On the classical dynamics of strongly driven anharmonic oscillators

Physica D 46 (1990) 317-341 North-Holland

ON T H E CLASSICAL DYNAMICS OF STRONGLY DRIVEN A N H A R M O N I C OSCILLATORS H.P. B R E U E R , K. D I E T Z and M. H O L T H A U S Physikalisches Institut, Unil~ersitiit Bonn, Nussallee 12, D-5300 Bonn 1, Germany

Received 19 June 1990 Revised manuscript received 27 July 1990 Accepted 27 July 1990 Communicated by F.H. Busse

We investigate the dynamics of periodically driven anharmonic oscillators. In particular, we consider values of the coupling strength which are orders of magnitude higher than those required for the overlap of primary resonances. We observe a division of phase space into a regular and a stochastic region. Both regions are separated by a sharp chaos border which sets an upper limit to the stochastic heating of particles; its dependence on the coupling strength is studied. We construct perpetual adiabatic invariants governing regular motion. A bifurcation mechanism leading to the annihilation of resonances is explained.

1. Introduction Research on chaotic motion in Hamiltonian systems has led to an increasingly deepened understanding of their long-time behaviour. Emphasis was given to the analysis of the onset of stochastic motion [1-3] in near-integrable systems [4]; new theoretical insights were stimulated by an impressive wealth of numerical data. We simply mention the scaling behaviour of dynamic variables [5, 6] in the vicinity of K o l m o g o r o v A r n o l d - M o s e r (KAM) tori [7], renormalisation group theory [8, 9] and universality classes of invariant curves as well as the analysis of Hamiltonian flows through invariant Cantor s e t s cantori [10, 11]. In this p a p e r we shall not pursue any of these theoretical ans~itze which are essentially connected to the notion of near-integrability. Our main goal is rather to study the structure of phase space for systems which are 'far' from integrable limiting cases: introducing a parameter, /3, say,

which, when 'small', controls the near-integrable case and, continued to 'large' values, brings the system to a p a r a m e t e r region where perturbation theory is no longer applicable to the originally defined separable system, we are led, again, to the problem of describing non-perturbative phenomena, an urging problem in many parts of physics indeed. Of course, varying fi from 'small' to 'large' values might lead to a point where the system is close to a separable system again and we are treating not a problem requiring non-perturbative methods but rather a more judicious choice of variables reflecting separability. The model systems we are going to consider do show such a behaviour; nonetheless, as we shall see, we do not run into a tautology since between the two limiting cases the phase space structure displays features which are not observed in near-integrable cases. Of central importance for our investigation of phase space is the construction of adiabatic invariants which, in particular, lead to a detailed

0167-2789/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

318

H.P. Breuer et al. / S t r o n g l y drit'en anharmonic oscillators

understanding of a characteristic resonance structure and the bifurcation of periodic orbits to be observed upon varying external parameters. In going beyond a merely cursory discussion we have to choose a certain type of physical system: we shall restrict our deliberations to Paul trap-like experimental a r r a n g e m e n t s with periodically driven sequestered particles which we model as a one-dimensional system subjected to a single-mode periodic dipole interaction and an anharmonic, static trapping potential of variable steepness. Theoretically speaking, the particular interest of such systems lies in the fact that permanent spatial localisation of states is guaranteed and, quantum-mechanically, an increasing energetic separation of states alleviates e.g. the discussion of the semi-classical limit. In the non-perturbative regime the relevant phase space of these models shows a clear separation of stochastic and regular motion: chaotic motion is confined to a torus-like region which is surrounded by the invariant tori of regular motion; an extremely sharp boundary separates both regions. We shall construct the adiabatic invariants responsible for regular motion. The sharpness of the boundary appears to be connected to the occurrence of very narrow resonances, too narrow, it seems, for a Chirikov mechanism to be viable in building up chaotic motion. As a particularly interesting feature we shall discuss the intrusion of resonances into the chaotic region and their bifurcation. The question of developing a physically relevant picture of adiabatic motion is only broached in a very curtailed manner in our examination of phase space structure and its dependence on external parameters. An interesting application appears nonetheless: the heating of an (almost ideal) gas in our trap considered as a function of the strength of the single-mode radiation driving it periodically is limited and scales approximately with a certain power. In presenting our material we shall not refrain from repeating some notions from the theory of near-integrable systems which seem necessary for

clearly stating our derivations and results (section 2). The discussion of the non-perturbative regime then follows in section 3. In section 4, finally, we summarize our results.

2. The model and its weak coupling limit

We shall discuss the physics of a particle trapped in a one-dimensional anharmonic potential experiencing a single-mode periodic driving force deriving from a dipole interaction. Its action functional is

g[21 = £'dt' L( 2,)r,t') =

f,

tdt'

5

-

K2 q

}

_ A2cos co,,),

(2.1)

where q=4,6,8 .... and m, K, A, co are the particle's mass, the strength of the trapping and the strength and frequency of the driving force, respectively. Classically only one p a r a m e t e r is relevant and is chosen here as a scaled coupling strength: defining 7 :=

( m _ ~ ) 1/(2 q)

(2.2)

and

9

c~ : = m w y ~,

(2.3)

A 1 3 "-- mco 2 T ,

(2.4)

we find = Q(d

,

_ x,, -

cos

,,)

=aS[x] =a

drL(x,2,'r')

(2.5)

H.P. Breuer et al. /Strongly driven anharmonic oscillators

with

flow

2=yx,

~L

13 = ~

r=cot

@~: r - + r

= mcoyp

OL 02"

for each coupling constant /3. Since the Hamiltonian is 2~r-periodic, information on the dynamics can be obtained by studying the iterations of the monodromy map [13]

(2.7)

A~ := q~t~~.

It is clear that c~ carries the dimension of an action and does not enter into the classical equation of motion

8S=0

or

8S=0;

its role in the corresponding quantum theory will be discussed elsewhere. In the following we shall exclusively treat the scaled model with action functional S [ x ] and the Hamiltonian H = lp2 + xq +/3x cos r.

(2.8)

Questions concerning the existence of periodic orbits in systems of this type have been extensively studied in the mathematical literature [12]. The limit q - ~ ~ corresponds to the case of a particle in a box. In this case, we write

---,V(x) = 0 ,

Ixl < a

=~,

Ixl > a

(2.9)

and obtain 7 =a

(2.11)

(2.6)

and

P-

319

(2.10)

from (2.2). There are two ways of investigating the dynamical system defined by the Hamiltonian (2.8): (i) It generates a non-autonomous dynamical system in phase space F = {(p, x)} = ~ 2 with a symplectic structure d p A d x and a Hamiltonian

(2.12)

However, it should be borne in mind that At3 is known explicitly only in exceptional cases like the kicked rotator [1, 14]; in the generic case the equations of motion have to be integrated numerically. (ii) It can be extended to generate a higherdimensional autonomous system by introducing a generalized momentum I~ conjugate to 7: The quasienergy ~lp2

~:=H+I,=

+xq+/3xcosr+l~

(2.13)

is a constant of motion and induces a flow q~ in the extended phase space

F= FX R x S'= ( ( p , x , ~ , r ) } .

