fig. 5(a)–(b)) and spiral pattern (Fig. 5(c)) emerge because of different random initial values. 4. Conclusion In this Letter, Turing instability conditions have been illustrated by linearization method and inner product technique for the two-
ε = 0.34. (a) t = 100 000. (b) t = 100 000. (c) t = 50 000.
dimensional LCML (4). In the parameters region of Turing instability, many complex structures such as space–time periodic structures, periodic or quasiperiodic traveling wave, stationary wave, spiral waves, and spatiotemporal chaos, etc. may be observed by numerical simulation. In particular, different patterns can be obtained for same parameters. Thus, the discrete Turing patterns depend upon initial values.
Acknowledgements The authors thank the reviewers for the valuable comments and suggestions. This work was financially supported by Tianjin University of Commerce with the grant number of X0803.
References [1] K. Kaneko, Physica D 37 (1989) 60. [2] K. Kaneko, Theory and Application of Coupled Map Lattices, first ed., John Wiley & Sons, New York, 1993. [3] G. He, Commun. Nonlinear Sci. Numer. Simul. 1 (3) (1996) 16. [4] G. He, A. Lambert, R. Lima, Physica D 103 (1997) 404. [5] V. Franceschini, C. Giberti, C. Vernia, Physica D 164 (2002) 28. [6] Z.R. Cherati, M.R.J. Motlagh, Phys. Lett. A 370 (2007) 302. [7] A. Jakobsen, Physica D 237 (2008) 3382. [8] D.S. Herrera, J.S. Martin, Phys. Lett. A 373 (2009) 2704. [9] P. Muruganandam, G. Francisco, M. Menezes, F.F. Ferreira, Chaos Solitons Fractals 41 (2009) 997. [10] J.D. Murray, Mathematical Biology, Springer, Berlin, 1989. [11] A.M. Turing, Trans. R. Soc. Lond. B 237 (1952) 37. [12] L.A. Segel, J.L. Jackson, J. Theor. Biol. 37 (1972) 545. [13] A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics, vol. 10, Springer, Berlin, 1980. [14] J.F. Heagy, L.M. Pecora, T.L. Carrol, Phys. Rev. Lett. 74 (1995) 4185. [15] Y. Chen, G. Rangarajan, M. Ding, Commun. Nonlinear Sci. Numer. Simul. 11 (2006) 934. [16] L. Bai, G. Zhang, Appl. Math. Comput. 210 (2009) 321. [17] H.X. Hu, Q.S. Li, S. Li, Chem. Phys. Lett. A 447 (2007) 364. [18] X.J. Sun, Q.S. Lu, J. Kurths, Physica A 387 (2008) 6679.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Turing instability for a two-dimensional Logistic coupled map lattice L. Xu, G. Zhang ∗ , B. Han, L. Zhang, M.F. Li, Y.T. Han School of Science, Tianjin University of Commerce, Tianjin 300134, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 5 February 2010 Received in revised form 22 June 2010 Accepted 30 June 2010 Available online 9 July 2010 Communicated by A.R. Bishop
In this Letter, stability analysis is applied to a two-dimensional Logistic coupled map lattice with the periodic boundary conditions. The conditions of Turing instability are obtained, and various patterns can be exhibited by numerical simulations in the Turing instability region. For example, space–time periodic structures, periodic or quasiperiodic traveling wave solutions, stationary wave solutions, spiral waves, and spatiotemporal chaos, etc. have been observed. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The similar patterns have also been seen in relevant references. However, the present Letter owes to pattern formation via diffusion-driven instabilities because the system is stable in the absence of diffusion. © 2010 Elsevier B.V. All rights reserved.
