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Regular and chaotic spatial patterns in a coupled map lattice
Volume 148, number 8,9
PHYSICSLETTERSA
3 September 1990
Regular and chaotic spatial patterns in a coupled map lattice O.A. D r u z h i n i n Space Research Instttute, 117810Moscow, USSR
and A.S. M i k h a i l o v MoscowState Umverslty, PhystcalDepartment, 117234Moscow, USSR Received 14 February 1990; revised manuscript received 15 June 1990; accepted for publication 5 July 1990 Communicatedby A.P. Fordy
A coupled map lattice is considered in which diffusive couplingbetween elements affects a bifurcation parameter of the map. It ~sshown that the formation of stable spatial soliton-likepatterns is possible. In the vicinity of the bifurcation point an explicit form of the solution is found. Analyticalresults are confirmedby numerical simulationsof the logisticcoupled maps.
1. Coupled maps (or one-dimensional coupled map lattices) have been widely used recently as models of complex nonlinear distributed systems [ 1-5 ]. The state of any element of such a system in a point with discrete coordinate r at the time moment n is characterized by the variable x . ( r ) . Evolution in discrete time is specified by the map
xn+l (r) =f(xn(r), z, DO2x~(r) ) ,
( 1)
where the term DOex=(r) accounts for diffusive coupling of the elements,
02x=(r)=½[x~(r+l)+x~(r-1)-2x~(r)] .
(2)
The case w h e n f ( x ) is the logistic function and the dependence of the map ( 1 ) on the diffusive term is additive,
Xn+l =zxn( 1 --X~) + DO2x~(r) ,
(3)
has been thoroughly investigated. The formation of spatial patterns (kink solutions) in such systems and some scaling properties of their spatio-temporal dynamics have been studied [1,2,5]. In this paper we consider the coupled map lattice in which the coupling o f t b e elements affects only the bifurcation parameter z of the map ( 1 ). Namely, we assume that ElsevierSciencePublishers B.V. (North-Holland)
Zn(r) =Zo + Dd2xn(r) ,
(4)
where Zo corresponds to the homogeneous state of the entire system, i.e. to x~(r) = const for any r. Such type of diffusive coupling appears in cellular automata models of some chemical and biological systems [6 ]. In the limit of continuous media it corresponds to the reaction-diffusi0n equation dx/dt=DO2x/ 0 2r +3'( x ) with coefficient D = D [x (r, t ) ] depending on the state of the system x(r) and, thus, varying both in space and time. We find that in the system of coupled maps (1) with the diffusive coupling specified by (4) the formarion of steady stable spatial patterns is possible. I f the parameter Zo is close to the value z¢ at which the bifurcation of the attractive luted point of the map (1) takes place (i.e. the cycle of period 2 sets in), the steady spatial pattern x ( r ) can be found in an explicit form. There are also some cases when small inhomogeneous perturbations of the system bring about an unbounded growth of disordered spatiotemporal oscillations. Our analytical results are confirmed by numerical simulations for the logistic map
Xn+l(r)=x~(r)[zo+DO2x~(r))][1--x~(r)].
(5) 429
Volume 148, number 8,9
PHYSICS LETTERS A
3 September 1990
2. Let us consider the case when the parameter Zo exceeds its bifurcation value zc and the fixed point xc =f(x~, z¢) of the map becomes unstable. Thus, the cycle of period 2 sets in. For the parameter value which is close to Zo the points x~ and x2 of the cycle are still close to the bifurcation point x¢, i.e. xt = x c-~, x2=Xc+ ~ with some small correction ~. If the system is weakly inhomogeneous, its state near the cycle point xn is given by
equation of motion for a fictitious particle in a certain potential U,
x,(r)=x~ +u,(r) ,
f ~ ( x o , Zc) S= 4Df~(Xc, z¢)"
(6)
where the term un(r) specifying the inhomogeneity is small, i.e. l un I ~ x °. Substitution of expression (5) into the twice iterated map ( 1 ) with the parameter zn(r) given by (4) yields
x, +un+2(r) =f{f(x, +u,(r), z,],
Zn+ l
},
(7)
where z, =Zo +DOZu,(r), z,+] =zo +DO2u,+n (r) .