(2.14)

We will use both ways of description: Whereas the first point of view leads to the construction of Poincar6 surfaces of section, the general advantage of the second is that it makes possible the application of theoretical constructs and theorems for autonomous systems. For /3 = 0, the system (2.8) obviously is integrable and the theory of near-integrable systems applies for/3 << 1. For the Hamiltonian H0 := 4p2 + x q

(2.15)

we introduce action-angle variables

1

1= ~ p

t"

dx,

(2.16)

0: its canonical conjugate,

(2.17)

320

H.P. Breuer et al. /Strongly drit'en anharrnonic oscillators

and decompose x = x(l, O) into a Fourier series

x(l,O) =

E

- 2Vl(1)cos(lO ).

ing /3 we meet successively overlapping primary resonance chains until at

(2.18)

/=1,3,5 ....

/3~" = ~ Expressing the Hamiltonian (2.8) in terms of these variables we obtain (the summation runs over

l=2k-l,

keZ)

H = H o ( I ) -/3~Vl(1)cos(lO -.r)

(2.19)

l

and the corresponding quasienergy ~Y/#=H + I v .

(2.20)

For q > 2 (q even), the condition 32Ho

~al

* 0

(2.21)

guarantees iso-quasienergetic nondegeneracy in the sense of K A M theory [4]. As is well known, the K A M theorem does not give a realistic estimate for a critical coupling constant/3c characterizing the onset of chaotic motion nor does it imply an estimate for the measure of invariant tori in phase space; criteria like the one for resonance overlap developed by Chirikov [1, 14] are of much greater help. For example, for q = oo eq. (2.19) reads explicitly H--~Tr212

/3 Y'. ~ 4

cos(/0 - ~-)

(2.22)

l

and 3(o _ c

1 4 ( / + 1) 2

(2.23)

is the critical coupling constant where the lth and the (I + 2)th primary resonance start overlapping. For small /3 we thus have the following picture which qualitatively holds for all q: U p o n increas-

(2.24)

even the main resonances for I = 3 and l = 1 overlap: global stochastic motion has set in the region 0 _ < I ~ 1 1 = 4 / w 2 of all primary resonances. This mechanism for the onset of chaotic motion has been studied for a weakly driven particle in a square-well potential in ref. [15] and quantitatively been compared with the Chirikov criterion as well as with renormalisation group methods and can be considered as essentially understood (we mention the relation between the decay of KAM tori and the destabilization of the approximating long cycles [5], renormalisation group analysis of maps [6, 8] and Hamiltonian flows [9]). However, this cannot be the whole story. What happens, for instance, if we increase /3 further into a region where this kind of perturbation theory no longer applies? What are the coordinates there which allow for an effective description of the system? Before we turn to such questions in the following section we shall give an example for a feature which reflects a non-perturbative restructuring of phase space typical for p a r a m e t e r regions far away from 'H0-dynamics'. The notions developed for the theory of the onset of chaotic motion discussed above mainly rely on the phase space structure for small perturbations. The latter implies that periodic orbits approximating a given KAM curve do not bifurcate when the perturbation is increased until the critical value is reached, an assumption explicitly entering into the derivation of the renormalisation group equations. The occurrence of bifurcations is thus a typical non-perturbative feature. As an example, we again consider the driven particle in a box ( q - o~) and show in fig. 1 the bifurcation of the highest primary resonance, i.e., of the most stable elliptic 1-cycle of the mono-

H.P. Breuer et al. /Strongly driuen anharmonic oscillators 2

321

I I I 1 ~ 1 ! 1 1 1 , : 1 [ [ 1 1 1 1

liIIIIII

'~i~ii: i!~i~!ii i~!::~i:i:i!iii!~i i:',:::/:.i¸i ,.:i.,::i p

iili' iiili !!ii ¸ii i

o

i:~,?:!~:i::(::i:i ~i:i:~::~i:i(i iiii%; ~i~ (a)! -I 0

0.5

0.0

0.5

(b) ~,ILII, -1 0

1.0

,

,

i

CC

0.5

~

~

~

=

I

0,5

:

:

,

,

1,0

X

X

P

-1,0

-0,5

O0

0.5

1.0

X Fig. 1. Bifurcation of the stable 1-cycle for the periodically driven particle in the box. We display 10 orbits integrated over 1000 periods. (a)/3 = 0.64, (b)/3 = 0.68, (c)/3 = 0.72.

322

tt.P. Breuer et al. /Strongly driven anharmonic oscillators

dromy map Ate. The position of the fixed point can be calculated exactly to be pf=_+2(1+/3), xf=-I

Hkicked--

(left end of the box)

02Ho

-- 0 Ol 2

Ol

v21 : 8

(2.25)

(at the box walls at x = + 1 we have elastic collisions p--* - p and thus have to identify + p and - p : (2.25), hence, represents one fixed point). In figs. la and lb we observe this elliptic fixed point to be still intact, at /3 = 0.68 it decays into a hyperbolic (unstable) and two elliptic (stable) ones which appear in fig. lc as clearly visible islands. We conclude that /3---0.7 can be taken as marking the beginning of a non-perturbative regime: the most stable resonance for l = 1 already bifurcated by detaching from the left end of the box in a way characteristic for a pitchfork bifurcation. Bifurcations typically connected with an isoquasienergetic degeneracy

O~o(I)

In action-angle variables the Hamiltonian (2.22) has to be replaced by

(2,26)

will be discussed in the following section. Before turning to questions of strong coupling it is of interest to compare the phase space structure for single-mode systems discussed above with kicked systems in the near-integrable region [16]. The latter are obtained by replacing the singlemode-driving term by a series of successive kicks:

~

4 cos(lO-mr)

/ 3 E E q.r2l~__ 1 m=0

(2.29) which is the Hamiltonian for a kicked particle in a box. Primary resonances are now located at 4m

l/m

W2 1 '

(2.30)

in contrast to the single-mode system, where they are found at

I1=111-

4 1 v 2

(2.31)

1"

The phase space of the kicked system thus shows a periodic resonance structure: the 10th primary resonance appears infinitely often with equidistant copies separated by Alt0 = 4/w210. We will come back to this system and show that also in the non-perturbative regime they describe qualitatively different physical situations.