Keywords: Logistic coupled map lattice Turing instability Diffusion-driven
1. Introduction Recently, Logistic Coupled Map Lattices (LCML) have been widely studied by a number of authors, for example, see [1–9] and the listed references. A one-dimensional CML with nearest neighbors coupling can be defined as
ε
ε
xti +1 = (1 − ε ) f xti +
f xti+1 + f xti−1 2
(1)
or
xti +1 = f xti +
2
f xti+1 − 2 f xti + f xti−1
,
(2)
where ε is the coupling parameter, the mapping function f (x) = λx(1 − x), and λ ∈ (0, 4]. In the two-dimensional lattices, each map is coupled to four of its nearest neighbors and given by
1 uti + = (1 − ε ) f uti j + j
ε
f uti+1, j + f uti, j +1 4 + f uti−1, j + f uti, j −1
or
(3)
1 uti + = f uti j + ε ∇ 2 f uti j , j
(4)
where
∇ 2 f uti j = f uti+1, j + f uti, j +1 + f uti−1, j + f uti, j −1 − 4 f uti j , and for the sake of convenience, we denote ε4 by
*
ε in (4).
Corresponding author. Tel.: +86 22 26675703; fax: +86 22 26686507. E-mail address: [email protected] (G. Zhang).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.06.065
For LCMLs (1) or (2), some exact periodic traveling wave solutions are obtained in [3], and the existence of quasiperiodic traveling wave solutions is considered in [5]. The analytical solutions are studied in [8], the control of spatiotemporal chaos is investigated in [6], and the symmetry breaking bifurcations are discussed in [7]. In particular, pattern dynamics of (1) or (3) have also attracted considerable attention [1,2,4,9]. It has been found that LCMLs exhibit a variety of space–time patterns: kink–antikink, space–time periodic structures or wavelike patterns, space–time intermittency and spatiotemporal chaos [1,2]. The numerical simulations and theory analysis of wavelike patterns are considered in [4]. The strong coupling cases are investigated in [9]. It is well known that Turing patterns are very important for a reaction–diffusion system. They have been proposed as mechanisms for biological pattern formation in embryological and ecological context [10]. All such works are based on the pioneering work of Turing [11]. In [12], the authors were the first to call attention to the fact that Turing’s idea would be applicable in ecological situation also. They conjectured that the nature of the equations which describe chemical interaction does not seem fundamentally different from the nature of those which describe ecological interaction among the species. Again, the idea that dispersal could give rise to instabilities and hence to spatial pattern was due to a number of authors [13]. A reaction–diffusion system exhibits diffusion-driven instability or Turing instability if the homogeneous steady state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present [10]. Unfortunately, a single reaction–diffusion partial differential equation cannot produce Turing instability. In fact, the systems (2) and (4) can been seen as discrete systems with diffusion terms. Then, can such systems produce Turing instability? Further, can Turing patterns be obtained? In fact, the frozen random, the pattern selection
3448
L. Xu et al. / Physics Letters A 374 (2010) 3447–3450
and the fully developed spatiotemporal chaos via spatiotemporal intermittency have been observed in [5] and [6]. The generalized Turing patterns emerging from global synchronized limit cycles or chaotic states have been studied in [14,15]. However, there has been little attempt to study Turing patterns for CMLs. In this Letter, we will answer the above problems for the system (4). To this end, we need to add the periodic boundary conditions
uts,0 = uts,m ,
uti,1 = uti,m+1
(5)
and t ut0,k = um ,k ,
t ut1,k = um +1,k
(6) Z + , where m is a positive
for k, s ∈ {1, 2, . . . , m} = [1, m] and t ∈ integer. The Letter is organized as follows. After a brief presentation of the logistic model, the Turing instability theory analysis will be given for the two-dimensional Logistic coupled map lattices. Turing instability conditions can then be deduced combining linearization method and inner product technique in Section 2. Based on the results of Section 2, a series of numerical simulations are performed and different patterns such as space–time periodic structures, periodic or quasiperiodic traveling wave, wave, spiral waves, and spatiotemporal chaos, etc. can be selectively realized in Section 3. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The final section is the conclusion. 2. Turing instability
(7)
has already been investigated in detail. With the increase of nonlinear parameter λ, the attractor of the isolated logistic map shows the bifurcation from the fixed point x∗ = (λ − 1)/λ to period 2, and then the period-doubling cascade to the onset of chaos. For discussing Turing instability of the problem (4)–(6), we will assume that the nonlinear parameter λ satisfies the condition 1 < λ < 3. At this time, the positive fixed point of (7) exists and is asymptotically stable. We also need to consider eigenvalues of the following equation
∇ 2 X i j + λ X i j = 0,
(8)
with the periodic boundary conditions
X i ,1 = X i ,m+1
X i ,0 = X i ,m ,
(9)
and
X 1 , j = X m +1 , j .
