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Two-frequency fermi mapping
Phys~:a 5D (1982) 243-257. North-HGlland Publishing Company
TWO-FREQUENCY FERMI MAPPING* J. E. HOWARD, A. J. LICHTENBERG and M. A. LIEBERMAN Department of Electrical Engineering and Computer Sciences and lhe Eleclronics Research Laboratory, Universily of Calilornia, Berkeley, Califormia94720, USA Received 2 December 1981
We examine the properties of the Fermi mapping with two driving terms, ..+, = u. + sin s(~. - + ~ sin r+,,
~.+I = 6. +
~'~
4+rM
(r ÷ s)u.,,'
where r and s are coprime integers,/L is the amplitude ratio of the driving terms, and M is a constaqt. Linear stability criteria and bifurcation thresholds are derived and confirmed numerically. Global stability limits are obtained by generalizing the cciterion derived for the loss of stability of the last KAM surface between two neighboring island chains to the case of unequal size islands. The analytic estimates are compared with the numericai calculations and found to be in good agreement for ~ - I. The results show that using two frequencies gives considerable dostruction of local stability and an approximately twofold increase in energy (u 2) for the position of the lowest KAM barrier to stochastic heating.
1. Introduction
In this paper we study the effect of two driving frequencies on the stability properties of the Fermi mapping [1, 2]. This two-dimensional area-preserving mapping represents the motion of a ligh* ball colliding elastically with a fixed and an oscillating wall, as depicted in fig. 1. The case of a monochromatic sinusoidal wall velocity has been treated previously [3-6] and its stability properties are now well known. In such systems there exists a lowest (in velocity) invariant phase space curve that separates regions of primarily regular motion from regions of connected stochasticity. A particle initially below such a Kolmogorov-Arnold-Moser (KAM) curve [7,8] cannot penetrate to higher velo":*;""
I~,1 l,,llg, ~I.
As
,~,', :gl il g
kILIII~ ,= mk,,.,,,,,~ 3 l l U ' ~ t ll'll
*l,llglg;, k,.
,~,,IA=,:A,,.
4ll,~ll'Ikllll,,l~+Jrllt
,.~4P a
17l
second wall frequency can significantly increase the velocity at which the first adiabatic barrier appears. *Research sponsored by the Office of Naval Research under Contract N00014.7~,C-0674 and by the Department of Energy Contract DE-ATOE-76ET53059.
0167-2789/82/0000-0000/$02.75 © 1982 North-Holland
Adiabatic limits can be of practical importance in the cyclotron heating of plasmas in magnetic mirror devices [9]. The mapping method has been applied to electron cyclotron resonance heating (ECRH) by Jaeger et al. [10] and by Lieberman and Lichtenberg [11]. Ion ~'!clotron resonance heating (ICRH) has been similarly modelled by Howard and Kesner [12]. These treatments ,-equire that the particle-field interaction be sufficiently localized in order to justify an impulsive mapping model [13]. The theoretical results for ECRH have been verified experimentally for collisionless plasmas [ 14]. Recently it has been suggested that ECRH might be enhanced by using two or more closely-spaced frequencies, with the total RF power held fixed. Recent experiments by Lazar et u,. _I [15] ~ . .v. ., ., .,
.u substantiate t h i s hope, a|-
though the precise reasons for the (approximately factor of two) increase in particle energy are not yet clear. Me'~yuk and Lee [16] have studied the effect of introducing a finite bandwidth of the driving frequency and found that all KAM curves could be effectively destroyed with a bandwidth equal to the orbital
244
J. E. Howard el ai. I Two.frequency Fermi mapping
corresponding to a twofold increase in kinetic energy, in qualitative agreement with experimental results on ECRH [15]. l.J
!
2. Mapping equations I
l f~eO
ql.
v
I wo~i
moving
woll
Fig. I. The Fermi model. A light ball of velocity u makes elastic collisions with a fixed wall and an oscillating wall, ,eparateu b.~ the average distance L.
bounce frequency. However, this technique is difficult to realize experimentally. Here we model the rather comprex ECRH dynamics with the analogous, but far simpler Fermi model, as has previously been done for single frequency heating [5]. In order to produce a periodic mapping., we choose the two frequencies, co, and ~o,. in the ratio of some rational number cod,o, = rls. where r and s are c,~t.rime integers. Taking r and s large, we can closely approximate any irrational number, or, ~ith r = s + 1. obtain closely spaced frequencies. Corresponding to each frequency we find sets of island chains which can overlap, thereby destroying nonlinear stability and increasing the adiat "';~.,....harrier.. In section 2 we derive the basic mapping equations, and in section 3 examine the linear stability of the island chains formed in the phase plane. Bifur;:ation thresholds follow naturally from the linear stability conditions. The principal results are contained in section 4, where the nonlinear stability criterion [I7] is extended to the case ot :mequal size islands. Numerical results are presenteo ,~L;ch substantiate both the linear and nonlinear stability criteria, and the bifurcation thresholds. Tyg,ically we find about a 50% increase in the adia'oatic barrier velocity,
The Fermi-Ulam model [1, 2] consists of a light ball bouncing between a fixed and an oscillating wall, as illustrated in fig. 1. In the simplified model [5], the oscillating wall transfers momentum to the ball instantaneously, without changing its position. The simplified model retains the essential features of the exact model [3, 4] with the advantage of yielding an explicit area-preserving mapping. Here we consider a momentum transfer having two frequency components normalized to maintain the sum of the amplitudes squared constant. (For uncorrelated sources, this gives coustant input power.) The momentum change per collision is then v,,~,- t,, = V, sin ~o,t, + V, sin(0~,t, + 8),
(1)
where V:, + V"~= V:
=
c(,nstant.