3. The dynamics for large coupling strength Let us now turn to the investigation of the system defined by

H-½P2+X'+/3xcosr,

q = 4,6,8 ....

(3.1)

/~x cos ~- + / 3 x ' ~ a 2 ~ ( ~ - )

=/3x~ E a ( , - 2 ~ n ) , It =

(2.27)

~c

where

X:/31/(q--l)~, ~c

va>_(r)

in the case of strong couplings (/3 > 1). For reasons that will become obvious later, it is convenient to rescale the coordinates

=

51 +

Y', c o s m r

(2.28)

m=l

is the 2-rr-periodic a-function which contains infinitely many modes with equal coupling strength.

p=/3q/2(q 1)~,

r = S'lf

(3.2)

such that the Hamiltonian reads /4 = ~'/5 2 + 2 q + 2 cos .¢2~:,

(3.3)

H.P. Breuer et al. /Strongly driven anharrnonic oscillators

where := ~ - ( q

(3.4)

2>/2(q-1>

We note that (3.2) is not a canonical transformation but rather a similarity transformation which contracts the area in {(p,x)}-phase space by a factor of [2~(q+2)/2(q 1) The starting point of our analysis is the observation that the dynamics of the system is governed by two frequencies: The fixed external driving frequency S2 and the frequency of the phase oscillations in the potential V(2, r) = 2 q + 2 cos r, which becomes arbitrarily large for large initial momenta. This fact suggests the introduction of the action variable 1

J = ~-~-(~p~d2,

(3.5)

From eq. (3.4) we have S2---) 0 for /3 ~ ~. The adiabatic theorem of classical mechanics [13] now states that for times of the order of 4 = ~(S2 -~) the action variable remains constant up to terms of the order ~(.Q). However, this estimate is rather useless since it controls the behaviour of J only over one period of the external field. As we shall now show, a much sharper assertion can be obtained by explicitly taking into account the time-periodic character of the external field. Therefore, we perform a further canonical transformation

( J , ~ ) --, (J,,O),

J' =J,

(

O=q~+~l

(/5,2) --* ( J , ~ )

(3.7)

is achieved by the generating function S(J,2,K27r) =

d21~?=~(j,m>

N(j).~._

(3.6)

and ~: is kept fixed. Therefore, in the phase space region of high momenta and for large couplings/3 the phase ~ conjugated to the action J is a variable which is oscillating fast compared to the driving term cos ~2~=. The time-dependent canonical transformation

J0"

d'r'w(J,"c'

>)

," (3.12)

in this formula

w(J,~')-

OJ

(3.13)

denotes the instantaneous frequency of the q~oscillations and

~(J)=~J0

(3.8)

(3.11)

where

where /5 = + [ 2 ( H - ~ v - 2 c o s a ~ : ) ] '/2

323

1

r2-~

d~-w(J,~-)

(3.14)

its zero-mode. We thereby obtain the new Hamiltonian

by means of OS

P=~-'

K(J,~,s~) = H ( J ) +S~H,(J,~,S~÷), (3.15)

OS

(3.9)

~ = -07'

where

and the new Hamiltonian reads 0S OS =/-?(J, O4) + s ~ .

1

H ( J ) = ~-~-£

(3.1o)

2~

d~-/4(J,~-)

(3.16)

describes an integrable part of the dynamics. As a

324

H.P. Breuer et al. /Strongly dricen anharmonic oscillators

result of our scaling (3.2) this part is independent of `0 (resp. /7), a fact which greatly simplifies the following discussion. Furthermore,

I

7"

F

I

//

~b ~S

Hi( J, tO, .OT) = ~ ( J, ¢, ~7)I+

/

=,~,-,>1,±

/"

,/

/

/

(3.17)

//

//

// // /'

is a perturbation which is 2~v-periodic in the phase variables tO and r. The particular importance of the representation (3.15) lies in the fact that it allows the application of a special version [4] of the KAM theorem in the extended phase space

/

/

L, /: /i

{(J,

I~., to m o d 2 v ,

r

/

mod 2 v ) }

where the flow is generated by the quasi-energy

......

2____

I

{[

~=K

+ `0I¢.

~ ( J ) = ~ (`0 J)

(3.19)

The non-degeneracy condition necessary for an application of the KAM theorem in our case reads a~ 4= O.

(3.20)

_

7

.

!

1

1

1

]

J

(3.18)

This theorem states that the action variable J is a perpetual adiabatic invariant, i.e., is conserved (up to terms of order `0) for all times and a large subset of the phase space is filled out by invariant tori close to the unperturbed tori J = const. Therefore, the conservation of adiabatic invariants is, by virtue of the KAM theorem, extended to infinite times. Precisely this extension also explains the perpetual confinement of charged particles in adiabatic traps in plasma physics [4, 17]. As usual in KAM theory, invariant tori are characterized by a sufficiently irrational winding number

!

Fig. 2. The function ~ =~(J)for q = 10; we find J * = 0.425 and w* = 1.487.

In fig. 2 we plot the numerically determined function ~ ( J ) for q = 10. For the other values of q, the functions ~ ( J ) behave very similar and show the same characteristic minimum [18]; the non-linearity condition (3.20) is thus fulfilled everywhere except at a single value J = J *" ~0~

=0,

~ ( J * ) =: oJ*.

(3.21)

J*

In order to demonstrate the fact that in a large subset of phase space the above constructed action variables J are indeed the correct perpetual adiabatic invariants determining the dynamics of our system, we compare a Poincar6 surface of section for the system (3.3) (fig. 3, `0 = 0.27) with a set of unperturbed tori J = const, calculated according to eq. (3.5) (fig. 4). In the surface of section we observe an extremely sharp division of the phase space into two qualitatively different regions. For large initial momenta we find the phase space almost completely stratified by in-

H.P. Breuer et al. /Strongly driven anharmonic oscillators 2 i

~ l l , l i i i 1 1 1 1 1 1 1

,

I

~/j

-1

325

I

I

I

I

I

r

i

; J i J 'J // i

1 1

-2 -i

0

1

I

1

0

Fig. 3. P o i n c a r ~ s u r f a c e o f s e c t i o n for t h e s y s t e m (3.3) w i t h q = 10 a n d f2 = 0.27. W e s h o w 10 o r b i t s i n t e g r a t e d o v e r 1000 periods.