X 0, j = X m , j ,
(10)
In view of [16], the eigenvalue problem (8)–(10) has the eigenvalues
(l − 1)π (s − 1)π = kls2 λl,s = 4 sin2 + sin2 m
m
for l, s ∈ [1, m].
(11)
The linearization equation of (4) can be written by
1 uti + = (2 − λ) uti j + ε ∇ 2 uti j . j
(12)
Then taking the inner product of (12) with the corresponding ij eigenfunction X ls of the eigenvalue λl,s , we see that m
i , j =1
ij 1 X ls uti + j
= (2 − λ)
m
i , j =1
ij X ls uti j
+ε
m
i , j =1
ij X ls ∇ 2 uti j
.
(13)
ij
2 U t +1 = (2 − λ) 1 − εkls Ut.
(14) ij
If U t is a solution of (14), then uti j = U t Xls is also clearly a solution of (12) with the periodic boundary conditions (5) and (6). Thus, the unstable system (14) will imply that the problem (4)–(6) is unstable. By the above discussion, we easily see that the conditions of Turing instability for the problem (4)–(6) are: there exist l, s ∈ [1, m] and ε > 0 such that
ε>
λ−1 (l−1)π 4(λ − 2)(sin2 m
+ sin2
( s −1 ) π
+ sin2
( s −1 ) π
m
)
for 2 < λ < 3
(15)
for 1 < λ < 2.
(16)
or
ε>
3−λ 4(2 − λ)(sin2
(l−1)π m
m
)
3. Numerical simulation and discussion For the problem (4)–(6), we employed the periodic Neumann boundary conditions with a system size of 200 × 200 space units. Here, we performed a series of two-dimensional simulations, in each, the initial condition was always a small amplitude random perturbation 1% around the steady state. As a numerical example, we firstly consider the distribution of time evolution. Let λ = 2.9, then (2.9 − 1)/2.9 ≈ 0.6552 and
In this section, we will consider Turing instability for the problem (4)–(6). It is well known that the logistic model of the form 1 xti + = λxti j 1 − xti j j
m
Let U t = i , j =1 Xls uti j and use the periodic boundary conditions (5) and (6), then we have
max
l,s∈[1,200]
sin2
(l − 1)π 200
+ sin2
(s − 1)π 200
= 2.
Thus, when ε > 0.2639, the problem (4)–(6) is Turing instable. Fig. 1(a)–(f) exhibits in detail the distribution of time evolution. In Fig. 1(a), the symmetry breaking around the fixed point can be observed. Fig. 1(b)–(d) shows the self-organization process of the system, space–time periodic characteristics begin to appear, and then steady periodic structures emerge in both Fig. 1(e) and Fig. 1(f). To explore clearly the dynamical behavior of 2-dimensional coupled map lattices, we investigate the effect of coupling parameter by keeping the nonlinearity parameter λ of the system fixed. Following the arrows we can realize different wavelength patterns by adjusting the coupled strength increasingly, as depicted in Fig. 2. A stable pattern of rhombi shapes for ε = 0.32 is given in Fig. 2(a). Fig. 2(b)–(c) shows the stable traveling wave pattern for ε = 0.33 and ε = 0.34, in particular, clear stripe pattern appears in Fig. 2(b), similar patterns can also be seen in [15,17]. Comparing Fig. 2(b) with Fig. 2(c), we can obtain the similar quasiperiodic traveling wave but different spatial behaviors. In Fig. 2(b), the direction of spatial propagation of the waves is the vertical axis, but diagonal direction in Fig. 2(c). An interesting situation is depicted in Fig. 3 where an ordered circular wave pattern [18] first is seen. Then with the evolution time proceeding, a stable pattern of square shapes, namely, stationary wave is observed. If we let ε = 0.35, spiral structure will be obtained in Fig. 4(a). With the further increase of ε , the system will gradually lack longtime, large-distance coherence in spite of an organized regular behavior at the local scale. Spatiotemporal chaos which absolutely arises from spatial diffusion will emerge. Fig. 4(b)–(c) just puts to the proof. The above simulations show that the dynamics of a lattice site changes from the fixed point, with the increasing of ε in the Turing instability region. We have seen:
L. Xu et al. / Physics Letters A 374 (2010) 3447–3450
Fig. 1. Snapshots of contour pictures of the time evolution of CML system at some instants when 199 900. (f) t = 200 000.
