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CHAOS 16, 023110 2006

Fractional dynamics of coupled oscillators with long-range interaction
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

George M. Zaslavsky
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 and Department of Physics, New York University, New York, New York 10003

Received 29 November 2005; accepted 27 March 2006; published online 11 May 2006 We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1/ n - m +1. It is shown that the equation of motion in the infrared limit can be transformed into the 2. We consider a medium equation with the Riesz fractional derivative of order , when 0 few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on . The presence of a fractional derivative also leads to the occurrence of localized structures. Particular solutions for fractional timedependent complex Ginzburg-Landau or nonlinear SchrÆdinger equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium. © 2006 American Institute of Physics. DOI: 10.1063/1.2197167 Although the fractional calculus is known for more than 200 years and its development is an active area of mathematics, appearance and use of it in physical literature is fairly recent and sometimes is considered as exotic. In fact, there are many different areas where fractional equations, i.e., equations with fractional integrodifferentiation, describe real processes. Between the most related areas are chaotic dynamics,1 random walk in fractal space-time,2 and random processes of the Levy-type.3­6 The physical reasons for the appearance of fractional equations are intermittancy, dissipation, wave propagation in complex media, long memory, and others. This article deals with long-range interaction that can work in some way as a long memory. A unified approach to the origin of fractional dynamics from the long-range interaction of nonlinear oscillators or other objects permits us to consider such phenomena as synchronization, breathers formation, space-time structures by the same formalism using new tools from the fractional calculus.
I. INTRODUCTION

Collective oscillation and synchronization are the fundamental phenomena in physics, chemistry, biology, and neuroscience, which are actively studied recently,7­9 having both important theoretical and applied significance. Beginning with the pioneering contributions by Winfree10 and Kuramoto,11 studies of synchronization in populations of coupled oscillators became an active field of research in biology and chemistry. An oscillatory medium is an extended system, where each site element performs self-sustained oscillations. A good physical and chemical example is the oscillatory Belousov-Zhabotinsky reaction11­13 in a medium where different sites can oscillate with different periods and phases. Typically, the reaction is accompanied by a color variation of the medium. Complex Ginzburg-Landau
1054-1500/2006/16 2 /023110/13/$23.00

equation51­53 is canonical model for oscillatory systems with local coupling near the Hopf bifurcation. Recently, Tanaka and Kuramoto14 have shown how, in the vicinity of the bifurcation, the description of an array of nonlocally coupled oscillators can be reduced to the complex Ginzburg-Landau equation. In Ref. 15, a model of population of diffusively coupled oscillators with limit cycles is described by the complex Ginzburg-Landau equation with nonlocal interaction. Nonlocal coupling is considered in Refs. 15­17. The long2 is range interaction that decreases as 1 / x +1 with 0 considered in Refs. 18 ­22 with respect to the system's thermodynamics and phase transition. It is also shown in Ref. 23 that using the Fourier transform and limit for the wave number k 0, the long-range term interaction leads under special conditions to the fractional dynamics. In the last decade it is found that many physical processes can be adequately described by equations that consist of derivatives of fractional order. In a fairly short period of time the list of such applications becomes long and the area of applications is broad. Even in a concise form, the applications include material sciences,24 ­27 chaotic dynamics,1 quantum theory,28 ­31 physical kinetics,1,3,32,33 fluids and plasma physics,34,35 and many other physical topics related to wave propagation,36 long-range dissipation,37 anomalous diffusion and transport theory see reviews in Refs. 1, 2, 4, 24, and 38 . Some historical comments on the origin of fractional calculus can be found in Ref. 39. It is known that the appearance of fractional derivatives in equations of motion can be linked to nonlocal properties of dynamics. Fractional Ginzburg-Landau equation has been suggested in Refs. 40­ 42. In this paper, we consider the synchronization for oscillators with long-range interaction that in continuous limit leads to the fractional complex GinzburgLandau equation. We confirm the result obtained in Ref. 23
© 2006 American Institute of Physics

16, 023110-1

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023110-2

V. E. Tarasov and G. M. Zaslavsky
+ +

Chaos 16, 023110 2006

that the infrared limit wave number k 0 of an infinite chain of oscillators with the long-range interaction can be described by equations with the fractional Riesz coordinate 2. This result permits us to apply derivative of order different tools of the fractional calculus to the considered systems, and to interpret different systems' features in a unified way. In Sec. II, we consider a systems of oscillators with linear long-range interaction. For infrared behavior of the oscillatory medium, we obtain the equations that have coordinate derivatives of fractional order. In Sec. III, some particular solutions are derived with a constant wave number for the fractional Ginzburg-Landau equation. These solutions are interpreted as synchronization in the oscillatory medium. In Sec. IV, we derive solutions of the fractional GinzburgLandau equation near a limit cycle. These solutions are interpreted as coherent structures in the oscillatory medium with long-range interaction. In Sec. V, we consider the nonlinear long-range interaction of oscillators and corresponding equations for the spin field. Finally, discussion of the results and conclusion are given in Sec. VI.