The transit time between collisions with the moving wall is
t.+= -
=
t.
2L
~ ,
Dn+l
(2)
where L is the wall separation and (O r
=-
(O~
r S
=
o.
(3)
""
with r and s coprime integers. The phase ~ does not have an appreciable effect on global stability and will be set equal to zero in what follows. Defining the phase & = ,o,t/s and amplitude ratio ~ = %/V,, (I) and (2) can be written in terms
J. E. Howard et al. I Two.frequency Fermi mapping
of the scaled velocity u = v / V as
u.÷,=]u.+sins'~.+~sinr'l',I V ' l + ~,:
'
4~'M 6.÷, = 6. + (r + s)u.+,'
(4)
(5)
where the absolute value has been introduced in (4) to account for negative velocities. In deriving (5) we first symmetrized (2) in r and s by means of the identity
to,
-r
.% =
S
=
203 r+s'
(6)
where 03 = ~(~,, + ,o,)
(7)
and then introduced the parameter M ~ L031~r.
245
which all period-one primary fixed points are unstable, and us, the adiabatic barrier, which bounds the region of connected stochasticity. For ~t = 0 there is a family of s-fold isiand chains at u = 50, 33.33, 25, etc., with smaller chains of secondary islands at the harmonics of tos. When /~ > 0 additional island chains associated with to, appear at u = 37.5, 30, etc. A secondary chain of five islands associated with to, + to, can be clearly seen at u ~ 41 w h e n / z = 1.0. The linear stability limit, u~., is seen to increase slowly with ~t, while us makes large jumps at particular values of/z. W h e n / z = I, the chains of primary islands at u = 50 and 25 have bifurcated from two to three islands, owing to the increasing influence of the co, driving term in (1). Theoretical expressions for the dependence of ut. and us on p. will be derived in sections 3 and 4 and compared with numerical mappings in section 4. Bifurcations are discussed briefly in section 3 and in more detail in ref. 20.
(8)
Eqs. (4) and (5) constitute the desired normalized two-frequency mapping, fully symmetric in the integers r and s. It is straightforward to generalize them to include an arbitrary number of frequency components. In working with this mapping we shall fix L and 03 while varying r, s and/~. The limits/z = 0 and oo yield simple Fermi-sine maps with known stability properties. It is the region /z ~ !, for which no nonlinear stability theory exists, that is of primary interest here. Although one usually requires r ~- s in cyclotron heating applications, we are not limited by this restriction for the Fermi mapping. Fig. 2 shows a series of numerical surface~ of section (u vs. cb) for rls = 312 and M = 125, and several values of ~. As usual, there are three distinct regions; a stochastic region at low velocities, an inter.mediate region with islands embedded in a stochastic sea, and a primarily regular region at high velocities. Two critical velocities delimit these three zones: UL below
3. Linear stability and bifurcation of period one fixed points
From (4) and (5), the period one ( 6 ~ b + 2-irk) fixed points common to both frequencies are given by _
Mc~
Uok- k '
k=
1,2,3 .....
sin s60 + ~ sin r60 = O,
(9) (lO)
where for convenience we have defined 2M
M,~ = ~ .
(ll)
r+s
When ~ = 0 there exists an additional set of fixed points given bv s6 ~ s6 + 2wk~, which includes the common set (9),
UOs =
sMe~ ks '
k~ = I, 2, 3, ....
(12)
al. I Two-lrequency Fermi
J. E. H o w a r d el
24~
rls=3/2
mapping
M=!25
#.l. = 0
6£
k I
k s k~ :~
5¢
3-
I.I
41 4 3-
5-
3~ ~--U
2
4
5- 7 8 5 4 5
6 7 8 iO
B
6-
2~
9 12~5
~-.-- UL
I~
i v
-Tr
0
~r
(a)
G O -,
-----:i
:. . . . . .
- ~~.~.~.~..
. . . . . . . . . . . .
~.' •
--~--.",
i
......
~
". . . . . . . . . .
' ....
•
'~:,~,.~..,.~-.........~-_~
,,'~_'. ~
'
i
-~T
..~;.,'"
'i
0
(b)
.
.
.
.
.
.
.
~..
..,,.~_-iiii.i~"~ . . . . . . . . . . .
• ,.,i~ ' ! i l < ~ ~ .
;~-
~i
,-r
¸247¸
i
i
~P--u L
-Tr
O
7/"
(c)
/.I. -I.O
I
4
-'n"
0
Tr
(d) Fig. 2. Surfaces of section for rls = 312, M = 125 and four values of ~. The locations of several comn~on, r-fold and s-fo~d island chains are given in the first plot. The linear stability limit UL and adiabatic barrier us are indicated in each case.