Fig. 2. Pattern selection with the increase of
3449
ε = 0.335. (a) t = 100. (b) t = 3100. (c) t = 3200. (d) t = 21 300. (e) t =
ε in the Turing instability region. (a) t = 50 000, ε = 0.32. (b) t = 50 000, ε = 0.33. (c) t = 100 000, ε = 0.34.
Fig. 3. Snapshots of contour pictures of the time evolution of CML system at some instants when (e) t = 32 670. (f) t = 32 870.
ε = 0.345. (a) t = 50. (b) t = 9700. (c) t = 26 450. (d) t = 32 130.
3450
L. Xu et al. / Physics Letters A 374 (2010) 3447–3450
Fig. 4. Pattern selection with the increase of
ε in the Turing instability region. (a) t = 200 000, ε = 0.35. (b) t = 200 000, ε = 0.36. (c) t = 200 000, ε = 0.37.
Fig. 5. Selective patterns resulting from random initial values when
(i) (ii) (iii) (iv) (v)
space–time periodic structures; quasiperiodic traveling wave; stationary wave; spiral wave; spatiotemporal chaos.
If the coupling parameter ε is further increased, transient pattern is observed. A domain is unstable and its boundary moves in time till a single domain covers the space. For kinds of purposes a more realistic hypothesis is obviously to allow for parameter fluctuations along heterogeneous lattice. The main point here is to report that many patterns may be easily produced by varying the parameters controlling local diffusion. The results of extensive numerical simulations also show that the type of the system dynamics is determined by the values of λ and ε . If λ has been selected, we run the simulations until they reach a stationary state or until they show a behavior that does not seem to change its characteristics anymore. For different sets of parameters ε , the features of the spatial patterns become essentially different. Of course, similar kinetics behaviors can be exhibited with the variation of λ in the parameter space. But the more patterns can be found if the parameter λ is closer to 3. It is plausible to argue that patterns observed in natural phenomena may be modeled as a combination of two complementary procedures, (i) fluctuations of system parameters; (ii) variation of the initial conditions. Although the affection of random initial values is not the object of the present writing, but it deserves attention and the mathematical mechanism should be explained in a future work. Here, an example has been shown for ε = 0.34 and λ = 2.9 in Fig. 5. Even if all parameters are fixed, different patterns such as stripe pattern (Fig. 5(a)–(b)) and spiral pattern (Fig. 5(c)) emerge because of different random initial values. 4. Conclusion In this Letter, Turing instability conditions have been illustrated by linearization method and inner product technique for the two-
ε = 0.34. (a) t = 100 000. (b) t = 100 000. (c) t = 50 000.
dimensional LCML (4). In the parameters region of Turing instability, many complex structures such as space–time periodic structures, periodic or quasiperiodic traveling wave, stationary wave, spiral waves, and spatiotemporal chaos, etc. may be observed by numerical simulation. In particular, different patterns can be obtained for same parameters. Thus, the discrete Turing patterns depend upon initial values.
Acknowledgements The authors thank the reviewers for the valuable comments and suggestions. This work was financially supported by Tianjin University of Commerce with the grant number of X0803.
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