Fig. 4. N u m e r i c a l l y d e t e r m i n e d t e m (3.3) at ~ = 0 ( q = 10).

variant tori whose shapes coincide excellently with the tori J = const, displayed in fig. 4. This quasiregular region of phase space surrounds a zone of connected stochasticity which, on the scale of fig. 3, does not show any structure as, for example, stable islands of periodic orbits. What could not be expected from the general theory is the formation of a very sharp boundary, dubbed 'chaos border' hereafter, between these two regions. In order to understand the division of phase space into a regular and a stochastic zone more closely we again consider the case q = ~ where the transformation (/5, £) ~ (J, q~) can be performed analytically. In the appendix we construct the Hamiltonian in the adiabatic coordinates,

where

/~(y, ~, 0~) = ~q(y, a~:) + OH,(J,~,O~), (3.22)

1

c u r v e s J = const, for t h e sys-

0S

-

D .~o sin. rrS2~= . ( 1 +. (2lq~l x ~( I~1 - rr)

~r) "rr 12o~ c°s D 2? ) (3.23)

and w - o g ( J , ~(24). From eq. (3.23) we see that the perturbation OH1 is proportional to the inverse of the instantaneous winding number

a( J, Y2~) = w( J, S27r) (3.24) D This fact could have been expected from our considerations leading to the introduction of the coordinates (J, ~): The larger a, the faster is q~ compared to the time variable (. Thus, keeping Y2 fixed and increasing J the perturbation O H 1 gradually decreases until finally the region of

H.P. Breuer et al. /Strongly driven attharmonic oscillators

326

KAM-convergence for the system in the extended phase space is reached. Hence, the region of phase space corresponding to these large values of J shows predominantly quasi-periodic motion on invariant tori the relative measure of which approaches unity for increasing J. This explains the fact that the stochastic region seen in fig. 3 is confined to a compact subset of phase space: It is surrounded by impenetrable perpetual adiabatic invariants. On the other hand, for r = S24 = - r r / 2 the frequency co is small for small J since co(J, Tr/2) = co0(J)

=

The main resonances (l = 1) which are located at J =J,, = (4X2/-rrZ)n n = 2k + 1 = 1,3,5 . . . .

are described locally by pendulum Hamiltonians:

Kn=(w2/8)J2+F,,(J,,)cos(O-nY24),

and hence for small J the perturbation is large. We conclude that the perturbation glHl differs by orders of magnitude in different regions of phase space, which is reflected in the qualitatively different structures of the stochastic and the quasi-regular region. In order to characterize the near-integrable region of phase space we now turn to a detailed investigation of its resonance structure. Again we exemplify the generic behaviour by the driven particle in the box. For J considerably larger than J * ] q _ : ~ = 0.508 (that is, J > 2) the transformation (3.11) and a Fourier decomposition finally yield approximately (see the appendix)

(3.29)

where

(3.25)

('n-2/4)J

(3.28)

] J, [ j k ( A J n ) - j k + l ( ` ~ j n ) ] ~ n = ~;~J(J,,) = a/3"rrz/2J,{ -

(3.30)

From these equations we obtain for half the separatrix width of the nth resonance

AJn =

3 4j

(3.3 1

These resonances are extremely sharp as can be seen from the following numerical example: Let Jn = 3 for winding number n = 9 (i.e. k = 4) which means ,(2 = -rr2/12 = 0.822. We then have A J 9 -~

3 × 10 -8

K( J,O,g24) = H ( J ) {214n

+ [-@] 7- ~-'gk(J)c°s[lO -- ( 2 k +

1)g2~:],

l,k

whereas the next higher main resonance with winding number n = 11 ( k - 5 ) is located at J ~ = 11 × 4 / 2 / 7 2 = 3 + 3 and has a half width of

(3.26) A J l l ~ 8 × 10

11.

where In this case the ratio l = 1,3,5 . . . . .

k~2, AJll + AJ 9 J l l - J9

H ( J ) = (,Tr2/g)J 2, 1 []k(laV" ) - ] k + , ( l ~ l ) ] V ~ ( J ) = ~3 ]k = k t h Bessel function of the first kind, 5g¢'= ~ ( J )

= l/3,rrZ~OJ 3.

(3.27)

4.5 × 10 s

(3.32)

is far too small to make possible an overlap of resonances according to the Chirikov mechanism, which requires a ratio of approximately 1. Furthermore, we remark that the width of the reso-

327

H.P. Breuer et al. /Strongly driven anharmonic oscillators

nances decreases winding number:

very rapidly with increasing

AJ~ = B,,. n ",

(3.33)

where B n d e p e n d s only exponentially on n. W e conclude that in the quasi-regular region no overlap of main resonances occurs, which excludes c o n n e c t e d stochasticity in this part of phase space. T h e results of this analysis p e r f o r m e d for q = ~c are qualitatively also valid for all values of q as long as J >_ 2. We now investigate what h a p p e n s when resonances a p p r o a c h the stochastic region if 12 decreases, i.e., if the coupling strength /3 increases. As noted before, the r e s o n a n t tori are determined by the condition

~(J)

-

(~(J)

12

-

kl

,

(3.34)

k2

and the main resonances are given by the equation ~(J)

= n12.

(3.35)

Now an i m p o r t a n t difference b e t w e e n the descriptions of the dynamics in terms of the a c t i o n - a n g l e variables (I, 0) of the u n p e r t u r b e d system (cf. eqs. (2.16), (2.17)) and, on the other hand, in t e r m s of the adiabatic coordinates (J, ~0) b e c o m e s apparent: W h e r e a s w 0 ( I ) = OHo/OI is simply a monotonically increasing function of I, the frequency ~ ( J ) = OH/OJ shows a n o n - m o n o tonic behaviour (see fig. 2). By virtue of this fact eq. (3.35) has two solutions J~]) and j(2) for n12 < ~ ( 0 ) as indicated in fig. 5. If 12 is decreased, the two solutions a p p r o a c h each o t h e r and finally collide at J = J * when n O = w*

(3.36)

W e thus have no solution of (3.35) for n O < o~*: Both resonances have d i s a p p e a r e d as a result of a

~5(o)

nf~ i l

L'> .;* J<.'>

J'

Fig. 5. Sketch of the function ~(J) with the two solutions j~],2) of the resonance condition (3.35).