Z x, t =

1 2

dke
-

ikx n=-

e

-ikn

zn t .

4

Multiplying Eq. 1 by exp -ikn , and summing over n from - to + , we obtain y k, t t =
n=- +

e
n=- +

-ikn

d zn t dt
+ +

e

-ikn

F zn + g

0 n=- m=- ,m n

e

-ikn

zn - z n-m

m +1

, 5

where
+

y k, t =
n=-

e

-ikn

zn t .

6

Using the notation
+ +

~ k= J
n=- ,n 0

e

-ikn

J n=
n=- ,n 0

e

-ikn

1 n
+1

,

7

II. LONG-RANGE INTERACTION OF OSCILLATORS A. Derivation of equation for the continuous oscillatory medium

the interaction term in 5 can be presented as
+ n=- +

e
m=- ,m n + +

-ikn

1 n-m e
-ikn

+1

zn - z

m

In this section we consider a simplified version of a chain of N oscillators N that have a long-range interaction of the power type. The corresponding equation of motion can be written as d zn t = F zn + g dt

=
n=- + m=- ,m n +

1 n-m

+1 zn

-
0 m=- ,m n

e
n=- m=- ,m n

-ikn

J n - m zn - zm ,

1
+ n=-

1 n-m

+1 zm

.

8

For the first term in the right-hand side of 8 :
+

where zn is the position of the nth oscillator in the complex plane, and F is a force. As an example, for the oscillators with a limit cycle, F can be taken as F z = 1+ ia z - 1+ ib z 2z . The nonlocal interaction is given by the power function 2

e
m=- ,m n + -ikn

-ikn

1 n-m
+

+1 zn

=
n=-

e

z

1
0

n m =- ,m

m

+1

= y k, t ~ 0 , J

9

where J n=n
- -1

.

3 ~ 0= J

+ n=- ,n 0

1 n
+1

This coupling in the limit is a nearest-neighbor interaction. This type of interaction was introduced by Dyson18 to study phase transitions and then was considered in numerous papers related to magnetic systems.19­22 Power type longrange interaction can appear as an effective interaction in dispersive or complex systems.26,36,40 The complexity of the system reveals in a noninteger that is defined by a specific type of the material. Let us provide also two examples from fluid dynamics where the dispersion, and nonlinear properties of the media define the order of fractional derivatives: tracer dynamics in the presence of convective rolls,43 and the equation for surface wave interaction.44 Let us derive the equation for continuous medium limit of system 1 with long-range interaction 3 . For this goal it is convenient to introduce the field

=2
n=1

1 n
+1

=2

+1 ,

10

and z is the Riemann zeta function. For the second term in the RHS of 8 :
+ n=- +

e
m=- ,m n +

-ikn

1 n-m e
-ikn

+1 zm

+

=
m=- +

z

m n=- ,n m

1 n-m e
-ikn

+1

+

=
m=-

z me

-ikm n =- ,n 0

1 n
+1

= y k, t ~ k . J

11

As the result, Eq. 5 yields

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023110-3

Fractional dynamics of CO with LR

Chaos 16, 023110 2006

t

y k, t = F F z

n

J + g0 ~ 0 - ~ k y k, t , J

12

where F F zn is an operator notation for the Fourier transform of F zn :
+

FFz

n

=
n=-

e

-ikn

F zn .

The function ~ k introduced in 7 can be transformed as J
+

~ k= J
n=- ,n 0 + -ikn

e

-ikn

1 n
+1 -

=
n=1

e

1 n
+1

+
n=-1

e

-ikn

1 n
+1

=
n=1

1 n
+1

e

-ikn

+e

ikn

= Li

+1

e

ik

+ Li

+1

e

-ik

, 13

where Li z is a polylogarithm function. This presentation was also obtained in Ref. 23, and it plays an important role in the following transition to fractional dynamics. Using the expansion -n n z, n!

Li ez = we obtain ~ k =2 J

1-

-z

-1

+
n=0

z

2,

14
FIG. 1. The function ~ k for orders J

= 1.1, and

= 1.9.

-

cos

/2 k +2
n=0

+1-2n -k 2n !

2n

, 15 Z x, t = 1 2

+

eikxy k, t dk ,
-

19

~ 0 =2 J

+1 .