(8)
Using the Taylor expansion of (7) at the point x~ up to the third order of u, we obtain the map for weak inhomogeneities in the vicinity of the point x~, i.e.
O2U
OU
"Or2 -
Ou'
(12)
where
U=Su2(u-2E) 2
(13)
and
Using this analogy, we can easily find different solutions of eq. (10). When the coefficient of diffusive coupling between elements is negative, i.e. D < 0 , U(u) has the form of a potential well (fig. 1 ). Hence, there is a class of periodic solutions u (r) which, for small amplitudes, have the spatial period k= 2r~Sn/2/E. When the amplitude is increased, this period grows and finally we come to a special kink solution that corresponds to the motion of the "particle" from xn to x2. This kink solution is given by
u(r) =3e{1 +exp[2eSt/2(r-ro) ]} - n
(14)
When D> 0, as follows from the form of the potential (dashed line in fig. 1 ), there are soliton-type solutions and solutions corresponding to small oscillations in the vicinity ofx~ or x2. No kinks are possible in this case.
Un+2(r) =fx(X2, Zo)fx(Xl, Zo)Un "4- [fx(Xl, ZO)"l'fx(X2, ZO)lfz(xn, go)DO2un
+ ~fxx(x,, zo)f~(x2, Zo)U~ + ~f,~(x2, Zo) Ifx(x,,.Zo)U. + ~f,=(x,, Zo)U~. +... ] 2 .
(9)
0.5
For the function u(r), which is invariant under the map (9), so that u , + 2 ( r ) = u , ( r ) = u ( r ) , the following equation holds, D O2u -t- a u -- flu 2 -t- ~,13~- O ,
a=a~ 2, #--~ae,
"" x. x.x.
/I
\
/ n
.
I
*
i
i
U/£
( l 1)
Eq. (10) describes the weak stationary inhomogeneities in the vicinity of the bifurcation point of the map (1). If these inhomogeneities are sufficiently smooth, we can treat r as a continuous variable. Then (10) can formally be interpreted as an 430
/ / f ""
-d.5
7=½a,
a=fl=(xc, z,) /2f~(xo, zo) .
D>O \
(10)
where the constant coefficients are
U
-0 5 Fig. 1. Potential U(u) for D>0 (dashed line) and D<0 (solid line); E=0.07.
Volume 148,number8,9
PHYSICSLETTERSA
Stability of the stationary solutions obtained should be checked. Using (9) for small perturbations u (r) = u (r) + 8u (r), where 18u [ ~: u and performing the Fourier transform of it, we can obtain the following stability condition in the form
I 1+2Dk2f,(x~, zc)J ~<1,
(15)
where the k's are wavenumbers of the Fourier modes of the perturbation 8u(r). It follows from (15) that for D > 0 all stationary solutions should be unstable. When D < 0, only perturbation modes with k< k~, k¢ = ~ ( x c , z~)IDI ]-1/2,
(16)
damp. Hence, in a truly continuous case no stable solution can exist, because there would always be unstable modes with k>k~. However, if coordinate r changes discretely (which is actually the case), stable solutions are still possible. We can expect that the kink solution will be stable in the case of discrete r's if the value of/% given by (16) is larger than ~t (i.e. the maximum possible wavenumber in the discrete case r=0, 1, 2, 3, ...). When the absolute value iDI of the diffusive coupling coefficient is increased, the stability threshold kc may become less than ~, so that the unstable modes k c < k < n will grow and spatiotemporal oscillations will set in. The above results should hold for an arbitrary map OC
1 O0
I
0 50
0,5C
0 O0
n=18 0
--
~ FO
~1 100
3 September 1990
0 O0
~=:c
b
0
~0
100
o
sb r
~6o
1 O0
1 O0
i ~
0 50
0 50
brae n=22 0 O0
c
sb r
160
o oo
Fig. 2. Development of inslability of the kink solution for positive D = 1.0. Various time moments are shown: (a) n = 18; (b) n=20; (c) n=22; (d) nffi24.
