The width of the stochastic web and particle diffusion along the web

Volume 144, number 4,5

PHYSICS LETTERS A

5 March 1990

T H E W I D T H O F T H E S T O C H A S T I C W E B A N D P A R T I C L E D I F F U S I O N A L O N G T H E WEB V.V. AFANASIEV, A.A. C H E R N I K O V , R.Z. SAGDEEV and G.M. ZASLAVSKY

SpaceResearchInstitute,Academyof Sciencesof the USSR,Profsoyusnaya84/32, Moscow117810, USSR Received 29 October 1989; accepted for publication 28 November 1989 Communicated by V.M. Agranovich

The width of the stochastic web is calculated for different cases of its symmetry. Exponential and preexponential factors are found. The correction of the asymptotic expansion due to the deformation of the stochastic web is included. The comparison of analytical results with the results of numerical simulation is made. The diffusion of panicles along the stochasticweb is investigated:

1. There is an infinite net of stochastic layers in the phase space of a Hamiltonian system with N = 1½ degrees of freedom which is called a stochastic web [1,2]. It occurs in the case when the unperturbed Hamiltonian is degenerate (linear) and resonance conditions are satisfied. An example of such a system is the linear oscillator forced by a periodical sequence o f a-pulses: /~+o~2v=esinv

~

a(t-n),

(1)

n ~ --oo

where v and o~ are the dimensionless coordinate and frequency of the oscillator respectively, e is the perturbation amplitude and the period of the external force is set to be unity. The condition of resonance is given by a=2=/q, where q is integer. A periodic two-dimensional web arises when qe {q¢}= {3, 4, 6}. Eq. (1) also describes the motion o f a charged particle in a magnetic field directed in the Z-direction and in the field of a wave packet propagating along the v-axis which is perpendicular to Z. The equation of motion ( 1 ) can be written in a difference form. So the following mapping appears [11, u,+l = (u, + K s i n vn) cos or+ v, sin or, v,+l = - (u~ + K s i n v,) sin o t + v , cos ce,

lems in plasma physics and the general theory of dynamical systems [ 1-7 ]. It represents a new kind of measure preserving mapping and leads to the possibility of infinite acceleration of particles along the channels of the stochastic web at arbitrarily small finite values of the parameter K. In the present work the problem o f the stochastic web thickness at K<< 1 and qE {qc} will be investigated. The first estimates of the web thickness were obtained in refs. [ 2,5 ]. Similar results for q = 4 were obtained in ref. [ 6 ] for model mapping close to (2). The diffusion of particles as a function of the parameter K was also considered there. In the case of precise resonance a=2~/q (q integer) the Hamiltonian of the system ( 1 ), (2) can be written in the following form [2]: K

H=Hq--}-Vq, Hq=-- qJ==~1 c o s ( R - e j ) , Vq = -

-2 K ~ cos(R.e,) q

j=l

~

{2urn ( t - j ) ) , cos(---~--

m=l

(3) where the vector R = (v, u) determines a point in the phase space and the system of q unit vectors

ej= (cos(2rt/q)j, -sin(2n/q)j) (2)

where K = E/or is a perturbation parameter; u = b/a and u , = u ( t = n - O ) , v,=v(t=n-O). Mapping (2) was investigated in connection with different prob-

(j= 1,..., q)

(4)

forms a right star {ej}. The expression for Hq coincides with one for the complete Hamiltonian H averaged in time. It describes a tiling of the plane (u, v) with different symmetry at different values of q.