From 13 we can see that ~ k +2 m = ~ k , J J 16

and the connection between Riesz fractional derivative and its Fourier transform:45
2

k -

where m is an integer. For =2, ~ k is the Clausen function J Cl3 k .54 The plots of ~ k for = 1.1, and = 1.9 are preJ sented in Fig. 1. After substituting 15 into 12 , we obtain y k, t = F F z -2g where a =2 - cos /2 0 2, 1. 18 - g 0a k y k , t +1-2n -k 2n !
2n

x

,

k2 -

x

2

.

20

The properties of the Riesz derivative can be found in Refs. 45­ 48. Another expression is Z x, t =- 1 2 cos /2 D+ Z x, t + D- Z x, t , 21

x t
n

where 1, 3, 5, ..., and D± are Riemann-Liouville left and right fractional derivatives y k, t , 17 D+ Z x, t = 1 n- -1 n-
n n x -

0 n=1

x

n

Z ,t d , x - -n+1 22 Z ,t d . - x -n+1

n

D- Z x, t =

To derive the equation for field 4 , we can use definition 6

x

n x

Substitution of Eqs. 22 into Eq. 21 gives

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023110-4

V. E. Tarasov and G. M. Zaslavsky

Chaos 16, 023110 2006

x

Z x, t =

2 cos
n

-1 /2
x -

n- Z ,t d x - -n
+1

t +
x

Z = F Z + g 0a

x

Z

0

2,

1.

28

x

n

-1 nZ , t d - x -n+1

. 23

Multiplying Eq. 17 on exp ikx , and integrating over k from - to + , we obtain Z = ~ Z + g 0a F +1-2n 2n !
2n

Equation 28 can be considered an equation for continuous 2 in the infrared k 0 aposcillatory medium with proximation. As an example, for F z = 0, Eq. 28 gives the fractional kinetic equation t Z = g 0a x Z 0 2, 1 29

t

x

Z -2g

0 n=1

x

2n

Z, 24

Z = Z x, t

0, 1, 2, . . . ,
n

where ~ Z is the inverse Fourier transform of F F z F ~Z= 1 F 2
+

:

dkeikxF F z
-

n

.

that describes the fractional superdiffusion.3,4,32 For F z defined by 2 , Eq. 28 is a fractional Ginzburg-Landau equation that has been suggested in Ref. 40 see also Refs. 41 and 2 and k 0 42 , and will be considered in Sec. III. For the main term in 15 is proportional to k2 and in 28 and 29 , we have a second derivative instead of the fractional one. The existence of the critical value = 2 was obtained in Ref. 23.
III. FRACTIONAL GINZBURG-LANDAU EQUATION

For x = n "n one can see that ~ Z x, t = F Z n, t = F z t . F n 25

A. Synchronized states for the Ginzburg-Landau equation

This is a standard procedure for the replacement of a discrete chain by the continuous one and in the following we will write F Z instead of ~ Z . F -1 2Z. Let us The first term n =1 of the sum is x compare the coefficients of terms with fractional and second derivatives in Eq. 24 . For 2, one can use the asymptotics -1 1 +O 1 , -2 a 1 +O 1 -2 2.

The one-dimensional lattice of weakly coupled nonlinear oscillators is described by d zn t = 1+ ia zn - 1+ ib zn 2z dt + c1 + ic
2 n

z

n+1

-2zn + z

n-1

,

30

As an example, for -1 Therefore

= 1.99, a - 100.929 21 . 1.

where we assume that all oscillators have the same parameters. A transition to the continuous medium assumes8 that the difference zn+1 - zn is of the order x, and the interaction constants c1 and c2 are large. Setting c1 = g x -2, and c2 = gc x -2, we get
2

- 99.423 51, -1 / a

t

Z = 1+ ia Z - 1+ ib Z 2Z + g 1+ ic

x

2

Z,

31

1 for 2 -

B. Infrared approximation

which is a complex time-dependent Ginzburg-Landau equation.51­53 Here Z n x , t coincides with 4 if we put x = 1. The simplest coherent structures for this equation are plane-wave solutions,8 Z x, t = R K exp iKx - i where R K = 1- gK and for
0 2 1/2

In this section, we derive the main relation that permits us to transfer the system of discrete oscillators into a fractional differential equation. This transform will be called the 2, 1, and k 0, the fractional infrared limit. For 0 power of k is a leading asymptotic term in Eq. 17 , and ~ 0 -~ k J J ak 0 2, 1. 26

K t+

0

,

32

,

K = b - a + c - b gK2 ,

33

is an arbitrary constant phase. These solutions exist
2

Equation 26 can be considered as an infrared approximation of 17 . Substitution of 26 into 12 gives y k, t = F F z 0 Then - g 0a k y k , t 2, 1. 27

gK

1.