431
Volume 148, number 8,9 1.00
3September 1990
PHYSICS LETTERS A
(1), provided that it has a period-doubling bifurcation. Particularly, they can be applied to the case of the logistic map.
X:
3. We performed numerical simulations of the logistic map x.+,(r)=x.(r)[zo+Dd2x.(r)l[1-x.(r)]. (17)
0.50
with discrete coordinate taking values r = 0 ..... N + 1 and
02x.(r)=½[x.(r+l)+x.(r-1)-2x.(r)] 0.00
5b
0
160
F
4
1 O0
0 50
0 50
~=28,Z9
50
a 160
I O0
0 50
0 5O
n=38 o
C 5b
~o
b
n=32 000
1 O0
0 O0
(18)
Flg. 3. T h e s t a b l e k m k f o r D = - 0 . 4 . S o h d a n d d a s h e d h n e s c o r respond to two consecutive u m e moments.
1 O0
0 O0
.
0 O0
o
sb f
5'o
~6o
-
Jl
r
Fig. 4. Development of instability o f the kink solution for D = - 3.0. Various Ume m o m e n t s are shown: ( a ) n = 28 ( solid line ) and n = 29 (dashed line); (b) n = 3 2 ; (e) n = 3 8 ; (d) n = 4 0 .
432
Volume !48, number 8,9
PHYSICSLETTERSA
1.00
1.00
0.50
0.50
3 September1990
a 0.00
5b
,60
o.oo 0
r
5b
b ~6o
r
Fig. 5. Transformation of small oscillations (a) n=30,31 into the set of stable kinks (b); D = -0.4.
The boundary conditions were x , ( 0 ) = x , ( 2 )
and
x,,(N- l ) =x. (N+ l ). The first period-doubling bifurcation for this logisticm a p occurs at zc= 3. In numerical simulations the value Zo= 3.05 was chosen. Then the cycle points x, =0.59 and x2=0.738 are close to the bifurcation point xc=2/3 and e~,0.07. In numerical experiments we had N = 100. For D > 0 we find that there are no stable stationary kink solutions. Small perturbations always result in growth of modes with large wavenumbers, so that disorder spatio-temporal oscillations with increasing amplitudes set in (fig. 2). If the diffusion constant is negative and its absolute value is sufficiently small, a steady kink pattern forms, which is described fairly well by the analytic solution ( 11 ) (fig. 3 ). If the absolute value of D is increased further, the stability threshold ~ becomes less than km~ = ~t and no stable solutions exist. The growth of unstable modes with wavenumbers ~ < k < x leads to irregular spatio-temporal oscillations which grow almost exponentially in time, as it is evidenced by fig. 4. Small-amplitude stationary periodic patterns are unstable. Numerical simulations show that after some time they transform into a sequence of steady kinks
Therefore, we conclude, that in the one-dimensional coupled map lattice with diffusive coupling specified by (4) steady spatial kink patterns can form. In the vicinity o f the bifurcation point the explicit kink solution (14) is found analytically. Numerical analysis performed for the logistic coupled maps shows good agreement with analytical results. Disordered spatio-temporal patterns are also observed. Detailed numerical analysis is to be presented elsewhere.
References [l]lL Kaneko, Prog. Theor. Phys. 72 (1984) 480; 74 (1985) 1033. [2] IC Kaneko,PhysicaD 37 (1989) 60. [3] C.A. Bunimovichand Ya.G. Sinai, Nonlinearity 1 (1989) 491. [4] S.P. Kuznetsovand A.S. Pikovsky,Radiofizik, 28 (1985) 308. [5] A.P. Kuznetsov,S.P. Kuznetsovand I.R. Sataev, in: Proc. 5th Int. Workshopon Nonlinear and turbulent processesin physics,Vol. 2 (NaukovaDumkR;leninmad, 1989) p. 383. [6] O.A. Druzhinin and A.S. Mikhailov, Spatio-temporalchaos and synchronization in probabilistic cellularautomata, preprint N1626, SpaceReSearchInstitute, Moscow(1989).