J. E. Howard el al. ! Two-lrequo.ncy Fermi mapping
248
These fixed points have period P. where P / Q is s/k, reduced to lowest terms. Similarly, w h e n ~t --+ :r. setting r
-"
~r
k, = !, "L 3,
""
"
(13) ....
which also includes (9) as a subset. At intermediate values of/.t the r- and s-fold islands at d~0 = 0 and 7;- are u n p e r t u r b e d , but the other fixed points move according to (10), which locates the angular coordinate ~bo independent of uo. In general, (10) must be solved numerically for cho(/a, r, s). 60 = O, +_ rr are obvious but not exhaustive solutions. We now investigate cases for which (10) may be solved analytically. (a) ~t ~ I. Regarding the /,t term as a perturbation, the unperturbed solut|ons of (10) are &0.=0 +.rr ___2,+r .+ s - - I ) s" s . . . . . . (---~--
zr, _.+rr,
(14)
k!,5. u ) = sin s6 + u sin r ~
&,,--
~ sin r~So s cos stho + k,r cos r~b4(
i
)
t'+!
_"+ ~h,,::::O,
......
-
_
_
.
.
.
.
x.+ I = T • x,,
'20) p r i m a r y fixed
(2 ! )
The stability in the neighborhood of x0 is then determined by the lint:arized m a p p i n g , (16)
Ax,+ ~= ~ A x , ,
(22)
where ~ is the Jacobian matrix of T. The motion near xo is stable if and only if ITr ~el < 2.
(23)
From (4) and (5) we find (17) Tr ,~ = 2 -
4.rrM
(p + I)UoV
which yields tile two families of solutions,
x (cos Sd~o I- g cos rd~o). ± 2w _*~4 n , ,+,"+",+= O, .~'+ . . . . .S . . . .r-t-S ...........
(19)
There are a total of 1 +[~(r + s)] ~b. roots, and [ ~ ( r - s + I)] d,t, roots, where [ ] d e n o t e s the integer part. In addition, there are several cases where (10) is e x a c t l y solvable for all p., namely r - 5 . We shall e n c o u n t e r these explicit solutions in the c o n t e x t of linear stability and bifurcation. The linear stability theory of fixed points of arbitrary order is given in ref. 5. H e r e we calculate the stability limits of the p r i m a r y periodone fixed points of the t w o - f r e q u e n c y Fermi mapping. As we shall see, these limits enable us to delineate the bifurcations the p r i m a r y island chains must m a k e with increasing tz as they change from s-fold to r-fold s y m m e t r y . Writing (4) and (5) in matrix f o r m ,
xo = T • xo.
A similar application of N e w t o n ' s method is of course possible in the limit p.--, ~c. (b) la = !. For equal a m p l i t u d e s (10) transforms |o sin r 2+ ~ +,,cos
r-
(15)
then gives ,+-8= +bo- FJFt'+, or th(~t) ~
+-3rr s' r - s '
__.zr -
with x - - ( u , cb), the period-one points are given by
a total of + _s + ! rot, Is in [-rr, rr]. Writing
F( +b,,. ~ ) -+ F'(~Su. ~t ) ( + - ~bo) = O,
ebb =
(24)
(18)
If 0 < cos S4+o+ ~ cos r4>o< I + tip, a sufficient
J. E. Howard et al. ! Two-frequency Fermi mapping stability condition is Tr .T > - 2 , uo2> u~ = V ~
or
,"
(25)
It should b e noted th~ stable secondary fixed points, or higher period primary fixed points may exist below UL. Also note that UL has m a x i m u m at ~ = p. For T r . ~ > 2, a sufficient condition for instability is clearly cos S4,o+ t~P cos r4,o < O,
(26)
which is to be solved for ~ and 4,0 simultaneously with (10). Since (26) is independent of u0, we see that if u0> ut., a stability threshold exists when cos sOo + UP cos r4,o = O,
(27)
sin s4,o + tx sin r4,o = O.
(10)
249
for 4,0 = 0 and we conclude that this point never bifurcates for u > u L. For 4,0- m however, (27) yields (-1)" + ~ * p ( - l ) ' =0, which has the solution t~*= lip, provided that r - s is odd. In particular, if r - s 1, this is the only bifurcation, giving birth to two new fixed points, exactly the required number to change from s-fold to (s + l)-fold symmetry. This bifurcation is clearly seen in fig. 2d, Another broad class of bifurcations is revealed by setting 4,0 = -+¢d2 in (10) and (27). Setting r=-s+2m, m = 1 , 2 , 3 , . . . . (27) is satisfied, since r and s must be odd in order to be coprime. Eq. (10) becomes ! +/x cos wm = 0, which is satisfied by Ix* = 1 and m odd. Thus, a pair of tangent bifurcations occurs at ¢b* = _-t_rd2 when t ~ * = l for r = s + 2 , s + 6 , . . . . Again, when r - s = 2 , these are the only Vossible bifurcations. For s < r _<5, complete algebraic solutions of (10) and (27) may be obtained by means of the factorization
These equations admit a simple geometrical interpretation if we observe that they are of the form F(~,4,0) = aF/04,o=O. As this is just the condition for a double root of (10), it follows that (10) and (27) together determine the bifurcations of the common fixed points as the mapping changes (with increasing ~) from s-fold to r-fold symmetry. That is, for fixed r and s, one or more common fixed points 4,* bifurcate at critical values ~*. In general there are three types of bifurcation: the pitchfork bifurcation (P) in which a stable fixed point destabilizes and gives birth to two nearby stable fixed points; the anti-pitchfork bifurcation(P') in which an unstable fixed
The critical values of ~ are then given by the vanishing of the discriminant of (29), except ~n a few degenerate cases. The bifurcation ang!e 4,* usually follows b : inspection or, if necessary, by numerical solution of (10). The calculated values of 4,~ and ~* and the bifurcation type are listed in table I. Formal solutions of (29) are
l.~UiliS,
~lV~li
al.l~Uillil:.~,.,9
Oilt.i
i~'191dlg:3
l l J I I Ikll
LWUI
Ulll~itillJll~
fixed points; and lastly, the tangent bifurcation (T), in which a stable-unstable hair of fixed points is created. Although all this information is in principle contained ~n (10) alone, (27) is often useful in practice. For example, (10) yields the immediate fixed points ,ko = 0, 11".Eq. (27) gives I +/~p = 0
sin nd, = sin 4,P, ,(cos ~),
(28)
where P, l(cos 6) is a polynomial of degree n - 1. Thus, (10) becomes
P, 1(cos 4,)+ u.P, 1(cos 6)=0.