34

The solution 32 can be interpreted as a synchronized state.8
B. Particular solution for the fractional Ginzburg-Landau equation

t

n

Let us come back to the equation for nonlinear oscillators 1 with F z in Eq. 2 and long-range coupling 3 ,

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023110-5

Fractional dynamics of CO with LR

Chaos 16, 023110 2006

d zn = 1+ ia zn - 1+ ib zn 2z dt +g
0 mn

K = b-a + c-b gK ,
n

1-gK

0.

46

1 n-m

+1

zn - zm ,

35

This solution can be interpreted as a coherent structure in nonlinear oscillatory medium with long-range interaction. If R2 =1- g K , 0 gK 1,

where zn = zn t is the position of the nth oscillator in the 2. The corresponding equation in the complex plane, 1 continuous limit and infrared approximation can be obtained in the same way as 28 t Z = 1+ ia Z - 1+ ib Z 2Z + g 1+ ic x Z, 36

then Eqs. 44 and 45 give R t = R 0, t =- K t+
0

.

47

Solution 47 means that on the limit cycle 42 the angle K . As the variable rotates with a constant velocity result, we have the plane-wave solution Z x, t = 1- g K
1/2 iKx-i

where g 1+ ic = g0a , and 1 2. Equation 36 is a fractional generalization of the complex time-dependent Ginzburg-Landau equation 31 compare to 28 . Here, this equation is derived in a specific approximation for the oscillatory medium. We seek a particular solution of 36 in the form Z x, t = A K, t e
iKx

e

K t+i 0

,

1-gK

0,

48

,

37

which allows us to use x e
iKx

which can be interpreted as a synchronized state of the oscillatory medium. For initial amplitude that deviates from 42 , i.e., R2 0 1- g K , an additional phase shift occurs due to the term which is proportional to b in 45 . The oscillatory medium can be characterized by a single generalized phase variable. To define it, let us rewrite 41 as d ln R = 1- g K dt d = a - cg K dt - R2 , 49

=- K e

iKx

.

38

Equation 37 represents a particular solution of 36 with a fixed wave number K. The substitution of 37 into 36 gives t A K, t = 1+ ia A - 1+ ib A 2A - g 1+ ic K A . 39

- bR2 .

50

Substitution of R2 from 49 into 50 gives d dt - b ln R = a - cg K - b 1- g K . 51

Rewriting this equation in polar coordinates, A K, t = R K, t e we obtain dR = 1- g K dt d = a - cg K dt R - R3 , 41 - bR2 .
i K,t

,

40

Thus, the generalized phase8 can be defined by R, = - b ln R . 52

From 51 , we get d =- dt K. 53

The limit cycle here is a circle with the radius R = 1- g K
1/2

,

gK

1.

42

The solution of 41 with arbitrary initial conditions R K,0 = R0, is R t = R0 1- g K e
-2 1-g K 1/2

This equation means that generalized phase R , rotates uniformly with constant velocity. For g K = b - a / b - c 1, we have the lines of the constant generalized phase. On the R , plane these lines are logarithmic spirals - b ln R = const. The decrease of corresponds to the increase of K. For the case b = 0 instead of spirals we have straight lines =.
C. Group and phase velocity of plane waves

K,0 =

0

43 Energy propagation can be characterized by the group velocity 44
v
,g

R2 + 1- g K - R 0 ,
-1

2 0

t -1/2

=

b t =- ln 1- g K 2 - where K t+
0

K . K

54

R2 + 1- g K - R2 e 0 0

-2at

From Eq. 46 , we obtain 45 For
v
,g

,

=

c-b gK

-1

.

55

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023110-6

V. E. Tarasov and G. M. Zaslavsky

Chaos 16, 023110 2006

K we get
v
,g

K1 =

/2

2-

,

56

A11 + A22 =-2 1- g K

,

1-gK

0,

66

v

2,g

.

57

and the first condition of 65 is valid. Substitution of Eqs. 62 and 64 into 65 gives A11A22 - A12A21 = b 1- g K - a - gc K - a - gc K . 67

The phase velocity is
v
,ph

3b 1- g K
-1

=

K /K = c - b g K

.

58

Then the second condition of 65 has the form V -1 V -3 0, 68

For K we have K2 =2
-2

,

59

where V= a - gc K b 1- g K . 69

v

,ph

v

2,ph

.