tangent bifurcation of the corresponding periodic orbits. W e illustrate this m e c h a n i s m with a series of Poincar6 surfaces of section for q = 10 in figs. 6a to 6f. In fig. 6a we observe a resonance with winding n u m b e r n = 7; its width is relatively large n e a r the stochastic zone. Using the notation of fig. 5 this r e s o n a n c e corresponds to the solution j(1) of eq. (3.35). T h e other solution Jn(2), however, still lies in the stochastic region and is not discernible as a stable elliptic island. For slightly lower values of .(2 (fig. 6b) both solutions are visible: R e s o n a n c e J~]) starts to intrude into the chaotic zone and resonance j~2), c e n t e r e d on the /5 = 0 axis, has emerged. W h e n .(2 is decreased further the two resonances a p p r o a c h a c o m m o n value of J and give rise to strongly d e f o r m e d invariant curves until finally the bifurcation of the elliptic fixed point of resonance j~2) and the hyperbolic fixed point of resonance j~l) takes place n e a r 12 = 0.212 (fig. 6f). T o u n d e r s t a n d the bifurcation m e c h a n i s m m o r e closely we now develop a local description of the dynamics n e a r J = J * . T o this aim we expand H ( J ) to third o r d e r in J - J * :

H(J) = 1 / ( j _ j , ) 3 + . , , ( j _ j , )

+ H * + ¢(IJ-J*J4),

(3.37)

H.P. Breuer et al. /Strongly dricen anharmonic oscillators

328

]

i --•T--~ ,

..,.,

!

I

1

P

,(a),

',...-5 ........

(b) I

L

1

!

I

I

,

1 ,

I

t

0

,

i

I

i

,

i

k

i

]

i

i

~

i

i

i

i I

i

i

1

i

~1 ~

i

!

i !

,

t

:

7

, ", "

'~\

"-,, 't\\

~¢X\ ,

t

, • ,

X

.

k

.

•

;

\

,

•

k

;

I

] :

.

,

. ..

-

'

•

•

,,

:

::

t

~ £ / / ' ./

i

/

r t

!

1

(c)

"

' ... +

/

(d) i

1

0

i

i

i

i

i

]

i

I

J

i

J

i

i

;

0

Fig. 6. P o i n c a r 6 s u r f a c e s o f s e c t i o n for q = 10 s h o w i n g t h e i n t r u s i o n o f t h e r e s o n a n c e w i t h w i n d i n g n u m b e r s t o c h a s t i c r e g i o n . ( a ) .(2 = 0.218, ( b ) J2 = 0.216, (c) ,Q = 0.215, (d) .(2 = 0.214, ( e ) J2 = 0.213, (f) .(2 = 0.212.

n = 7 into t h e

329

H.P. Breuer et al. /Strongly drit,en anharmonic oscillators ,I

i

i

i

i

I

i ,

!

2t '

i

I

i

i

'

'

I

r

,

i

r

I

i

1

i

\ I

f 1//

<;!, @t. :~+,;/, ,.:,) !£; !~. . . .

/ -1

:S/ i(e)

(f) I

~

~

-1

~

I

0

!

0

1

Fig. 6. Continued where

where

H* = H(J*),

o- = -/(J~') - J * ) .

~o* = N ( J * )

= O~j

The resonance condition reads

J

C')3/~ J* , 7 = 3--3-jT

(3.38)

a n d a d d a r e s o n a n t t e r m with w i n d i n g n u m b e r n. W e thus o b t a i n the m o d e l H a m i l t o n i a n h ( J, ~ , 4) = H ( J ) - M c o s ( ¢

(3.41)

- nS2~:).

P e r f o r m i n g a c a n o n i c a l t r a n s f o r m a t i o n (J,q~) ( J ' , ¢ ' ) by m e a n s of the g e n e r a t i n g function

~(j)

_ l~/(j_j,)2

+ o)* =F/.Q

(3.42)

with solutions j(I,2). F o r n O = o9" we have J~l) = a n d thus the q u a d r a t i c t e r m in the H a m i l t o n i a n (3.40) vanishes since ~r = 0. F o r ~r > 0 the flow g e n e r a t e d by h ' is c h a r a c t e r i z e d by four fixed p o i n t s X i = (Ji', q~'):

j(2) = j ,

X 1 = (0, 0),

X2 = ( 0 , + r r ) , S(J',q~,()

=(q~-ng27r)J'+J(ni)q~

(3.39)

X 3 = ( - 2~r/7,0), (3.43)

yields the new H a m i l t o n i a n (up to c o n s t a n t )

X 4 = (-2~r/7,_+ rr),

I -,z -Mcosg/, h' = h + ~OS = gl y J ,3 + ~o'a

w h e r e X 1 a n d X 4 a r e of elliptic a n d X 2 a n d X 3 of h y p e r b o l i c type. F o r o- ~ 0 X 1 a n d X 3 and, on

(3.40)

330

tt.P. Breuer et al. /Strongly driuen anharmonic oscillators

i

----]-

i

I

I

~-

i--°-

j/° --

<£<

...>. ) >

i

-2

3

(a)

3

--2

1

0

2

~pt

I

I

I

t /

.."

". ...

"

j! o •

(

~

\

}

< /"

-""" -

2

.

3

(b)

/

-'"""

/

--2

I

0

1

2

(491

Fig. 7. Phase portraits for the model system (3.40) showing the bifurcation of fixed points. Parameters are 7 = M = 1. (a) ~r = I.(), (b) cr = 0.5, (c) o- = 0.0.

H.P. Breuer et al. / Strongly dricen anharmonic oscillators I

I

i

I

i

331

i

i

r

..........

.................

...

..-..._.........

.

..

1

"ii -3 I --3

~

I

--2

~

I

--1

r

I

0

,

I

1

~

I

2

t

I

3

(c) Fig. 7. C o n t i n u e d

the other hand, X 2 and X 4 collide and disappear in a tangent bifurcation. In figs. 7a, 7b and 7c we show the phase portrait of the model system for three different values of o-. Comparing especially figs. 6d and 6e with 7a and 7b we see that the simple model describes the bifurcation mechanism quite well indeed. We remark, however, that the two pairs of fixed points do not collide simultaneously in the actual system since there, contrary to our model, the amplitude M of the resonant term does depend on J. As a further illustration of the bifurcation scenario we display in figs. 8a to 8f the annihilation of the resonances with winding n u m b e r n = 9. From eq. (3.35) and the value of to* = 1.487 read off from fig. 2 we determine the annihilation frequency to be ,(2 = t o * / 9 = 0.165, which coincides well with the numerically observed value. Even for those parameters for which main resonances intrude into the chaotic region, there is a remarkably sharp chaos border and thus a dichotomy of the phase space. For fixed ~ a strongly

driven particle in an anharmonic potential either performs quasi-periodic motion or, if it starts in the chaotic region, moves stochastically but there is an u p p e r limit to the kinetic energy it can gain. This limit is provided by the chaos border or, more precisely, by the maximal m o m e n t u m /Smax(.Q) on this border. In fig. 9 we show this upper limit ]~max as a function of ~ as obtained from Poincar6 surfaces of section. The intrusion of the resonances n = 5 , 7 and 9 appears as clearly visible spikes. However, apart from these spikes /~.... tends to decrease slightly with decreasing Y2, a fact which is easy to understand: If we consider a fixed interval Jl < y K J2 of the action variable, decreasing Y2 results in a decreasing reciprocal winding number, which is the relevant perturbation parameter. Scaling back to the coordinates ( p , x ) of the original system (3.1), we have P m a x = ~ q/2(q =~'~-q/(q

')])max(.Q) 2)1~. . . . ( ~ ) '

(3.44)

H.P. Breuer et al. /Strongly driven anharrnonic oscillators

332

~[,,

i , ,

,~

i

. . . .

i

,

2,

,

I

~

I

I

i

j

i

i

i

i

j

i

i j

/

1

io

F

1

q Z

q~l

1

S

I

I

3

1

0

!