!!1 I ' l l .
(29)
ZU.
Additional insight into the structure of the bifurcations may be gleaned from ttle averaged Hamiltonian (derived in the text section) /# = ~C;u 2 + v ( 4 , ) ,
(30)
where G is a constant and, for a common island
250
3.
Howard et ai. I Two-frequency F e r n i mapping
E
r/s:3/2
Table I
rls
~b~
ix*
Type
2/1 371 312 4/!
180 ° 90 ° 180 ° 180 ° 65.91 ° 180 ° 52.24°'1, 127.76 = J 180 ° 55.80 ° 90 o 180 °
1/2 ! 213 I/4
P'
413
3t' 5/2 5/3 5!4 (n ~- I)ln (n + 21!n
T
415
P P' T P+ T T P T T P
180 °
nl(n + I)
P o r P'
90 °
I
T
3V618 314 ,t/5 2/5 0.9414
I
-
2 -'n"
o
~ 0
--/'N t,
-~
-rr
(el ~r
.
I
3"rr
ju.: I
7,- ,.,-t
.
.
,I
0
Tr
2
~
O
27r
rr
.
.
i
2~
,
I
37r
4,
(d) 2rr
3Tr
F:5
r/s:3/2 V Ok_:~/,~- - - / ' ' k ~ : - ~ ' ~ : :'7"~--'7'~"----~
21 -
t -~
,,t
0
i
rr
.t
2~r
/
3."
to)
u ~ I.
I. .
0
.
.
.
1.
....
~
i .
.
2u
.
.
.
.
3rr
H.=O
Fig. 3. Theoretical level c u r v e , and effective potential for r/s = 3/2 and four values of ,. Note the p i t c h f o r k bifurcation of the elliptic fixed po!,,t at ~b = ~. w h e n tt = 2/3, and the corresponding change in ' , ' ( ~
|
-v
...........
~
.......
0
~
........
u
chain,
~ ..................... _j
2rr
3Tr
v(~,) =
i- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b)
~ r I.
- ""
.
.
.
.
.
L. . . . . . . . . . .
0
1 ........................
~
l
....................
2Tr
3~r
0
~*:2/3
, (,
Vl-g~
c o s s,~ + r c o s r,~
)
(31)
is the effective potential. At a pitchfork or antipitchfork bifurcation, V ' - V " - V " - O, while V'# 0 for a tangent bifurcation. Thus, tangent bifurcations arise from inflection points in V(~), while pitchfork h i f l l r o ~ t ; ~ n © e t , ~ m f~.e,m tk:~.,t order critical points. More exotic bifurcations may be produced from potentials having higher order singular points [19]. A set of calculated levels curves and their associated effective potential~ are shown in fig. 3 for the case r l s = 3/2. Note the pitchfork bifurcation at 0 = 180~ when ~t - 2/3, as predicted.
I. E. H o w a r d e l a|. /Two.[requencyFermimapping
251
where O= s~ and 4.1. Derivation of averaged equations
Uo~ - sM,~ik~,
Oo, = 2¢tk:~,
(33)
terms o f n as locally approximates the Fermi map The destruction of KAM curves may be a,,t.erstood intuitively as arising from the interaction of neighboring island chains. Each island possesses a stochastic layer about its separatrix, so that for sufficiently large island widths these layers overlap, permitting diffusion over the region of velocity space occupied by the two islands. The island width is easily calculated from an approximate Hamiltonian in which only the dominant resonant term is retained, with all other terms averaged out. Analy:ic and numerical calculations show that the last KAM curve lying between two contiguous island~ is destroyed when the sum of the island h~df-widths equals approximately two-thirds the distance between the island centers measured in action space (velocity in our problem). This criterion has been shown to be quite sharp for equalwidth island [18] and works reasonably well if the widths are not too disparate [5]. Escande and Doveil [21] have derived a criterion valid for two islands with unequal widths, but their method is difficult to apply owing to the influence of neighboring large islands. We m©r©[or© © m p m y here m e simpler --twothirds rule," which agrees well with numerical calculations for comparable-sized islands. We begin by defining slowly changing variables near the s-fold elliptic fixed points, u, = u - uo~,
O~ ffi 0 - 0o,,
(32)
dd, = sin O + ~ sin(eOis) ~ 8(n - m), d. .o_. dO, 2wsM~ dn = u0~ u.