(r~ 5).
433
PHYSICSLETTERSA
3 September 1990
Regular and chaotic spatial patterns in a coupled map lattice O.A. D r u z h i n i n Space Research Instttute, 117810Moscow, USSR
and A.S. M i k h a i l o v MoscowState Umverslty, PhystcalDepartment, 117234Moscow, USSR Received 14 February 1990; revised manuscript received 15 June 1990; accepted for publication 5 July 1990 Communicatedby A.P. Fordy
A coupled map lattice is considered in which diffusive couplingbetween elements affects a bifurcation parameter of the map. It ~sshown that the formation of stable spatial soliton-likepatterns is possible. In the vicinity of the bifurcation point an explicit form of the solution is found. Analyticalresults are confirmedby numerical simulationsof the logisticcoupled maps.
1. Coupled maps (or one-dimensional coupled map lattices) have been widely used recently as models of complex nonlinear distributed systems [ 1-5 ]. The state of any element of such a system in a point with discrete coordinate r at the time moment n is characterized by the variable x . ( r ) . Evolution in discrete time is specified by the map
xn+l (r) =f(xn(r), z, DO2x~(r) ) ,
( 1)
where the term DOex=(r) accounts for diffusive coupling of the elements,
02x=(r)=½[x~(r+l)+x~(r-1)-2x~(r)] .
(2)
The case w h e n f ( x ) is the logistic function and the dependence of the map ( 1 ) on the diffusive term is additive,
Xn+l =zxn( 1 --X~) + DO2x~(r) ,
(3)
has been thoroughly investigated. The formation of spatial patterns (kink solutions) in such systems and some scaling properties of their spatio-temporal dynamics have been studied [1,2,5]. In this paper we consider the coupled map lattice in which the coupling o f t b e elements affects only the bifurcation parameter z of the map ( 1 ). Namely, we assume that ElsevierSciencePublishers B.V. (North-Holland)
Zn(r) =Zo + Dd2xn(r) ,
(4)
where Zo corresponds to the homogeneous state of the entire system, i.e. to x~(r) = const for any r. Such type of diffusive coupling appears in cellular automata models of some chemical and biological systems [6 ]. In the limit of continuous media it corresponds to the reaction-diffusi0n equation dx/dt=DO2x/ 0 2r +3'( x ) with coefficient D = D [x (r, t ) ] depending on the state of the system x(r) and, thus, varying both in space and time. We find that in the system of coupled maps (1) with the diffusive coupling specified by (4) the formarion of steady stable spatial patterns is possible. I f the parameter Zo is close to the value z¢ at which the bifurcation of the attractive luted point of the map (1) takes place (i.e. the cycle of period 2 sets in), the steady spatial pattern x ( r ) can be found in an explicit form. There are also some cases when small inhomogeneous perturbations of the system bring about an unbounded growth of disordered spatiotemporal oscillations. Our analytical results are confirmed by numerical simulations for the logistic map
Xn+l(r)=x~(r)[zo+DO2x~(r))][1--x~(r)].
(5) 429
Volume 148, number 8,9
PHYSICS LETTERS A
3 September 1990
2. Let us consider the case when the parameter Zo exceeds its bifurcation value zc and the fixed point xc =f(x~, z¢) of the map becomes unstable. Thus, the cycle of period 2 sets in. For the parameter value which is close to Zo the points x~ and x2 of the cycle are still close to the bifurcation point x¢, i.e. xt = x c-~, x2=Xc+ ~ with some small correction ~. If the system is weakly inhomogeneous, its state near the cycle point xn is given by
equation of motion for a fictitious particle in a certain potential U,
x,(r)=x~ +u,(r) ,
f ~ ( x o , Zc) S= 4Df~(Xc, z¢)"
(6)
where the term un(r) specifying the inhomogeneity is small, i.e. l un I ~ x °. Substitution of expression (5) into the twice iterated map ( 1 ) with the parameter zn(r) given by (4) yields
x, +un+2(r) =f{f(x, +u,(r), z,],
Zn+ l
},
(7)
where z, =Zo +DOZu,(r), z,+] =zo +DO2u,+n (r) .