I

1 F

-1

(d)

(c) 0

!

-i

Fig. 8. P o i n c a r d s u r f a c e s o f s e c t i o n f o r q = 10 s h o w i n g t h e i n t r u s i o n o f t h e r e s o n a n c e w i t h w i n d i n g s t o c h a s t i c r e g i o n . ( a ) ,Q = 0 . 1 6 8 , (b) &~ = 0.167, (c) -Q = 0 . 1 6 6 , ( d ) ~2 = 0 . 1 6 5 , (e) ~'~ = 0 . 1 6 4 , (f) -Q = 0.160.

number

n = 9 into the

H.P. Breuer et al. /Strongly drit,en anharmonic oscillators

I

i

I

i

i

I

333

I

I

i

i

I

i

I

-1

(f)

(e) -?

I

i

1

I

I

I

I

o

I

I

i

i

I

I

i

r

I

i

I

I

I

0

1

i

i

i

I

[

i

1

Fig. 8. Continued

However, as the example of fig. 9 indicates, the dependence of /gmax o n ~2 is weak compared to that of the scaling factor q/q

2 = j~q/2(q-- 1) (X ,~q/2(q- 1).

(3.45)

In fig. 9 ,6.... decreases from roughly 1.5 to 1.2; the scaling factor and ,Q 5/4 (q = 10), however, changes much more rapidly from 3.7 to 42.3. Thus, the slight decrease of /Sr~,× with increasing /3 is negligible compared to the increase of the scaling factor and therefore the maximal momentum of stochastically moving particles increases approximately as Pmax C~ j~q/2(q

1)

(3.46)

For example, for the driven particle in the box we have P . . . . (3( j~ 1/2.

Finally, we illustrate the sharpness of the transition from regular to stochastic motion from another point of view. We consider a short line segment of initial values transversal to the chaos border and show in figs. 10a, 10b and 10c its behaviour under the flow after one, two and three periods. Whereas the part of the initial line segment lying above the chaos border in the quasi-regular region is merely twisted, the part lying in the stochastic region is strongly stretched and folded even after two periods and nearly fills the whole stochastic zone after only three periods. This behaviour indicates very strong stochasticity, which deserves further investigations.

(3.47)

4. Conclusions

We investigated the structure of phase space for our model Hamiltonian (2.8) in a large range

H.P. Breuer et al. /Strongly driuen anharmonic oscillators

334 1

.

6

I

Prnax f.

/ / if---

1

.

~ . 2

_J

y~

3

J I

0.1

~__r

I

I

0.2

Q.~

Fig. 9. M a x i m a l m o m e n t u m o f p a r t i c l e s in the c h a o t i c r e g i o n as f u n c t i o n o f f2 for q = 10. T h e s p i k e s reflect the i n t r u s i o n o f the r e s o n a n c e s n = 5, 7 a n d 9.

of the coupling strength /3. We observe: (i) for small /3, i.e. /3 of the order of 10 1, the transition from the integrable case to the onset of global stochasticity in the range of small mom e n t a according to the well known theory; (ii) for large /3 (/3 > 1) an intriguing dichotomy of phase space: a solid torus (imbedded in extended phase space) of stochastic motion is surrounded by invariant tori corresponding to regular motion. These tori are based on the existence of perpetual adiabatic invariants confining stochastic motion for infinite times and the boundary between the two regions is remarkably sharp; (iii) a rapid increase of the width of a reson a n c e - extremely narrow farther off the stochastic r e g i o n - a p p r o a c h i n g the stochastic boundary and the intrusion of intact resonance zones into the chaotic region, a process which should "be interpreted as an interaction of an isolated resonance with a large region of stochastic motion and not as the overlap of stochastic layers along separatrices. The non-monotonous character of

the function ~ ( J ) leads to the vanishing of resonance zones at J = J * where elliptic and hyperbolic orbits coalesce. To make these features more transparent we employed subsequently two scaling transformations (2.6) and (3.2). The first of these scalings was performed to introduce the two parameters, c~ and/3, which are the two relevant combinations of the original four parameters in the model: The first one simply scales the symplectic form d/3 A do? + d l , A dt = a ( d p A d x + dl~ A d r ) (4.1) and, hence, the action integral, whereas the second represents the properly scaled intensity of the driving field. The second scaling (3.2) again rescales the symplectic form in extended phase space dp A dx +dI, A dr

=/3(q+2)/2(q

l)(d/5 A d2 + d l ÷ A d e ) .

(4.2)

Obviously [13] these scalings are non-canonical transformations although they leave Hamilton's

H.P. Breuer et al. /Strongly dril,en anharrnonic oscillators

i

I

i

i

i

i

I

i

i

i

i

[

f

i

I

i

i

335

~

i

I

I

i

i

;

,

i

I

I

<

I

I

I

i

I

I

=!

:Fi

~~.~.~

(a)

(hi r

-I

0

I

I

I

2l ,

I

I

'

I

I

1

'

'

'

I

'

'

'

'

0

I

1

I

(c) _ 2 1 1 1 1 1 ! l l i l l i l r l 1 0

1

Fig. 10. Behaviour of a small line segment of initial values (20000 points) under the flow generated by (3.3) for S~ = 0.27 and q = 10. (a) Initial line segment (upper left corner) and its image after one period; (b) image of the initial segment after two periods; (c) image of the initial segment after three periods.

H.P. Breuer et al. /Strongly dricen anharrnonic oscillators

336

equations in phase space {(/3, 2)} invariant. This non-canonical character, especially of (3.2), is of particular importance for the discussion of the large intensity limit /3-~ ~: Keeping the action defined in (3.5) fixed we see that the action defined on the original phase space tends to infinity: J = ¢/3 d2 ~ ~c

(4.3)

for

~-~c,

J=¢~d2

fixed.