(34)
(35)
where (5) has been expanded about Uo~. Fourier-expanding the train of delta functions, 8(n-m)= m~-x
~ e 2'~i'",
(36)
I=-x
and averaging (34) over n, we find that the first term survives for I - -+k, while the second term is slowly varying only if ! ~ +_rkJs, which is only true near the common fixed points, not considered here. Thus, the averaged form of (34; 's simply du__z~= sin O, dn X/1--~'"
(37)
Eqs. (35) and (37) are derivable from the averaged Hamiltonian ~ , = 21rsM~ d~ cos 0"-~, uo., 2 VI--+~"
(38)
which yields the island amplitude A = (l +
2uL \~rSMe~]
•
(39)
Similarly, expanding u and 0 about an r-fold
J. E. Howard el at. I Two-~nrquency Fermi mapping
252 chain gives ,~
.x, = \ / ~ 1
+ t~
,,~:
,
I,:2
.
(40)
Eqs. (39) and (40) are generally valid only for 0t <~ I and ~ ~ 1, respectively, when the fixed r~oints are well approximated by (10). Neverthe',¢ss. a useful overlap criterion may be obtained r-v focusing on the 0o, = 0 (rood 2,-r) fixed point, which i~ unperturbed for all ~ and elliptic for u ~ u~. The widths A, and A, are valid for these central islands for arbitrary ~,. The motion near the common fixed points is obtained in the same manner by keeping the terms in the 8-functior, satisfying I = +--k, and I = -'-rk,/s, ~dmultaneously: d,i _- sin s4; + g sin rd, dn \I1 + g
(41)
separatrices, the o u t e r m o s t of which first overlaps with the separatrices of neighboring island chains. In terms of the angle d~ the central island width is ~iven by
A:= G~/~4
[1` s i n "4\( s26 ¢/ 5) + Ers i)n " ( - ~s ) ] "
Taking ris = 3/2 as an example we find A2 = 81.t 3 + 18/.t2 ÷ (1 + 4 ~ " ) 3/: -- I .
.
.
12G~"X/1-'~"
.
.
.
.
.
.
(46)
.
'
which varies by only 30% as t.t varies from 0 to ~c and by only 2% in the range 0 < ~ < I. In general, for r > s, A(/x) is either monotonic decreasing or has a gentle maximum near Ix = 0.5. In all cases (45) gives m
A(0) = x f r dn =
ac ---i
uii
Integrating thc~,~ ,,quations then gives the averaged Hamiltonian for the c o m m o n fixed points
!-
,
(,
)
(47,
;
Thus, for closely spaced frequencies the variation of A(t.t~ is entirely negligible so that wP may take 4,..
.... ~cossd~+ ....... t:/ . . ._C;~i \ / t ~:-......... ~t ,~ r ~osr~b , (43)
A ~ A(0) = k/G-~s-
',,,'here
In extreme cases and use (45); for from (18). More may be found in
2 rr,',Ik.~ (; ...... ~. .... .
(44)
Calculation of the island width in this case is complicated by the sequence of bifurcations as ~t increases from 0 to x. As discussed in section 3. this calculation can be carried out explicitly ,,nlv for the particular values of r and s listed in l:thle I. A v,idth can be calculated explicitly for the central island at cb, = O, which is elliptic for all g. F,~r ~t ~ 1 a single separatrix crosses the , ::: u,, line at d~ = ± rr/s (mod 2rr), but at larger g: there i~ in general a complex sy~,,em of nested
(48) one must calculate the angle cb~ # = 1 one may easily obtain d~,. information on island widths ref. 20.
4.2. O v e r l a p c r i t e r i o n Having outlined a procedure for calculating approximate island widths, we now turn to the choice of an appropriate overlap criterion for estimating the loss of globai stability with increasing/~. A useful rule of thumb is the "twothirds rule" [18],
a, + a: >
luo,- Uo:l,
(49)
$.:E. l.lo,ward:et:aLI Two,frequency, F~i mapping where-uot and uo2 a r e the two resonance positions and A t and &2 are the r e s o n a n c e halfwidths & , As or &, as the case may be. In deriving (49) it is implicitly assumed that the islands are of comparable size. s o t h a t the secon-
agreement for p. = I b u t only qualitative predictions for ~ ~ 1 or tt ~' 1. In the latter cases, although (49) still predicts KAM destruction quite accurately, the precise value of ;t at which this occurs is predicted less accurately, ~t(or N -t) being a perturbation. We consider three controlled numerical experiments in examining the effect of two frequencies on stochasticity. In the first, M is held fixed, while ~ is varied from 0 to oo. As ~t is "turned on," the r-fold islands grow rapidly and interact with s-fold and common islands, large jumps in the adiabatic barrier us generally occurring at fairly small values of ~t. In the second numerical experiment, we hold ~ fixed and vary M. Here (49) will be seen to give accurate predictions of the jump in the adiabatic barrier. In the third experiment we hold M and tt fixed and vary the frequency ratio r/s. On the several possible combinations of overlapping r-fold, s-fold and common islands, we choose for illustration only two. For overlap of the k, and ks + 1 fold chains, using (9), (11), and (48) in (49) gives M < M*, with M* = ~t-~
(2ks + I ) : V l - ' ~ ,
(501
which is equivalent to the single frequency overlap criteria for p ,~ 1. For overlap of at', r and an s-fold chain, using (12), (13), (39) and (40) in (49) yields
9 (r+s~I
k,
l+rk~
skr
(51)
253
However, care must be exercised in using this result f o r ~ v e r y small, as overlap is then determined by neighboring s..ch,Jns and their perturbation by interspersed r-chains.