0375-9601/90/$ 03.50 © Elsevier Science Publishers B:V. (North-Holland)

229

Volume 144, number 4,5

PHYSICS LETTERS A

At q = 4 the separatrices of the dynamical system with Hamiltonian Hq(u, v) where v is a coordinate and u is a momentum, make the united net in the form of a square lattice. In case of q = 3 or q = 6 the separatrix web is a Kagome lattice and has a hexagonal symmetry [ 1 ]. The non-stationary part Vqo f the Hamiltonian H destroys this net and converts it into a stochastic web inside which a charged particle performs a random walk. At K<< 1 the thickness of the stochastic web is exponentially thin [2,5 ]. The regions inside the separatrix cells (islands) are occupied by invariant curves. The area o f these islands decreases as K increases. The stochastic web thickness is determined below analytically and numerically for q~ {qc} and the diffusion coefficients are determined with respect to dynamical correlations.

5 March 1990

tion (6) because of the influence of the high-frequency perturbation (7) can be determined from /I4 = K 2 sin u sin vcos nt.

(10)

Substituting the expression for the trajectory of the particle motion (9) in the right side o f eq. (10) and

2. In the simple case of fourth order resonance

(a=n/2) the mapping (2) reduces to the following one,

Un+l=vn, Vn+l=-(un+Ksinvn).

(5)

The expression for the averaged Hamiltonian (3) at q = 4 is H4 = - ½K(cos v + cos u)

(6)

and corresponds to the integrable system with one degree of freedom. The perturbation V4 ~--- - K(cos v - c o s u) cos 7tt+ ...,

(7)

adding one half degree of freedom, makes the system nonintegrable and leads to the appearance o f a stochastic web. The dots in eq. (7) include the rest terms in the sum (in m) in eq. (3) beginning from m = 3 . At K<< 1 the contribution o f these terms to the thickness of a web is negligibly small. The separatrices of the averaged motion (6) are determined as trajectories at H4 = 0:

v=+(u+x)+2rcm ( m = 0 , + l , . . . ) .

(8)

The motion along the separatrix v = u + re) is described by

(for example,

1

sin v= - s i n u = ch[

½K(t-to) ]

(9)

where t=t~ is the m o m e n t of passing o f the middle point. The change of energy of the undisturbed mo230

Fig. 1. Element of the stochastic web in the neighborhood of the saddle at: (a) K=0.6 and q=4; (b) K=0.6 and q=6 (parameters K and q are determined in expressions ( 1) and (2) ).

Volume 144, number 4,5

PHYSICS LETTERS A

integrating in time over a quarter of the period of the particle oscillation in a separatrix cell the change of the averaged energy E = H 4 can be determined: i

AE=-K2

COS %t

dtch2[½K(t_tn)]

--QO

--- - 8n 2 e x p ( - n 2 / K ) cos ntn.

(11 )

The period of the particle oscillation near the separatrix in a cell is equal to [ 1 ] T ( E ) = --SKIn(4K/[E[ ) .

(12)

Eqs. ( 11 ), (12) lead to the separatrix mapping describing the dynamics of the particle moving inside the stochastic web, En+ 1 E , - 8n 2 exp ( - n 2/ K ) cos n& , =

t.+, = 6 + 2 1 n ( 4 K / I E n + l I ) •

(13)

The stochastic motion appears when the condition of local phase instability is fulfilled:

[~tn+l/~ln- I I >

1,

(14)

from which the estimate for the stochastic web thickness follows, IEI ~ E s ( 4 ) : 16n 3

E(4) s

-

K

the solid line represents the theoretical dependence of the web thickness on the parameter K, calculated using eq. ( 15 ). 3. Let us determine the web thickness in case of three-order ( a = 2 n / 3 ) and six-order ( a = 2 n / 6 ) symmetries. These cases appear to be much more complicated than for a = 2n/4. While estimating the thickness at q= 4 eq. (6) was used for the averaged Hamiltonian, from which eq. (9) followed. Besides that, it was assumed that in a saddle point the separatrices intersect at the same angle as without perturbation. Defining a web thickness more precisely one should take into account the O ( K 2) corrections to the averaged Hamiltonian. In case of four-order symmetry these corrections are equal to zero because of the conservation of the angle between separatrices in the point of their intersection and they do not influence eq. ( 15 ). However, for q = 3 and q = 6 these corrections lead to essential deformation of the phase portrait of the averaged motion and to alteration of the angle between separatrices in the saddle point (fig. 3). Neglecting the O (K 2) corrections in the averaged Hamiltonian, analytical calculations similar to those in the previous part give ~(3) _F(6) 64,~/3 -s --s 9 ~exp

2

-

--

.