60

Therefore, the long-range interaction decreases as x - +1 with 1 2 leads to an increase in the group and phase velocities for small wave numbers K 0 . Note that the ratio v ,g / v ,ph between the group and phase velocities of plane waves is equal to .
D. Stability of the plane wave solution

As the result, we obtain 0 1- g K a/b - c/b g K 3 1- g K , 70

i.e., the plane wave solution 48 is stable if parameters a, b, c and g satisfy 70 . Condition 70 defines the region of parameters for plane waves where the synchronization exists.
E. Forced fractional Ginzburg-Landau equation for the isochronous case

The solution of 48 can be presented as X = R K, t cos Y = R K, t sin K, t + Kx , 61 K, t + Kx , where X = X K , t =Re Z x , t and Y = Y K , t =Im Z x , t , and R K , t and K , t are defined by 44 and 45 . For the plane waves X0 x, t = 1- g K Y 0 x, t = 1- g K 1- g K
1/2

In this section, we consider the fractional GinzburgLandau FGL equation 39 forced by a constant E the so-called forced isochronous case b =0 Ref. 8 t A = 1+ ia A - A 2A - g 1+ ic K A - iE Im E =0 , 71 where A = A K , t , and we put for simplicity b = 0, and K is a fixed wave number. Our main goal will be transition to synchronized states and its dependence on the order of the long-range interaction. The system of real equations is

cos Kx - sin Kx -

K t+ K t+

0

, , 62

1/2

0

0.

Not all of the plane waves are stable. To obtain the stability condition, consider the variation of 39 near the solution 62 d X K, t = A dt
11

d X = 1- g K dt d Y = 1- g K dt

X - a - gc K

Y - X2 + Y 2 X , 72

X+A

12

Y,

d Y K, t = A dt

Y + a - gc K

X - X + Y Y - E,

2

2

21

X+A

22

Y, 63

where X and Y are small variations of X and Y , and A11 =1- g K -2X0 X0 - bY 0 - X2 + Y 2 , 0 0 A12 =- a + gc K -2Y 0 X0 - bY 0 + b X2 + Y 2 , 0 0 64 A21 = a - gc K -2X0 Y 0 + bX0 - b X2 + Y 2 , 0 0 A22 =1- g K -2Y 0 Y 0 + bX0 - X2 + Y 2 . 0 0 The conditions of asymptotic stability for 63 are A11 + A
22

0,

A11A22 - A12A

21

0.

65

From Eqs. 62 and 64 , we get

where X = X K , t is real and Y = Y K , t are imaginary parts of A K , t . In the simulation of Eq. 72 , we will take the parameters close to the selected ones in Ref. 8, where the parameters a , g , c , e , K were selected to demonstrate the existence of the Hopf-type bifurcation and the appearance of synchronization. Some differences in our case are due to the fractional 2, while in Ref. 8 it value of the interaction exponent was =2. A numerical solution of Eq. 72 was performed with parameters a =1, g =1, c = 70, E = 0.9, K = 0.1, for within the interval 1;2 . The results are presented in Fig. 2, 2, where 0 1.51 , . . ., the only and Fig. 3. For 0 stable solution is a stable fixed point. This region is of perfect synchronization phase locking , where the synchronous oscillations have a constant amplitude and a constant phase shift with respect to the external force. For 0 the global

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023110-7

Fractional dynamics of CO with LR

Chaos 16, 023110 2006

FIG. 2. Approaching the bifurcation point = 0 = 1.51. . . of the solution of the forced FGL equation for the isochronous case with fixed wave number K = 0.1 is represented by real X K , t and imaginary Y K , t parts of A K , t . The plots for orders = 2.00, = 1.70, = 1.60, = 1.56.

FIG. 3. Transformation to the limit cycle of the solution of the forced FGL equation for the isochronous case with fixed wave number K = 0.1 is represented by real X K , t and imaginary Y K , t parts of A K , t . The plots for orders = 1.54, = 1.52, = 1.50, = 1.40.