of the island chains as given by (9), (11) and (13) for several values of k, k, and ks. The k~ = 2-8 islands are clearly visible, but the ks ffi 9 chain (uo = 11.1) falls beneath the linear stability limit and has disappeared. Numerically, we find the last KAM curve at about us = 27, as indicated in the figure, in agreement with the overlap criterion (50), which predicts that KAM surfaces exist between the ks = 3 and 4 island chains, but not between the ks = 4 and 5 chains. As we increase ~t to 0.1, the adiabatic barrier increases dramatically to uB = 37. This jump is due to the intercession of the kr = 5 islands at u0, = 30, whose stochastic layer overlaps those of the ks = 3 and 4 islands at uo~ = 33.33 and 25. Applying the overlap criterion, we find that the ks = 3 and k, = 5 islands overlap for arbitrarily small ~, but that overlap of the k, = 5 and k~ = 4 islands should not occur until ~t >0.5. The underlying reason for this premature overlap is the chain of four secondary islands at u0 ~-28, which may b~. seen in fig. 2b. There is also a chain of six secondary islands at u0 ~ 27, barely visible in fig. 2b. These two secondary chains effectively bridge the gap between the kr = 5 and ks = 4 islands, so that Or""~,rlap occurs at ~ ~ 0.065. As tt is increased from 0.1 to 0.25, uB once again suddenly increases, as the k, = 4 islands at u0,= 37.5 grow and overlap with the k: = 3 islands at u0s = 33.33. According to theory this jump should occur at tt = 0.0025, whereas it does not actually occur until ,t •0.15. The underlying reason for .this discrel~ancy is the great difference in size of the two islands, the smaller being distorted with no significant in-
J. E. Howard et ai. I Two.Irequency Fermi mapping
254
50
. . . . . . . . . -r.....
-
ADIABATIC
......... ~ . . . . . . . . .
; .......
BARRIER
,.'s=3 2
L
M.:t25
UB
.q----
~,,
: 4
NUM[RICAL
DATA
~ - - ~s--3 ~.
~
~
k,:5
L ..............
.9 M'--- s e c o n d o r ~ e s
!.
-,. ~-
--
k :2
! 2,." . . . . . . . . . . 0 0 0.~
'-
02
/.,t
- -~ . . . . . . . . . . . . . . . . 0.3 0.4
Ftg. 4 A d i a b a t i c barrio; as a f u n t t i o n o f ~. f o r rls = 3/2 and .%! = !25. T h e j u m p s in uB a r e a s , , o c i a t e d w i t h o v e r l a p o f the t,,land c h a i n s i n d i c a t e d in the right-hand ,'rmrgin.
termediate island chain to mediate the overlap. To complete this series of r/s = 3/2 cases we include the m a p p i n g for vt = !. Here the twothird~, rule successfully predicts that the k = 1 ch~,,in ..~t u,, = 50 cannot o~ erlap the k, = 4 chain at 37~ (Note that the c o m m o n island chain at u , = 5~) has bifurcated at d, = 180 ° and ~hat the r-fold chain at uo, = 37.5 has u n d e r g o n e a more complex division (trifurcation) into three is!ands.) The o b s e r v e d variation of uB with Ix is shown in fig. 4. The iimbility of the two-thirds rule to predict the exact value of ~ at which two I:,and chains overlap, for small ~, it is not surprising. Since ~ enters the o v e r l a p condition in (51!. as a perturbation, ~ significant change in t.t corresponds to only a small c h a n g e in the overlap criterion itself. l-or t.,: ~ 1 we expect much better agreement, ~ hich is in fat:t roun,! numerically. To illustrate thi,,, fig. 5 presents un as a function of M for t.L = 0 and i (eq. (49) is always solvable for M). To follow a given K A M curve with changing M, we note that in the absence of overlap, the island centers, and therefore ut~, are proportional to M. H o w e v e r . the island width A-.-
•
x_. .......
t, .....
,t.....~_,.~__x_,.t._,l,~d
50
....................
tOO
H,,O
.k. ......... , _ , t . ~ _ a . _ - _ . k - - - . L
500
x=~
_
IO00
M
Fig. 5. Adiabatic barrier as a function of M for ~ = 0 and I. The solid "'zig-zag'" curves were calculated using the twothirds rule, while the circles were obtained by numerically iterating the mapping I0" times.