(16)

(15)

exp(_n2/K)

Fig. la shows an element of the stochastic web for K = 0.6 near the destroyed saddle ( v = 0, u = n), where its thickness was calculated numerically. The results of the numerical simulation are shown in fig. 2, where 1.00

5 March 1990

10g E~4)

But numerical calculations show that the web in this case of six-order symmetry ( q = 6 ) at K<< 1 is several orders thicker than the web at q = 3. The results of numerical calculations are represented in fig. 4, where the solid curve represents the theoretical result calculated using eq. (16). To define eq. (16) more precisely let us take into account the O ( K 2) corrections in the averaged Hamiltonian. At q = 3 follow from (3)

-3.00

H3 = - ]K(cos v + 2 cos ½vcos ½V/3 u) , -5.00

-7.00

v3 = - ~ x [ (cos v - c o s ½v cos ½v/~ u)

/

-9.00 o.3d .......

X (cos }zt + cos 4nt + cos ]nt +...) K o.~;

......

'o'.~d ......

'o'6d .....

Fig. 2. Comparison of the theoretical and numerical results for the thickness of the web as a function of K at q = 4.

-

x/3 sin ½v sin ½~

+cos-~nt-...) ] .

u (sin }n t - sin ]nt (17)

Averaged in time the Hamiltonian ( H ) with O ( K 2) corrections is 231

Volume 144, number 4,5

PHYSICS LETTERS A

5 March 1990

8 •.,

:I'

,./

/

".,,.

~

J"

•

\

Fig. 3. (a) The web at q=3 and K=0.6. (b) The web at q=6 and K=0.6. (c) Lines of level of Hamiltonian (H) in the (U, V) plane. (d) Lines of level of Hamiltonian (H) in the (u, v) plane.

/or3

OV3 )

(H) =H3(U, V)+\-~Sp+ ~-~u ,

(18)

components of motion, which can be obtained from

ov3

8 b = - OU' where ( ) means averaging over a period, U = ( u ) and V= ( v ) are the slow c o m p o n e n t s of m o t i o n and 8v = v - V a n d 6u = u - U are the quickly oscillating 232

or3

8 9 = 0---V"

(19)

Carrying out the operation of averaging in (18) we obtain

Volume 144, n u m b e r 4,5

PHYSICS LETTERS A

5 March 1990

Similar analytical calculations at q= 6 lead to

log E~3).log El6/ . /

. f

K(6) =K(1 + 3K/4xf3 )

-3.00

(24)

elf

O

So, with respect to new effective values of Keffin the cases of q = 3 and q=6:

-8.00 /

/

7~3

D

E~3'6)= 64x/~9 K exp(

/

K -

13.00 I 0.30

I t , l l l l L i i

, i , i i

0.50

i b l l l l l

iii

0.70

Fig. 4. Comparison of theoretical and numerical results for the thickness of the web as a function of K at q = 3 (El) and q = 6

(o).

