Onset of turbulent pattern in a coupled map lattice. Case for soliton-like behavior

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11 April 1988

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ONSET OF TURBULENT PATTERN IN A COUPLED CASE FOR SOLITON-LIKE BEHAVIOR

MAP LATTICE.

K. AOKI and N. MUGIBAYASHI Department ofElectrical Engineering, Faculty OfEngineering, Kobe University, Rokkodai, Nada, Kobe 657, Japan Received 3 November 1987; revised manuscript Communicated by A.R. Bishop

received

9 January

1988; accepted

for publication

5 February

1988

The onset of turbulence in sohton-like behavior has been investigated in a specified coupled map lattice of Nequivalent switching elements with the firing activities S: (i= 1,2, .... N) at a discrete time t. In the looped lattice with one-way coupling of S:+ I = O(S:, .S-’ ), a soliton propagates into the direction of down flow with velocity U. By injecting another soliton which propagates assumingadummy loopp,,, =@(g,, p,:“)), a against the down flow with velocity Y’ (it is made by S:, I = @(S:,S:-‘,$+I), collision between the two solitons is realized in the original loop. As a function of v (v’ is kept constant), the collision becomes turbulent with the velocity lower than a critical value ZJ,.The onset of turbulence is discussed in terms of separation distance, Lyapunov exponent and symbolic dynamics

A great number of theoretical studies on lowdimensional chaos [ 11 have successively given a global description for the onset of turbulence and also the statistical properties of weak turbulence observed in a variety of nonlinear physical systems [ 2,3 ] which possess spatially high-dimensionality. When the elementary low-dimensional systems are nonlinearly coupled (e.g., the coupled logistic lattice (CLL) [ 41 or the coupled neuron-like lattice (CNL) [ 5]), the dynamical behavior becomes much more complex than that in a single map function, and the symbolic dynamics of the coupled map lattices (CML) behave like cellular automata [4-91. Each element in a CML is usually assumed to be identical and the nonlinear interaction is localized within a few lattice sites. So, CML’s may be considered as computational tools for the complex nature such as in spin glass [ 7 1, polyacetylene [ lo] and neuron nets [ll-131. As Kaneko [ 4 ] has mentioned, there are main topics in the study of CML’s: characterization of patterns, transition to chaos, characterization of spatial complexity, estimation of the fractal dimension, Lyapunov spectra and so on. The CML’s often show a transition from a regular to an irregular pattern when a control parameter (e.g., the coupling 0375-9601/88/$ (North-Holland

03.50 0 Elsevier Science Publishers Physics Publishing Division )

strength) is varied [ 5 3. In a specified CNL [ 141, soliton-like behavior (soliton collision) becomes irregular, showing a transition to non-soliton-like behavior. In this paper, our special concern is to characterize the transition from regular to irregular patterns and to find a criterion for the onset of turbulence in the specified CNL which exhibits soliton collision. In previous papers [ 5,141, we have described cellular automaton-like behvior [ 51 as well as soliton collision [ 141 in the CNL. The soliton model we study is given by [ 14 ] loop 1:

S;,,

=@(-4$:+2-Rsin(2r&)

-Fy(s:+‘), loop 2:

)

(1)

s;, , = @( -4S:+2-R

sin(27rSj)

(2)

-rfb(S:-‘,$+I))) where ) ,

(3)

fb(s;-‘,~+‘)=sin[4x(S:-‘+S:+‘)],

(4)

f”(S:+’

)=sin(4r&+’

and @(y)=[l+tanh(y)]/2. B.V.

(5) 349

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Here, we consider two loops of the CNL’s; 9, and S; are the order parameters of the ith element (i = 1, 2, .... N) at the discrete time t (t= 1, 2, ...) in loop 1 and loop 2, respectively, 8, R, T and T are control parameters, and functions? and f” denote local interactions with the nearest neighbours. Each element corresponds to a random nerve net with the firing activities .$EI= [O, 1 ] and S:EI= [0, 11 [ 11,151. The individual function, e.g., map -i,+, =@(JJ($)) with f=O in loop 1 has n-rotaS tional symmetry around the equilibrium point s’~O.5, wherey(S:)=-45,+2-lsin(2ng). The individual map function has the following properties as a function of R4, (i) OQE< l/n, 5’=0.5 is unstable and s”, is oscillatory with period 2. (ii) l/x 1.387, s”, shows a period-doubing bifurcation route to chaos. Fig. la shows an example of the map function for R”= 0.9, in which s’= 0.5 is a stable equilibrium point (curve b is discussed below). When N equivalent switching elements are coupled ( T# 0)) properties (i)- (iv) no longer hold for any value of R since each element always suffers the

%+ I

Fig. 1. Iterative map y(S,)= -49,+2-dsin(2n$,) +Z-~sin(2n~~)-~sin[4x(l-$,)] R~0.9 and T~0.2.