V ' M , so that an o v e r l a p must inevitably occur for a given island c h a i n as M is d e c r e a s e d , with the chain contint.,mg to sink into the stochastic sea as M decreases further. For t~ = 0 w e follow only the s-fold islands, while for ~ = I we m u s t examine all possible c o m b i n a t i o n s of rfold, s-fold and c o m m o n islands. The solid zigzag curves r e p r e s e n t the theoretical formula (49), with the numerical results s h o w n as individual pcint~. The dashed line indicates the average pn oportionality u, --- V ' M . The a g r e e m e n t b e t w e e n t h e o r y and n,~,merical values is seen to be quite good. We have also e x a m i n e d a , u m b e r of other cases, varying the value ol r and s to d e t e r m i n e first if the two-thirds rule for overlap holds and also to investigate the fractional j u m p in the action as r and s b e c o m e larger. Specifically, we take r = s + 1 and c o n s i d e r s = !, 2, 3, 4, . . . . For fixed M, this c o r r e s p o n d s to taking the two frequencies s u c c e s s i v e l y closer together. Of particular interest here is the case s = I, both theoretically b e c a u s e of the simplicity of the interspersed island structure, and practically b e c a u s e a second harmo.qic may be g e n e r a t e d naturally in a physical system. B e c a u s e the r. fold islands are interspersed h a l f w a y b e t w e e n the s-fold chains and reinforce the second h a r m o n i c s of the s - s y m m e t r y , we might expect
J. E. Howard et aLI Two-[requency :Fermi mapping 28 . . . . . . . . t "'
26
~'
I ....
~ ' •
For large s(lls <0.05) the r-fold and s-fold islands nearly cohlcide near us, so that we would expect no increase in us with increasing /z.
i
decreases at first with increasing s, but does not fall t o zero as e x p e c t e d , This residual change is apparently due to t h e mutual destruction of closely spaced r-fold and s-fold island chains and a small increase in island width due to the normalization. The oscillations in ~q at smaller s can be understood in terms of the resonance interspersal, the maxima occurring when the resonances are evenly spaced near the adiabatic barrier. From the fixed point conditions (12) and (13) the r-fold and s-fold islands are evenly interspersed where
ADIABATIC BARRIER r/s=2/I M=50
24
UB 22
20
.~4
18
L., , 16 0.0
0.~
I 0.2
i p. 0.3
*
secondaries
EL~_ks= 2
0.4
Fig. 6. Adiabatic barrier as a function of/~ for rls - 211 and M =50, The agreement between the theory and the numerical data is much better in this case, since the jumps occur at larger p..
particularly good agreement with the two thirds overlap criteria. This is indeed the case as seen in fig. 6, which graphs ua against/~ for rls = 2/i. The arrow at /~ = 0.39 indicates the predicted overlap of the k, = 3 and k~ = 2 island chains, in excellent agreement with the value of /z = 0.35 at which the last KAM surface between these islands disappears. The smaller jumps in the adiabatic ba, rier between these two major chains involve the interaction of secondary islands and are therefore not predicted by the first order theory. Generally the theory predicts whether, for given p, the last K A M surface between any two island chains is destroyed. To demonstrate this, we calculate the maximum change in Ua over a large ~'ange of !~, for r = s + I and fixed M. The results are shown in ,fig. 7, which plots tl,e relative change in ua,
2rM=ff = s M ~ + sMeff k, k, k~+l "
k,, = ~(2m
-
(54)
which implies s >> 1 for moderate m. If we evaluate u0, at the adiabatic barrier u = a ~ / 2 ~ - M , where Meif~ MIs and a is approximatery equal to the increase in u, clue to two O.
7 . . . . . . . . . . .
!
m': I
m'= 3
0.6 0.5 0.4 0.3
•
.-
'
j
m': 2 O.
0.301
as a function of 1Is, where us(I) ~ u,(/~ = I) WF~max.
I)s,
o.if- ---J--. .. (52)
(53)
Taking r = s + I and k, = k, + m then gives, for k , ~ 1,
1
u . ( I ) - u.(O) = us(O)
255
1
t
i
l l l J |
i
i
O.Ol
i
i
i i i i ~L...
O. i
~L
I
l ~ 1 =:J_
l.O
lls
Fig. 7. Fractional increase in u, as a function of lls for M = 500. According to theory the increase should vanish for s ~ 100. The residual change is probably due to the strong interaction of overlapping r- and s-fold island chains.
]. 17.. Howard a al. i Th'o-[reaaency Fermi mapping
256
frequencies, then (55) becomes
$
The maximum enhancement thus occurs at a set of odd integer values, m' = 2m - 1. Conversely, the s and r resonances are cemered on top of one another for even integers m ' = 2m, where we expect ~ to be a minimum, i.e.. ! 4,-r - = a 2m
s
(56)
2M"
From these results and the position of the s and r island chains from (12) and (13). we expect the last KAM curves to be destroyed at a = 1.4to give the peaks, and at a --- I. ! to give the minimum. These values of I/s are shown on fig. 7 as arrows, which indicme resonable agreement with the predictions of (55) and (56).