(8)
Using the Taylor expansion of (7) at the point x~ up to the third order of u, we obtain the map for weak inhomogeneities in the vicinity of the point x~, i.e.
O2U
OU
"Or2 -
Ou'
(12)
where
U=Su2(u-2E) 2
(13)
and
Using this analogy, we can easily find different solutions of eq. (10). When the coefficient of diffusive coupling between elements is negative, i.e. D < 0 , U(u) has the form of a potential well (fig. 1 ). Hence, there is a class of periodic solutions u (r) which, for small amplitudes, have the spatial period k= 2r~Sn/2/E. When the amplitude is increased, this period grows and finally we come to a special kink solution that corresponds to the motion of the "particle" from xn to x2. This kink solution is given by
u(r) =3e{1 +exp[2eSt/2(r-ro) ]} - n
(14)
When D> 0, as follows from the form of the potential (dashed line in fig. 1 ), there are soliton-type solutions and solutions corresponding to small oscillations in the vicinity ofx~ or x2. No kinks are possible in this case.
Un+2(r) =fx(X2, Zo)fx(Xl, Zo)Un "4- [fx(Xl, ZO)"l'fx(X2, ZO)lfz(xn, go)DO2un
+ ~fxx(x,, zo)f~(x2, Zo)U~ + ~f,~(x2, Zo) Ifx(x,,.Zo)U. + ~f,=(x,, Zo)U~. +... ] 2 .
(9)
0.5
For the function u(r), which is invariant under the map (9), so that u , + 2 ( r ) = u , ( r ) = u ( r ) , the following equation holds, D O2u -t- a u -- flu 2 -t- ~,13~- O ,
a=a~ 2, #--~ae,
"" x. x.x.
/I
\
/ n
.
I
*
i
i
U/£
( l 1)
Eq. (10) describes the weak stationary inhomogeneities in the vicinity of the bifurcation point of the map (1). If these inhomogeneities are sufficiently smooth, we can treat r as a continuous variable. Then (10) can formally be interpreted as an 430
/ / f ""
-d.5
7=½a,
a=fl=(xc, z,) /2f~(xo, zo) .
D>O \
(10)
where the constant coefficients are
U
-0 5 Fig. 1. Potential U(u) for D>0 (dashed line) and D<0 (solid line); E=0.07.
Volume 148,number8,9
PHYSICSLETTERSA
Stability of the stationary solutions obtained should be checked. Using (9) for small perturbations u (r) = u (r) + 8u (r), where 18u [ ~: u and performing the Fourier transform of it, we can obtain the following stability condition in the form
I 1+2Dk2f,(x~, zc)J ~<1,
(15)
where the k's are wavenumbers of the Fourier modes of the perturbation 8u(r). It follows from (15) that for D > 0 all stationary solutions should be unstable. When D < 0, only perturbation modes with k< k~, k¢ = ~ ( x c , z~)IDI ]-1/2,
(16)
damp. Hence, in a truly continuous case no stable solution can exist, because there would always be unstable modes with k>k~. However, if coordinate r changes discretely (which is actually the case), stable solutions are still possible. We can expect that the kink solution will be stable in the case of discrete r's if the value of/% given by (16) is larger than ~t (i.e. the maximum possible wavenumber in the discrete case r=0, 1, 2, 3, ...). When the absolute value iDI of the diffusive coupling coefficient is increased, the stability threshold kc may become less than ~, so that the unstable modes k c < k < n will grow and spatiotemporal oscillations will set in. The above results should hold for an arbitrary map OC
1 O0
I
0 50
0,5C
0 O0
n=18 0
--
~ FO
~1 100
3 September 1990
0 O0
~=:c
b
0
~0
100
o
sb r
~6o
1 O0
1 O0
i ~
0 50
0 50
brae n=22 0 O0
c
sb r
160
o oo
Fig. 2. Development of inslability of the kink solution for positive D = 1.0. Various time moments are shown: (a) n = 18; (b) n=20; (c) n=22; (d) nffi24.