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023110-8

V. E. Tarasov and G. M. Zaslavsky

Chaos 16, 023110 2006

attractor for 72 is a limit cycle. Here, the motion of the forced system is quasiperiodic. For = 2 there is a stable decreases, the stable mode transfers into a node. When stable focus. At the transition point it loses stability, and a stable limit cycle appears. As the result, we have the decrease of order from 2 to 1 leads to the loss of synchronization see Figs. 2 and 3 . 1, when the bifurcation and synchroniThe value of zation appears in our case can be easily understood from the results of Ref. 63, where it was shown that the fractional derivative in a nonlinear oscillations model leads to a dissifor 1 pation with a decrement of the order cos / 2. Our results show that the fractional derivative in Eq. 36 does not change the qualitative pattern of synchronization but, instead, brings a new parameter to control the process under consideration. Evidently, synchronization and bifurcation in the following simulations are at the dissipation parameter value of order one since the dissipation, frequency, and nonlinearity terms in 72 are all of order one. The choice of the wave number K can be arbitrary but we select it to be small in order to satisfy the infrared approximation. In Fig. 2 = 2.00, = 1.70, and = 1.60, = 1.56 , we see that in the synchronization region all trajectories are attracted to a stable node. = 1.54, = 1.52, and = 1.50, = 1.40 , a In Fig. 3 stable limit cycle appears via the Hopf bifurcation. For = 1.54, and = 1.52, near the boundary of synchronization the fixed point is a focus. For = 1.4, the amplitude of the limit cycle grows, and synchronization breaks down.
F. Phase and amplitude for the forced FGL equation

FIG. 4. Phase K , t for K = 0.1 and = 2.00, = 1.50, = 1.47, = 1.44, = 1.40, = 1.30, = 1.20, = 1.10. The decrease of order corresponds to the clockwise rotation of curves. For the upper curve = 2. For the most vertical curve = 1.1.

dR = 1- g K dt d = a - cg K dt

R - R3 - E sin , 78 E cos - . R

The oscillator medium can be characterized by a single generalized phase variable 52 . We can rewrite 52 as X, Y = arctan Y /X - b ln X2 + Y 2 , 2 73

where X and Y are defined by 61 . For E = 0, the phase rotates uniformly d =- dt K = a - gc K , 74

The numerical solution of 78 was performed with the same parameters as for Eq. 72 , i.e., a =1, g =1, c = 70, E = 0.9, 1,2 . The results are K = 0.1, and within the interval presented in Figs. 4 and 5. The time evolution of phase K , t is given in Fig. 4 for = 2.00, = 1.50, = 1.47, = 1.44, = 1.40, = 1.30, = 1.20, = 1.10. The decrease of from 2 to 1 leads to the oscillations of the phase K , t after the Hopf bifurcation at 0 = 1.51 ,..., then the amplitude of phase oscillation decreases and the velocity of phase rotations increases. The amplitude R K , t is shown in Fig. 5 for = 1.6, = 1.55, = 1.55, = 1.51, = 1.50, = 1.45, = 1.2. The appearance of oscillations in the plots means the loss of synchronization.
IV. SPACE-STRUCTURES FROM THE FGL EQUATION

where K is given by 46 with b = 0, and can be considered as a frequency of natural oscillations. For E 0, Eqs. 72 and 73 give d =- dt K - E cos . 75

This equation has an integral of motion. The integral is I1 =2
2

-E

2 -1/2

arctan tan

-E t /2 + t,
2

In previous sections, we considered mainly time evolution and "time structures" as solutions for the FGL equation. Particularly, the synchronization process was an example of the solution that converged to a time-coherent structure. Here, we focus on the space structures for the solution of the FGL equation 36 with b = c = 0 and the constants a1 and a2 ahead of the linear term, t Z = a1 + ia2 Z - Z 2Z + g x Z. 79

2

-E

2 -1/2

E2 ,

76

I2 =2 E2 - E-
2

2 -1/2

arctanh E - tan t /2 + t,
2

2 -1/2

E2 .

Let us seek a particular solution of 79 in the form 77 Z x, t = R x, t e
it

,

R* x, t = R x, t ,

*

t=

t. 80

These expressions help to obtain the solution in the form 40 for the forced case 71 keeping the same notations as in 40 . For polar coordinates we get

Substitution of 80 into 79 gives

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023110-9

Fractional dynamics of CO with LR

Chaos 16, 023110 2006

FIG. 5. Amplitude R K , t . The upper curve corresponds to = 2 for all plots. The lower curves correspond to = 1.2. The appearance of oscillations on the plots means the loss of synchronization.

= 1.6,

= 1.55,

= 1.51,

= 1.50,

= 1.45,

+

t

R = a 1R - R 3 - g

x

R,

t

t = a2 .

81

R1 x, t =
-

G x ,t

x - x dx .

85

Using t = a2t + 0 , we arrive at the existence of a limit cycle with R0 = a1/2. 1 A particular solution of 81 in the vicinity of the limit cycle can be found as an expansion R x, t = R0 + R1 +
2

Let us apply the Laplace transform for t and the Fourier transform for x,
+

~ G k, s = R2 + ... 1. 82 /x1
0

dt
-

dxe

-st+ikx

G x, t .

86

Zero approximation R0 = a1/2 satisfies 81 since 1 = 0, and for R1 = R1 x , t , we have R1 =-2a1R1 + g R1 .