( H ) =H3(U, V ) - K2a (sin 2 ( ½V) cos 2 ( ½x/3 U) -cos2(½ V) sin2(½V/3 U) (20)

where ~ is 1

1- 5 +

1

1

+

x

+ . . . . 3,¢6"

Lines of level of the Hamiltonian (20) at K = 0.5 in ( U, V) coordinates are represented in fig. 3c. To get a picture similar to that represented in fig. 3a it is necessary to return in (20) from ( U, V) coordinates back to (u, v) coordinates at any fixed moment of time. In fig. 3d the lines of level of Hamiltonian (20) are plotted for K=0.5 on the (u, v) plane at t = 1: Expanding Hamiltonian (20) in a polynomial series in the neighborhood of one of the saddle points (for example, V=0, U=2n/x/~) one can get the equations of the separatrices in the neighborhood of the saddle, V= _+x/~ (1 +K/2xf3 ) U + Z x

(21)

and the equation of motion along the separatrix, K (1 - 3K/4x/~ ) V. I)= - x/~ -~ -~

(22)

The factor in parentheses is caused just by O ( K 2) corrections to the averaged Hamiltonian and leads to effective decrease of the parameter K in formula (16):

K(3) eff =K( 1- 3K/4x/3 )

) (25)

i i i i i 1 ~ 1 1

0.90

- 2 sin Vsin ½Vcos ½v/3 U ) ,

3 /C(1-T3/C/4,/~ )

(23)

The dashed and dot-dashed curves in fig. 4 represent the results calculated using (23) and (24) for q= 3 and q= 6 respectively. So, the expressions (16) and (25) for the stochastic web thickness are obtained for small values of K. One should note that the estimates for the web for mapping (2) were carried out in refs. [ 1,5,8 ] and for a mapping obtained from (2) by iterating it four times at q=4, in ref. [6]. The method used in ref. [6] is limited and can be applied only in case of fourth order symmetry as the mapping (2) iterated four times is equivalent to (2) only to terms O(K). In the cases of q= 3, 6 the corrections of higher orders are important for determining the thickness of the web. If we rewrite (25) (at K<< 1) in the form

E~3,6) 64xf3 n 3 9 K exp(-Y-ln2) e x p ( _ ] x f ~ n 2 / K ) , one can see, that O ( K 2) corrections influence not the exponent, but the additional preexponential factor (compare with (16)). The difference between the theoretical and the numerical results in fig. 4 can be explained by the fact that the values of K used in the numerical calculations are not very small and the corrections of higher orders exert noticeable influence. The difficulties of numerical checking at smaller K are connected with the narrowness of the web ( < 10 -12) and with noticeable influence of rounding-0ff errors during the simulation. 4. The description of particle diffusion along the channels of the stochastic web is rather simple in the case when there are two different scales of time in a system: quick mixing in phase variables and slow diffusion in action variables. In this case the diffusion is described by an equation of Fokker-Planck233

5 March 1990

PHYSICS LETTERSA

Volume 144, number4,5

Kolmogorov type (see, for example, ref. [ 5 ] ). In case of the mapping (2), when K is rather large, mixing in phase passes much quicker than diffusing and a quasilinear consideration is possible leading to the Fokker-Planck-Kolmogorov equation [ 1 ]. This approach permits one to obtain a rather precise result in many practical applications. Taking into account the finite time of mixing in phase leads to the appearance of correlation corrections to a diffusion coefficient. These corrections for the standard mapping were obtained in ref. [9 ]. The method of their calculation is described in refs. [ 10-13 ]. Let us calculate correlation corrections for a mapping with twisting (2) at q = 4 using a different technique. To obtain a diffusion coefficient D = lira ( R ~ )

(26)

F ( k ; R j + I ) = E E Jm,(FcK)Jm,_,[(~-mj)K] mj m j _ 1

× F [ ~ - m j , 17-mj_1; --Rj_l ] •

(29)

Continuing the process of iteration and taking n even (for simplicity) we obtain

f(k;R~) = •

E "" E • Jm.-~(~GK)Jm~-=(L-2K)

mn--I ran--2

ml mo

x J,,._,(17,_2K)...&, (172K)Jmo(&K) ×F(ko; Ro( - 1 ) . / 2 ) ,

(30)

where

kn= (£, 17), kn_ 2 = ( ]~-mn_l,

k'-b mn-2)