431
Volume 148, number 8,9 1.00
3September 1990
PHYSICS LETTERS A
(1), provided that it has a period-doubling bifurcation. Particularly, they can be applied to the case of the logistic map.
X:
3. We performed numerical simulations of the logistic map x.+,(r)=x.(r)[zo+Dd2x.(r)l[1-x.(r)]. (17)
0.50
with discrete coordinate taking values r = 0 ..... N + 1 and
02x.(r)=½[x.(r+l)+x.(r-1)-2x.(r)] 0.00
5b
0
160
F
4
1 O0
0 50
0 50
~=28,Z9
50
a 160
I O0
0 50
0 5O
n=38 o
C 5b
~o
b
n=32 000
1 O0
0 O0
(18)
Flg. 3. T h e s t a b l e k m k f o r D = - 0 . 4 . S o h d a n d d a s h e d h n e s c o r respond to two consecutive u m e moments.
1 O0
0 O0
.
0 O0
o
sb f
5'o
~6o
-
Jl
r
Fig. 4. Development of instability o f the kink solution for D = - 3.0. Various Ume m o m e n t s are shown: ( a ) n = 28 ( solid line ) and n = 29 (dashed line); (b) n = 3 2 ; (e) n = 3 8 ; (d) n = 4 0 .
432
Volume !48, number 8,9
PHYSICSLETTERSA
1.00
1.00
0.50
0.50
3 September1990
a 0.00
5b
,60
o.oo 0
r
5b
b ~6o
r
Fig. 5. Transformation of small oscillations (a) n=30,31 into the set of stable kinks (b); D = -0.4.
The boundary conditions were x , ( 0 ) = x , ( 2 )
and
x,,(N- l ) =x. (N+ l ). The first period-doubling bifurcation for this logisticm a p occurs at zc= 3. In numerical simulations the value Zo= 3.05 was chosen. Then the cycle points x, =0.59 and x2=0.738 are close to the bifurcation point xc=2/3 and e~,0.07. In numerical experiments we had N = 100. For D > 0 we find that there are no stable stationary kink solutions. Small perturbations always result in growth of modes with large wavenumbers, so that disorder spatio-temporal oscillations with increasing amplitudes set in (fig. 2). If the diffusion constant is negative and its absolute value is sufficiently small, a steady kink pattern forms, which is described fairly well by the analytic solution ( 11 ) (fig. 3 ). If the absolute value of D is increased further, the stability threshold ~ becomes less than km~ = ~t and no stable solutions exist. The growth of unstable modes with wavenumbers ~ < k < x leads to irregular spatio-temporal oscillations which grow almost exponentially in time, as it is evidenced by fig. 4. Small-amplitude stationary periodic patterns are unstable. Numerical simulations show that after some time they transform into a sequence of steady kinks
Therefore, we conclude, that in the one-dimensional coupled map lattice with diffusive coupling specified by (4) steady spatial kink patterns can form. In the vicinity o f the bifurcation point the explicit kink solution (14) is found analytically. Numerical analysis performed for the logistic coupled maps shows good agreement with analytical results. Disordered spatio-temporal patterns are also observed. Detailed numerical analysis is to be presented elsewhere.
References [l]lL Kaneko, Prog. Theor. Phys. 72 (1984) 480; 74 (1985) 1033. [2] IC Kaneko,PhysicaD 37 (1989) 60. [3] C.A. Bunimovichand Ya.G. Sinai, Nonlinearity 1 (1989) 491. [4] S.P. Kuznetsovand A.S. Pikovsky,Radiofizik, 28 (1985) 308. [5] A.P. Kuznetsov,S.P. Kuznetsovand I.R. Sataev, in: Proc. 5th Int. Workshopon Nonlinear and turbulent processesin physics,Vol. 2 (NaukovaDumkR;leninmad, 1989) p. 383. [6] O.A. Druzhinin and A.S. Mikhailov, Spatio-temporalchaos and synchronization in probabilistic cellularautomata, preprint N1626, SpaceReSearchInstitute, Moscow(1989).
(r~ 5).
433