By the definition of the Riesz derivative, ~ G x, t - k G k, s ,

t

x

83

x

87

Consider the Cauchy problem for 83 with an initial condition R1 x,0 = x, 84

and for the Laplace transform with respect to time ~ G x, t sG k, s -1 .

t

88

and the Green function G x , t such that

Applying 86 ­ 88 to 83 , we obtain

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023110-10

V. E. Tarasov and G. M. Zaslavsky

Chaos 16, 023110 2006

G x, t = gt

-1/2 -2a1t e

1 2

e

-x2/ 4gt

.

97

For 1 2 the function L x can be presented as the convergent expansion L x =- 1 x -x
n=1 n

1+ n/ n!

sin n /2 .

98

The asymptotic x , 1 Lx - 1 x -1 nx
n=1 -n

2 is given by 1+ n n! sin n /2 , 99

x
FIG. 6. Gauss PDF =2 , Levy PDF = 1.6 , and Cauchy PDF = 1.0 . Levy for = 1.6 lies between the Cauchy and Gauss PDF. In the asymptotic x and x 3 on the plot, the upper curve is the Cauchy PDF, and the lower curve is the Gauss PDF.
-1

,

with the leading term Lx 1+ x
- -1

,

x

.

100

As the result, the solution of 79 is Z x, t = e
i a2t+ 0

a

~ ~ ~ sG k, s -1=-2a1G k, s - g k G k, s or ~ G k, s = 1 . s +2a1 + g k

1/2 1

+
-1/

gt

-1/

e

-2a1t

89
+

L x gt
-

x - x dx + O

2

. 101

90

Let us first invert the Laplace transform in 90 . Then, the Fourier transform of the Green function
+

^ G k, t =
-

dxeikxG x, t = e

- 2a1+g k

t

=e

-2a1t -g k t e

. 91

This solution can be considered as a space-time synchronization in the oscillatory medium with long-range interaction decreasing as x - +1 . For x = x - x0 , solution 101 has the form Z x, t = e
i a2t+ 0

a

1/2 1

+ gt

gt
-1/

-1/

e

-2a1t 2

L

x-x

0

+O

,

102

As the result, we get G x, t = gt where L x= 1 2
+ -1/

and the asymptotic is
-2a1t

e

L x gt

-1/

,

92

Z x, t = e

i a2t+ 0

a

1/2 1

+ gte +O

-2a1t

-1

1+ . 103

x-x dke
- -ikx -a k

0

- -1

2

,

x

e

93

This solution shows that the long-wave modes approach the limit cycle exponentially with time. For t =1/ 2a1 , we have the maximum of Z x , t with respect to time, max Z x, t = a
t0 1/2 1

is the Levy stable = 2.0, = 1.6, and As an example, tion with respect to e and G x, t = For e and
-k

PDF Ref. 55 . The PDF L x for = 1.0 are shown in Fig. 6. for = 1 we have the Cauchy distributhe coordinate 1 x2 +1 94

+g

1+ 2e

x-x

0

- -1

+O

2

. 104

L1 x =

1

As the result, we have the power law decay with respect to the coordinate for the space structures near the limit cycle Z = a1/2. 1
V. NONLINEAR LONG-RANGE INTERACTION AND FRACTIONAL PHASE EQUATION

1 gt -1e-2a1t . x2 gt -2 +1

95

= 2, we get the Gauss distribution
-k2

L2 x =

1 2

e

-x2/4

96

Here, we would like to show one more application of the replacement of dynamical equation by the fractional ones for a chain with long-range interaction. The model was first considered in Refs. 10, 11, and 49 with application in biology and chemistry. This model has additional interest since it can be reduced to a chain of interacting spins.

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023110-11

Fractional dynamics of CO with LR
+ + +

Chaos 16, 023110 2006

A. Nonlinear nonlocal phase coupling

Let us consider the phase equation d dt
+ n

1 2

dke
-

ikx n=-

e
+

-ikn m=- ,m n

1 n-m

* +1 snsm

t=

n

+g
m=- ,m n

J n - m sin

n

-

m

,
n

105 its natu-

= S* x, t

1 2 2

dk1a k1, t ~ k1 e J
-

ik1x

where n denotes the phase of the nth oscillator, ral frequency, and J n=n
- -1

= S* x, t

+1 S x, t - a
2n

x ,

S x, t

.