350

functions $+, =@(y(g,)), where in curve (a), and y($!) = -43, in curve (b). Parameters

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1I .4pril 1988

local stimuli from the nearest neighbours. In the coupled lattice map, the firing activity s”, is trapped in either one of two basins of attraction which form a symmetric pair around Ef=0.5. The symbolic dynamics in the one-way coupling of loop 1 (eq. ( 1) ) have been described in detail in ref. [ 5 1. With ?‘# 0, some initial disorders run through the lattice chain of antiferro-like structure, under the periodic boundary conditions (PBC) SF+’ =Sj Under the absorbing boundary conditions (ABC) L?r+’ = 0. the “soliton-like” disorders disappear when they arrive at the terminating lattice site (i= 1 ), the spatial configuration resulting in a stationary antiferro-like structure lOlO...lO 10. The antiferro-like structure can be explained qualitatively by curve b of fig. 1, which is obtained by assuming the antiferro-like configuration p, + s”:’ ’ = 1 with R= 0.9 and i7’=0.2 in eq. ( 1). Here, s’= 0.5 is destabilized and there appear two stable points 3 (1) and 9 ( t ) which satisfy p(~)+p(r)=l. If L?‘(l) is realized in the ith element, then ,!?““‘=S’(J) (m: even) and pi”‘=p(r) (m: odd) result. With T=O.2, the solidary domain is observable only with 0.66 5 R’L 1.2 I [ 51. The velocity of the soliton increases with increasing I?, e.g., tl,=15/117 at l?= 1.2 and u,= 1 at Z?=O.68. A similar situation can be seen in loop 2 if we put S:+ ’ = 0 in eq. (2), but the soliton runs towards the opposite direction due to f”( S:- ’ ). With the local interaction rule .f” (S’- ’ , g,+ ’ ) in loop 2, loop 2 is perturbed by the solitons in loop 1 towards the opposite direction (against the down flow). By the perturbation. loop 2 experiences a “soliton collision”. With loop 2 only, we failed to observe the soliton collision for any interaction rule f” (5’- ’ , S:’ ’ ), which is the reason to use two loops in the present paper. All the iterations were made with double precision. With single precision, some differences were found in the evolution of the patterns in the transient region, which is attributed to the sensitive dependence on initial conditions. Initial conditions were chosen to be Sb=&=0.2+5’, I<‘] ~0.0123 ( (5’) =O), where the 5”s are 2N random variables. In this paper, the local interactionsfa, .fb are taken into account from the second iteration. For both loops of N= 90, ABC with 3:;” =S:’ = 0 were initially used ( t < t,), then the loops were closed at 23 to -,v+ ’ =$ and Sp =S;“;. By doing so, several by PBC S,

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Fig. 2. Freeze effect of wavelength doubling ofS: obtained under ABCand with (a) R=0.80 (wavelength 2), (b) R=0.87 (wavelength 4), (c) R=0.875 (wavelength 8) and (d) R~0.90 (chaotic phase). The bifurcation diagram of S: (i=20-70, 1=301-600)is shown as a function ofR in (e).

domains die out before loop closing, and only a few solitons survive at t a to. The control parameter R was varied in the range 0.8
‘.