,

...,

ko = ( k - r a n - 1 .... - m l , 17 + m,_2 +...+mo) •

(31)

let us construct a generating function Using definitions (27), (29), (31), we obtain

Fn(k) = ( e x p ( i k ' R n ) ) ,

(27)

where Rn= (Un, Vn) is a radius vector in the point with coordinates (u, v) in n steps of the mapping, k = (/~,/7) is a wave vector and ( ) means averaging in initial conditions: 2n

if

( )--(2g)n

F(ko) = (F(ko; Ro) ) = (F(ko; Ro(

-

2~

1

f

-- ( 2 n ) n J

exp (iko'Ro) dRo = 6 ( ~ ) 6 ( ~ )

0

= d(l~- ran_ i .... - ml )d(17+ m n _ 2 ( )dRo.

0

( R 2 ) = -- ( 0 2/0/(72 dr" 0 2/0172 ) F n ( k )

I,=o.

(28)

Let us introduce the following function,

(32)

( F ( k , ; Rn) ) = F , ( k n ) = Z mn- 1

... • J~.-, (17.K)'"J~o (1~oK)6(~)6(17o)" trio

(33)

F(k; R i) = exp(ik.Rj). Then for the case q= 4 with respect to mapping (2),

F(k;Rj+~)=exp[i(Euj+~ +17vj+,) ] =exp [i ( ~vj-17uj-17K sin vj ) ] • Using the formula ~

exp(imO)Jm(z),

m : -oo

where Jm(Z) is the ruth order Bessel function, one can obtain 234

"+"...-1- m o ) •

Averaging (30) and taking into consideration (32) and (27) we obtain

Then

exp(izsin0)=

1 )n/z) )

Let us consider the term with m o = m l . . . . . rnn_l=0. From (33) follows F~(°) (k) = [Jo (EK) ]n/2 [Jo (17K) ],/2,

(34)

where the index zero means the zero-order approach for Fn(k). Then according to (27) and (28) we have ( R 2 ) = ½ K 2 n . For this approximation the diffusion coefficient is D (o) = 1K2' i.e. the known result for the quasilinear approximation [ 1 ]. At K>> 1 the main corrections for the diffusion

Volume 144, number 4,5

PHYSICS LETTERS A

5 March 1990

D 2 i .01

0 .g(

0.41

0.20

0.0( 0 ,';'0

%

0.25

K

°i~+

01~

'

01_~

'

01++

'

01~

i

°i+~

'

01,~

'

1100

,~

Fig. 5. Comparison of theoretical and numerical results for the diffusion coefficient as a function o f K a t q = 4 .

IO'L

~)

~,.00

g.5~

H-.5~

II

IIIIi

II

K..Or I

D .50

~.O0 0 .~0

o'.~t

~'.+2

+

11_,~

'

Fig. 6. The power index fl as a function of K at q = 4.

235

Volume 144, number 4,5

PHYSICS LETTERS A

coefficient can be obtained taking into consideration in (33) the terms with m i = + 1, mi+2=-T- 1, and all the rest m j = 0 ( j # i , i + 2 ) , and also the terms with m i = ___1, mi+4=-T- 1, and all the rest m i = 0 (/'#i, i+4). Leaving only the terms of order K we have D = ½K2 [ 1 + 2Jo (K) + 2Jo2(K) ] =Do[1 + 2 J 0 ( K ) + 2 J Z ( K ) ] .