106
11,49­51

+2
n=0

+1-2n 2n !

x

2n

S x, t

112

For = -1, Eq. 105 defines the Kuramoto model with sinusoidal nonlocal coupling infinite radius of interaction . We can rewrite Eq. 105 for classical spin-like variables sn t = e
i nt

where we use 15 for ~ k , and a is the same as in 18 . J For the term n, we use x= 1 2
+ +

dke
-

ikx n=-

e

-ikn

,

sin

n

-

m

1 * s ns m + s *s m . = n 2i

n

.
n

113 = , then

107

Then Eq. 105 is
* n

If all oscillators have the same natural frequency x= . As the result, Eq. 108 is transformed into 108 S* x, t =i t S x, t x - f S* x, t S x, t S* x, t x S x, t + S x, t x S* x, t

s

d sn = i dt

n

+

g 2

+ m=- ,m n

1 n-m

+1

* s ns m + s *s m . n

This equation describes the long-range interaction of spin variables. We also will call Eq. 108 as the phase coupling equation since sn 2 = const. Thermodynamics of the model of classical spins with long-range interactions have been studied for more than 30 years. An infinite one-dimensional Ising model with long-range interactions was considered by Dyson.18 The d-dimensional classical Heisenberg model with long-range interaction is described in Refs. 19 and 20, and its quantum generalization with long-range interaction decreases as n - can be found in Ref. 21.

-g

+g
n=1

+1-2n 2n !
2n 2n 2n

S* x, t where

x

S x, t + S x, t

x

2n

S* x, t ,

114

B. Phase-coupled oscillatory medium with nonlinear long-range interaction

f =2g

+1 ,

g = 1/2 a g = g

-

cos

/2 . 115

Let us derive an equation for the continuous medium that consists of oscillators of 105 or 108 type with nonlinear long-range interaction. The medium can be defined by the field 1 2
+ +

Equation 114 is a fractional equation for the oscillatory medium with long-range interacting spins 108 . We can call 114 the fractional phase equation. In the infrared approximation k 0 , we can use 15 ~k J 2 0 - cos 2, /2 k +2 1, +1 , 116

S x, t =

dke
-

ikx n=-

e

-ikn

sn t .

109

and Eq. 114 is reduced to S* x, t t S x, t = i x -f -g S* x, t x S x, t 117

We also will need the following momentum representations:
+

a k, t =
n=-

e

-ikn

sn t .

110 where 0 2,

+ S x, t 1.

x

S* x, t ,

For the left-hand side of 108 , we get 1 2
+ +

dke
-

ikx n=-

e

-ikn * sn

d d sn = S* x, t S x, t . dt dt

VI. CONCLUSION

111

For the interaction term, we similarly obtain 9 ­ 17 :

A one-dimensional chain of interacting objects, say oscillators, can be considered as a benchmark for numerous applications in physics, chemistry, biology, etc. All consid-

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023110-12

V. E. Tarasov and G. M. Zaslavsky

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ered models were related mainly to the oscillating objects with long-range powerwise interaction, i.e., with forces proportional to 1 / n - m s and 2 s 3. A remarkable feature of this interaction is the possibility of replacing the set of coupled individual oscillator equations into the continuous medium equation with the fractional space derivative of the 2, 1. Such a transformation order = s - 1, where 0 is an approximation and it appears in the infrared limit for the wave number k 0. This limit helps us to consider different models and related phenomena in a unified way applying different tools of fractional calculus. A nontrivial example of the general property of the fractional linear equation is its solution with a powerwise decay along the space coordinate. From the physical point of view that means a new type of space structure or coherent structure. The scheme of the equations with fractional derivatives includes either the effect of synchronization,8 breathers,56 ­58 fractional kinetics,1 and others. Discrete breathers are periodic space-localized oscillations that arise in discrete and continuous nonlinear systems. Their existence was proven in Ref. 59. Discrete breathers have been widely studied in systems with short-range interactions for a review, see Refs. 56 and 60 . Energy and decay properties of discrete breathers in systems with long-range interactions have also been studied in the framework of the Klein-Gordon,57,61 and the discrete nonlinear SchrÆdinger equations.62 Therefore, it is interesting to consider breathers solution in systems with long-range interactions in the infrared approximation. We also assume that the suggested replacement of the equations of interacting oscillators by the continuous medium equation can be used for improvement of simulations for equations with fractional derivatives.
ACKNOWLEDGMENTS

8

9

10

11

12

13

14

15

16

17

18

19

20

21

We are thankful to N. Laskin for useful discussions and comments. This work was supported by the Office of Naval Research, Grant No. N00014-02-1-0056; the U.S. Department of Energy, Grant No. DE-FG02-92ER54184; and the NSF, Grant No. DMS-0417800. V.E.T. thanks the Courant Institute of Mathematical Sciences for support and kind hospitality.
1

22

23

24

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