(4.4)

Thus, the region of regular motion in the phase space {(/3,2)} shrinks for large /3 to a small region around J = ~ or, to say the same thing, for f fixed and /3 ~ ~ one necessarily dives into the stochastic region of the {(/3, 2)} space. This brings out clearly the usefulness of the scaling transformations for uncovering the fact that the strong field limit is governed by a nearintegrable structure, a fact which could be overlooked when working only in the phase space of the original variables {(/3, 2)}. A crucial element of our discussion is the construction of the adiabatic coordinates (J, ¢) in section 3. The transformation (/5, 2 ) ~ (J, ¢) reveals that the relevant perturbation p a r a m e t e r is the inverse winding number. In the coordinates (/5, 2) the size of the stochastic zone of the phase space decreases with increasing /3 and the limit /3 - , ~c again corresponds to an integrable system; however, in the original coordinates the stochastic zone grows according to (3.44). The chaos border scparating stochastic and regular motion and defining a compact submanifold S of stochastic motion in phase space limits the energy of particles moving irregularly, and, thus, plays for single-mode periodically driven systems the role which is played in autonomous systems by the condition of fixing the energy. For instance, trajectories starting in S being confined to S, a chaos border is the reason for a folding mechanism which counteracts Lyapunov stretching: The stretching and folding of trajectories

instigated by this border very soon leads t( stochastic motion as is shown in fig. 10. We still have to discuss the important observa tion that the existence of a boundary for chaoti~ motion is closely connected to the Fourier spec trum of the periodic driving term; in the extrem~ case of a periodically 'kicked' particle the Fouric spectrum is unbounded and all modes have the same amplitude. For example, for q = zc this fac leads to the occurrence of 'accelerator modes' fo: arbitrary large field strength /3: Consider two tor with integer winding number k and k ' ( k ' > k) If

=(2/v~)(k ' k)

(4.51

a particle initially on the torus I k has gained th~ m o m e n t u m A p - , r r / 3 after a kick and thus ha, jumped to the torus Ik,. This process repeat, itself infinitely often, which means that there it no upper limit for the m o m e n t u m a kicked parti. cle in a box can acquire. Generally, for kicked systems there can be n(: perpetual adiabatic invariants since the FourieJ spectrum of the periodic 6-function is unboundec and thus the frequency N ( J ) o f the phase oscillations is nowhere in phase space large comparcc to the frequencies of the interaction. As a physical consequence of this fact an (almost) ideal gas stored in our trap will heat u F indefinitely when periodically kicked; driven witl~ a single-mode force, however, the heating will bc limited and increase at most as a power of the intensity when the latter is changed. More precisely, the energy transferred to the gas will increase with the irradiation time in the case of J kicked system whereas it will stay constant, after an initial heating time, for thc single-mode case. A closer discussion of the thermal properties ol trapped many-particle systems is certainly of great interest. The notion of temporal variation of a thermal state requires a discussion of adiabaticit) and its time scales, both on the classical and the quantum mechanical, respectively semiclassical level. We hope to come back to these questions soon.

H.P. Breuer et al. /Strongly dricen anharmonic oscillators'

337

Appendix In this appendix we perform the transformation to the adiabatic coordinates (J, q~) and (J, 6) tor the driven particle in the box [18]. The aim is to derive an explicit expression for the perturbation f2H~ = f2OS/~r and to obtain its Fourier decomposition with respect to the phase variables ~b and r. Since we are interested in cases of high momenta we restrict ourselves to the region of phase space defined by

tTI(J,r) > Icos rl for all r. This condition guarantees that the turning points of the particle motion are given by 2 = + 1 (left and right wall of the box) for all values of the phase r. We then obtain for the action variable j = ~ y Pr~ l - d2 = 3Tr23/2COST [(144-cOST)3/2-(I~-cOST)3/2]"

(a.1)

The action S generating the time-dependent transformation (/5, 2) ~ (J, q~) reads 2

2 3/2

[

S+(J,2,T) = f 1 / 5 d 2 - 3co-s-r ( / ~ + cos T) 3/2 - ( / 4 - 2 cos T) 3/2]

(A.2)

for the upper branch and

S_(J,2,r) =

J;

23/2

- ~ d 2 +~rJ- 3cos -- T

cos T)3/2 +

cos,)3J2] + . J

(A.3)

1

for the lower branch. We have: S+(J,-

S+(J,+l,r)='rrJ,

1,r)=0,

S (J,+l,r)=vJ,

S (J,-1,r)=2"rrJ.

By means of the relations aS+ g~+=

(A.4)

3J

we get ( / ~ _ _ 2 COS T ) I / 2

=

(/.~ + COS T ) I / 2

COST 21/20) q~+

(A.5)

and cos T . ( / ' ] - - - ~ COS T) 1/2 = (/t] __ COS T) 1/2 + 2 1 ~ 7 ~ ( ~

-- Tr).

(A.6)

338

H.P. Breuer et al. /Strongly drit,en anharmonic oscillators

Differentiating S with respect to 7- we obtain aS+ _ 2~/2 s i n s [ ( / ~ + cos 7 - ) 3 / 2 ( / ~ _ 2 073 cos27-

Jr- COS 7" ( /~ -I- COS 7-)1/2 ~0/-I -

cos '7")3/2]

sinT- -- (/~ -- 9~COS 7") 1/2

-

+ 2 sin 7-

(A.7)

and OS Or

_

23/2 sin ~ - [ _ ( ~ _ 3 c0s27-

+

cos ~')3/2+ ( / ~ _ 2 cos 7-)3/2]

c°sT-)'/: ~-7o# +sinT-

+(/t-2cos7-)

1/2 O/t + 2 sin r

.

(A.8)

From eqs. (A.5) and (A.6) one has (the upper and lower sign correspond to the u p p e r and lower branch, respectively; for the lower branch one has to replace q~+ by q~ - w, cf. eq. (A.6)):

COS 7-" ) 3 Jr-(/] -F COS 7-)3/2 ~ (/~_3~ COS 7-)3/2 = -1-(/~ -}- COS 7-)3/2 -T (/~-'1- COS T)1/2 -T 2 1 ~ - q ) + . We now insert this relation into (A.7) and (A.8) to arrive at as+ 07-

1/2 2 3 _(i~l+cosT-)2q~+sin7-_T_(lYi+cosT-)1/22 q~+ sin 7- q~+ sin ~- cos 7to cos 7o)2 + 3o23 +

2 I/2 [ cos

[ _ + ( / 4 +_ cos

]1/2(0/] t

-Y-sin

)

-Y-(/4-2cos

7`~1/2(0/~ ,

~ a7- + 2 s i n T -

]]

JJ

(A.9)

Solving (A.5) and (A.6) for £ x = T 1 - I - 2 1 / 2 ( / ~ + c o s 7 - ) 1/2@+ _ _ o)

co.ss_7" 2 2o)2 q0+

(A.10)

and inserting this expression into the last term of (A.9) yields: aS+ 07-

l(O/~ ) 1/2( -'/2sin7- 2 o) ~-~ ~+ sin 7- ~+_+ 2 / t +_ cos 7-) ~5-w2~ +

sin 7-c°s7- 3 6o)3 q~+.