(35)

To check this expression 103 trajectories were taken and 105 iterations were made for different values of K. The solid curve in fig. 5 represents the expression D / K 2. The experimental data agrees with (35) for these values of K. In fig. 6 the values of the power index fl in the following formula are represented, (R2> =Dn p , for K not too large. The averaging was made over 103 initial conditions. For K rather small the phase space has noticeable islands not covered by the stochastic web. Maybe that is the reason for deviation offl from 0.5 for small K. 5. In conclusion, we note that in general the problem of determining the thickness of the stochastic web in systems with degenerate Hamiltonians has nontrivial peculiarities, connected with the deformation of the averaged phase portrait under the action o f perturbation and, in the first place, with the alteration of the angle of intersection o f the separatrices in the saddles of the separatrix net. Moreover the thickness of the stochastic web and the diffusion rate of particles are determined not only by typical frequencies of the averaged motion, amplitude and frequency of external force, but mainly by the concrete type of the perturbation. Having calculated the thickness of the stochastic web in cases of three- and six-fold symmetry we showed that the final results differ greatly. In systems with nondegenerate Hamiltonians accounting for these deformations leads just to small corrections to the estimates of the stochastic web thickness. Another remark deals with a rigorous estimate of the stochastic layer and the stochastic web thickness. For a long time the rigorous result was connected only with the estimate of the angle of the splitting o f the separatrices [14]. To estimate the thickness of the 236

5 March 1990

stochastic layers the method of constructing the separatrix mapping was suggested [ 15 ] and the main results for the thickness of the layer, including also exponentionally small thickness, were cited (see also review [ 5 ] ). The rigorous estimates for the thickness o f the layer and for the angle o f splitting of the separatrices appeared not long ago [ 8,16,17 ], and confirm the results obtained with the help of separatrix mapping [ 5,15,18 ]. In the works by Lazutkin [5,17] the comparison of the angle of the splitting of the separatrices and the thickness of the stochastic layer is made. And though the thickness of the layer was determined to within a constant it was shown that its dependence on the parameter K is fractal. We express our sincere gratitude to V.F. Lazutkin for interesting discussions and for acquainting us with unpublished results.

[1 ] G.M. Zaslavsky, M.Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov and A.A. Chernikov, Zh. Eksp. Teor. Fiz. 91 ( 1986) 500. [2] G.M. Zaslavsky, M.Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov and A.A. Chernikov, Pis'ma Zh. Eksp. Teor. Fiz. 44 (1986) 349. [ 3 ] D.W. Longcopeand R.N. Sudan, Phys. Rev. Lett. 59 (1987) 1500. [4] S. Murakami, T. Sato and A. Hasegawa, Physica D 32 (1988) 269. [5] G.M. Zaslavsky, R.Z. Sagdeev, D.A. Usikov and A.A. Chernikov, Usp. Fiz. Nauk 156 (1988) 193. [6] A.J. Lichtenberg and B.P. Wood, Phys. Rev. A 39 (1989) 2153. [7] V.I. Arnold, Physica D 33 (1988) 21. [8]V.F. Lazutkin, Dokl. Akad. Nauk SSSR (1990), to be published. [9] A.B. Rechester, M.N. Rosenbluth and R.B. White, Phys. Rev. A 23 (1981) 2664. [ 10 ] H.D.I. Abarbanel, Physica D 4 ( 1981 ) 89. [ll]H.D.I. Abarbanel and J.D. Crawford, Phys. Lett. A 82 ( 1981 ) 378; Physica D 5 (1982) 307. [ 12] R.V. Jensen and C.R. Oberman, Physica D 4 (1982) 183. [ 13 ] R.V. Jensen and C.R. Oberman, Phys. Rev. Lett. 46 ( 1981 ) 1547. [ 14 ] V.K. Melnikov, Dokl. Akad. Nauk SSSR 148 ( 1963) 1257. [15] G.M. Zaslavsky and N.N. Filonenko, Zh. Eksp. Teor. Fiz. 54 (1968) 1590. [ 16 ] Ph. Holmes, J. Marsden and J. Scheurle, Contemp. Math. 81 (1988) 213. [17]V.F. Lazutkin, I.G. Schahmanskii and M.B. Tabanov, Physica D 40 (1989) 235. [ 18 ] B.V. Chirikov, Phys. Rep. 52 ( 1979) 263. [ 19] V.F. Lazutkin, Alg. Anal, 1 (1989) no. 2.