,’

‘_

:

pattern

,, ,,,“8

I’

2 w

o
Before describing the soliton colliston, an interesting phenomenon observed with ABC (fO+m) is briefly mentioned. With the ABC, the spatial patterns of S: are frozen (static) after the intial disorders die out, as is shown in figs. 2a-2d. As a function of R, the patterns in loop 2 experience wavelength doubling [ 161 except the antiferro-like structure at the terminating edge, where the spatial wavelength becomes longer and longer by increasing R, and finally the pattern becomes chaotic (fig. 2d). In fig. 2e, the bifurcation diagram is shown as a functionofRfor$ (i=20-70, t=301-600). Thewavelength doubling caused by the perturbation of wavelength 2 in loop 1 is very reminiscent of period doubling in the nonautonomous case of low-dimensional systems. The wavelength doubling is related with the onset of turbulence in the soliton collision observed with PBC, as is discussed later. A typical example of the soliton collision with PBC is shown in fig. 3a, which is the spatio-temporal pattern of S: obtained with R = 0.83 and to= 12 1. In fig. 3b, one-dimensional mapping of the firing activity at the middle element (i=45) is shown with t < 1.5 x 10’. The injected soliton denoted by “1” in fig. 3a runs with the velocity V,=0.65, while the velocity v2 of soliton 2 is 0.7 1. At the top of fig. 3a is the cross-sectional pattern at t=301. Two solitons collide stably, no disorders are created or annihilated. In fig. 3b, an equilibrium point which is located nearby an arrow is destabilized at each time when the solitons pass through the element. b.

a

Fig. 3. (a) Spatio-temporal

11April 1988

F

of S: obtained

,

r N=90

with R=0.83

%+I 0.7 r

i=45

and lo= 121. (b) One-dimensional

mapping

ofS:+,

versus S; (i=45,

1.5x 105).

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obtained by the nearby trajectory of fig. 4b with Sl’=p,+d, %‘=p:+d (d=lOmh), as is discussed below. In order to characterize the turbulence of the patterns as well as the orbital instability of the firing activities, the separation distance d, and the Lyapunov exponent I are first investigated, as usual [ 171. According to ref. [ 17 1, and regarding both loops as a ZN-dimensional system, the total separation distance is given by

b.

b: d,(f)=

(’ i

,=I

C.

(s;

2 (s:

-sy,y+

/=,

= [ (iw, y+ (iv?. )2]‘/2 )

Fig. 4. Symbolic presentation of the spatio-temporal patterns with (a) R=0.83. (b) R=0.88 and (c) R=0.93, where f,,= 121. Pattern (b’ ) was obtained as the nearby pattern of (b). Symbolic dynamics are investigated cated symbols of 1 or 0 according

by the trunto S: aO.5 or

S: -c 0.5, respectively. Fig. 4 shows the change of the cellular automata as a function of R. The soliton-like behavior becomes irregular with Rk0.87; an example for R=0.88 is shown in fig. 4b. The irregularity originates from the creation or annihilation of dusts and domains. Fig. 4b’ is the pattern which is

I,?

-R,)’

>

(6) (7)

where t’=t-t, (t,>to), S$‘=Sb+A, $‘=,?&+A, d,( 0) = $%A and A is the initial small deviation in each element at t’ = 0 (t = f, ). In the vector presentation, the initial distances are (SS, I = (S& ( =,%A. It was found that the separation distance I&S, 1 in the driving loop 1 rapidly converges to zero. So. the separation distance in eq. (7) gives a criterion for the onset of orbital instability in loop 2. Fig. 5a shows the asymptotic behavior of d,( t’ ) for R~0.84, 0.86 and 0.88, where A= lo-” and t, ~200. With R~0.88, the nearby trajectories apparently diverge exponentially, i.e., ds(t’ ) B ,,,%A, which means the orbital instability in loop 2. The result is consistent with the dusty pattern in figs. 4b, 4b’. With R=0.86, d,(t’ )

log(d,) O-I-

a. *#“u*w /” R= 0.88

{F

0

2800 e

t’