(A.11)

Since £ = + 1 for q~+=~r and 2 = - 1 for q~ = 2 w we deduce f r o m e q . ( A . 1 0 ) (/~+C0S7-)1/2_

o) (2+'rr2_cos7-) 21/2,rr 2o) 2 '

(A.12)

339

H.P. Breuer et al. /Strongly driven anharmonic oscillators

and ( / ~ __ COS ,T) 1/2 __

(2

0)__

2~/2,rr

w2-c°~sr ) 2m 2 •

(A.13)

Thus, we have from (A.11) 1 (0/t ) sinr 2 wsinrcosr 2 0~- - o) ~-r - sin r ~ + + ~ro) + + 4093 qo+

OS+

sinrcosr 3 6o93--- q~+

(A.14)

and OS_ 1(0/~ 0~--~ ~-+sinr sin r cos 6o93

)

sinr, (~¢ - w ) - ~ - ( ~ ¢

--Tl')2+

w sinr c o s t 4o93

(~

--Tl') 2

(A.15)

(~ --"iT) 3.

As a final step we eliminate OI-~I/Or from these expressions. In order to obtain an equation for 0/t/Or we differentiate eq. (A.1) with respect to r 0 = "rr 0"r0J_ 23/2 ] c~Trsin ~- [(/_~3 + COS ~.)3/2 + C-~-r ( I q +

r,

~ Or

(/_~_ COS r)3/2 sin

r

-- (/q-- cos r) '/2 -0H ~r

+ sin r

and use eqs. (A.12) and (A.13) to get 1 0/q o) 07

sinr(4w_j) COS ~" ~

"

(A.16)

If, on the other hand, eqs. (A.12) and (A.13) are inserted into eq. (A.1) one arrives at j = - ~4os + w

-'Tr 2-COS2T 12w 3

(A.17)

The last two equations imply 1 0/4 oJ Or

av2 sin r cos r 12o93

(A.18)

We thus obtain OS+ OS0r

sinr( w) w c o s r ) - ~-o2 1 - ( 2 ¢ + ~ (q~+--rr)¢+,

_

sin, rrwr ( 1 + [2(q~_- w) - 7] ~~cos~) (~p_- 2"rr)(q~_- w).

(A.19) (A.20)

H.P. Breuer et al. /Strongly drit,en anharmonic oscillators

340

According to its definition the phase variable ¢ varies within the interval [0, 2-rr]. In order to obtain a simple expression for aS/at valid for both the upper and the lower branch we replace ¢ by the new phase variable ¢ - w ~ [ - r r , rr]. Now aS/at can be written as a single-valued, smooth function on the phase plane {(J, ¢)}: aS ar-

sin r (1 +[21¢1 ~ ~r cos r ) v~ - w l 1-i-f~w2 ¢ ( 1 ¢ 1 - ~ r ) .

(A.21)

Eq. (A.17) gives rise to the following expansion of w: co(J, r) = (~r2/4)J [1 - ( B/J )4

__ 3 ( B / J ) 8

-...

],

(A.22)

where B = (4/'rr2)[(+r4/48) cos2r] ,/4

(A.23)

Because of the rapid convergence of this expansion if suffices to take into account only the first two terms as long as J >_ 2:

w~('rr2j/4)[1-(B/j)4].

(A.24)

We then have

KCJ, O,r)

=H(J)

+/2gl(J,t/,,r )

where sinr /2(1+[21¢1_ ,1vcosr) 7rr ~ "rrl 1 ~ - 2 ¢(l¢l-Tr)

aS /2Sl=/2ar-

(A.25)

and

r=/24,

¢=g,-,~g'sin2r,

(A.26) (A.2V)

& ¢ ' = 1 / 3 w r 2 / 2 J 3.

The Fourier decomposition of the periodic function P ( ¢ ) =

P(¢) -

8

w2

(1/wr)l¢ I¢ -

¢ yields

1

~

(A.28)

~ sin(/c),

/ = 1 , 3 , 5 ....

and hence sin/2+: P ( ¢ ) -

4 +rr2

~

/ = 1,3,5 .... ; k~zY

1 [ + ~ ( / ~ ) _ ~+l(lS~/)]cos[/0 _ (2k + 1)/2+:] l~

(A.29)

(]k denotes the kth Bessel function of the first kind). If we now disregard the second term in brackets in

341

H.P. Breuer et al. /Strongly driL'en anharmonic oscillators

eq. (A.25) and approximate o) by

K(J,O, s2:r) = H ( J )

"rr2J/4 in the

denominator of

S2/w, we

finally arrive at

/ 2 ]4~Q

+ ~-I

Vt, k(J ) cos[/~0 - (2k + 1)S2~:],

7-

(A.30)

l,k

where

v,,k(J) :=

[L(tJ)

(A.31)

In deriving eq. (3.31) from this expression we have approximated the Bessel functions according to

ira(x)

-~

(½x)m/m!.

It is easy to show that for J > 2 the condition m >> x necessary for this approximation is fulfilled.

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[11] D. Bensimon and I.p. Kadanoff, Physica D 13 (1984) 82-89. [12] A. Bahri and H. Berestycki, Commun. Pure Appl. Math. 37 (1984) 403; I. Ekeland, Convexity Methods in Hamiltonian Mechanics (Springer, Berlin, 1990). [13] V.I. Arnol'd, Mathematical Methods of Classical Mechanics (GTM/Springer, New York/Heidelberg, Berlin, 1980). [14] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (AMS/Springer, New York/Heidelberg, Berlin, 1983). [15] W.A. Lin and L.E. Reichl, Physica D 19 (1986) 145-152. [16] J. Heagy and J.M. Yuan, Phys. Rev. A 41 (1990) 571. [17] A.M. Liebermann and A.J. Lichtenberg, Phys. Rev. A 5 (1972) 1852. [18] H.P. Breuer, Dissertation, Bonn-lR-90-14 (1990).