Fig. 5. (a) Separation distance in the logarithmic scale (loglo ) for R=0.84. 0.86 and 0.88. Initial separation distance is indicated by dy ( =,‘?%?A), where A’=90 and A= 10eh. Loop closing and the initial deviation d: were made at ti,= 121 and t, =200. respectively. (b) Lyapunov exponents as a function of R (0.83
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When ABC is converted to PBC at t> t,,, the disorder at the terminating edge runs through the lattice chain as soliton 2. The background laminar structure of wavelength 2k (k= 1,2, ...) was found to be very sensitive to the external perturbation (e.g., to the soliton injection from loop 1). With R~0.87 (wavelength 4), the laminar structure is collapsed into the chaotic phase by the soliton injection, resulting in the dusty pattern. The situation is very similar to the external noise effect for period doubling in low-dimensional systems [ 191. In summary, we have investigated soliton collision and the onset of the turbulent pattern in a specific model of CNL. The turbulence is caused by the collapse of the background laminar (wavelength doubled) structure by soliton injection from a dummy loop rather than by the soliton collision itself. In order to find out the criterion for the onset of the turbulent pattern. we have introduced the separation distance of the nearby patterns in the symbolic dynamics (i.e., ) Y,) ). The ensemble averages of ( Y’; j and of the spatial measure entropy in the difference pattern behave like C[exp(n’s) - 11 with the characteristic exponent A’. The presented idea can be extended to the problem of turbulence in a wide class of CML’s. Shortly, the characteristic exponent gives a measure for the irregularity of the spatio-temporal attractor (we do not say for the complexity of patterns). Not mentioned is the spatial measure entropy of the original pattern X, (local entropy SL ( t ) in ref. [ 9 ] ) , which fluctuates around an average as a function of t in figs. 4a-4c. Since the irregularity is an intuitive concept, there should be different quantitative definitions, one of which will be also given by the standard deviation of SL ( t ). The freeze effect of the wavelength doubled structure and the chaotic structure in fig. 2 will be an interesting result, since the frozen pattern may be considered as a kind of “memory” in CNL’s. Finally the different

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behavior of the soliton turbulence observed in a coupled circle map lattice [ 201. e.g., many kinks with different velocities and “glider guns”. was not seen in the present model. References [I]J.-P.Eckmann,Rev.Mod.Phys.53(1981)643. [ 21H.L. Swinney and J.P. Gollub, Phys. Today 3 I ( 1978) 41, [3] J.S. Turner. J.C. Roux, W.D. McCormick and H.L. Swinney. Phys. Lett. A 85 ( 1981 ) 9: Chaos and statistical methods (Springer, Berlin, 1984). [4] K. Kaneko, Prog. Theor. Phys. 72 (1984) 480; Phys. Lett. I\ III (1985) 321. [ 51K. Aoki and N. Mugibayashi, Phys. Lett. A I14 ( 1986) 425. [6]S. Wolfram,PhysicaD IO (1984) 1. [7] S. Kirkpatrick and D. Sherrmgton. Phys. Rev. I7 ( 1978 ) 4384: G.Y. Vichniac. Physica D IO ( 1984) 96. [ 81J.K. Park, K. Steiglitz and P. Thurston. Physica D 19 ( 1986) 423. [9] G. Coscnza and F. Neri, Physica D 27 ( 1987) 357. [IO] W.P. Su. J.R. Schrieffer and A.J. Hecger, Phys. Rev. Lctt. 42 (1979) 1698. [ 1 I ] S. .Amari. IEEE Trans. Syst. Man Cybcm. SMC-2 ( 1972) 643.

[ 121 J.J. Hoplield and D.W. Tank. Biol. Cybern. 52 ( 1985) 141: J.J. Hopfield. Proc. Natl. .4cad. Sci. 79 ( 1982) 2554. [ 131 M.Y. Chat and B.A. Huberman. Phys. Rev. A 28 ( 1983) 1204: K.E. Kiirtcn and J.W. Clark. Phys. Lctt. .A I I4 ( I986 ) 4 13. [ 141 K. .4oki and N. Mugibayashi, Proc. 1st Int. Symp. Science on Form (KTK ScientiIic. Tokyo. 1986) p. I 13. [ 151 K. Aoki. 0. Ikezawa and K. Yamamoto. Phys. Lett. A 98 (1983) 217. [ 161 A. Lahiri and S.S. Ghosal. Phys. Lett. A I24 (1987) 47. [ 171 M. Imada, in: Chaos and statistical methods (Springer. Berlin. 1984) p. 176. [ 181 G. Benettin. L. Galgani and J. Strelcyn, Phys. Rev. A I4 (1976) 2338: K. Kaneko. Physica D 23 ( 1986) 436. [ 191 J.P. Crutchfield and B.A. Huberman, Phys. Lett. .A 77 (1980) 407. [20] J.P. Crutchfield and K. Kaneko. Phenomenology of spatio-temporal chaos, in: Direction in chaos. ed. Hao Bai-lin (World Scientific. Singapore. 1987).