5. Discussion
The addition of a second driving frequency to the Fermi mapping produces a considerable enrichment of the phase plane structure. This c:m be seen, for example, by comparison of figs. 2a and 2d. A new se~ of islands appear, corn~sponding to the fixed points of the second frequency. The islands associated with the fixed points common to both frequencies become considerably more involved, with bifurcations changir~g the island topology. Stochasticity near the separatrices of the bifurcated i~;lands decreases the area occupied by local KAM curves (islands). Interactions also occur belho ]n~ct
tw,~:'n ........................
csrcl+ar ;e|,~nrle •.,~.,
,a~,a,~uo
,~¢
~t
one
.......
which destroys the KAM surfaces between islands at lower driving amplitude. The twothirds rule, which was developed to predict when the last K A M surface between neighboring equal width island chains is destroyed, is applied to the two-freque.ncy Fermi map with its islands of unequal width. It is found that the rule works well unless the islands are very different in size. Comparing the results of figs. 2a and 2d it is s e e n that the last adiabatic barrier
increases from the single-frequency value u B 27, above the k, = 4 island centered at u - 25, to us = 37, above the k, = 3 island ,:entered at u = 33.3. These results are in agreement with the two-thirds rule. Considering the Fermi acceleration model as an analog to electron cyclotron resonance heating, our results have implications for the heating problem. If the resonances are fairly evenly interspersed near the adiabatic barrier, then for a given input power it is possible to increase the energy obtained by stochastic heating by a factor of two, by employing two heating frequencies rather than one. We can also relate the requirement for interspersed resonances found for the Fermi acceleration model to ECRH. From the definition of phase slip sA~ in (5) we see that
sMe~ = Uo,
2~oa'
(57)
where o , s : ¢ru/! is the "bounce frt.qeency" corresponding to t w o mapping periods and o', is the driving frequency. Substituting Me,/Uo~ from (12) into (57) and noting that 8,0 : o ' , - ,o, : o',/s, we have for interspersed resonances
....
~ynuti~'.tu y
and higher order islands of the second symmetry, giving rise to ve-y interesting island
svructures. The addition of a second set of islands, with their own stochastic layers interpersed among the first set of islands, leads to an interaction
8¢o
= (2m - l)o'..
(58)
This result is consistent with a numerical calculation by Rognlien [22], which indicates peaks in the ECRH induced heati:.g rate for frequency separations satisfying (57). A peak in ECRH
$.E. Howard c¢ al.i Two,Srequency Femi mopping Q heating Wa s also found experimentally to occur when ~ =cob [15]. There are additional constraints that must be satisfied in an ECRH experiment, such as the
257
Systems, Ann. ef Math. 77 (Princeton Univ. Press, Princeton, N.J., 1973). [9] T, H. ~:~x, Nucl. Fusion 15 (1975) 737. [10] E.F. ~Jaeger. A.J~ Lichtenberg and M. A. Lieberman, Plasma Phys. 14 (1972) 1073. sma Phys. in a Tanngineering
L,S,..vj.
:. l i ~ o i ;
t,,t;i|~titil;IiiLIVll3
UA
UUUI
LIICUi'CUI;til
and practical importance leave considerable scope for future work on this problem.
References [1] E. Fermi, Phys. Rev. 15 (1949) 1169. [2] S. Ulam, Proc. 4th Berkeley Syrup. on Math. Stat. e~nd Prob. 3 (Univ. of Calif Press, Berkeley, 1961), p. 315. [3J G. M. Zaslavskii and B. V. Chirikov, Soviet Physics Doklady 9 (1965) 989. [4] A. Brahic, Astr. and Astrophys. 12 (1971) 98. [5] M. A. Lieberman and A. J. Lichtenberg, Phy.~. Rev. A5 (1972) 1852. [6] A. J. Lichtenber8, M. A. Lieberman and R. H. Cohen, Physica ID (1980~ 291. [7] V. I. Arnold, Russian Math. ~urvey 18 (1963) 9. [8] J. Moser, Stable and Random Motions in Dynamical
,ieberman. Plasma Phys. 17 (~975) 679. [15l B. H. Quon et al., Bull. Am. Phys. Soc. 26 (1981) 893. [16] C, R. Menyuk and Y. C. Lee, Phys. Fluids 23 (1980) 2225. [171 B. V. Chirikov, Physics Reports 52 (1979) 263 [lS] A. J. Lichtenberg, in Intrinsic Stochasticity in Plasmas, C. Laval and D. Gresillon, eds. (Editions de Physique, Orsay, 1979), p. 13. [19] I. Stewart, Physica D 2 (1981) 245. [20] J. E. Howard, M. A. Lieberman and A. J. Lichtenberg, Bifurcations of the Two-Frequency Fermi Map, UCB/ERL Memorandum M81/56 (1981), University of California, Electronics Research Lab. [21] D, F. Escande and F. Doveil, Physics Lett. 83A (1981) 307. [22] T. D. Rognlien, Bull. Am. Phys. Soc. 26 (1981) 1035. [23] J. L. Tennyson, M. A. Lieberman and A. J. Lichter~berg, AIP Conf, Proc. No. 57, Nonlinear Dynamics avd the Beam-Beam Interaction, M. Month and J. C. Herrera, eds., (1979) 272.
TWO-FREQUENCY FERMI MAPPING* J. E. HOWARD, A. J. LICHTENBERG and M. A. LIEBERMAN Department of Electrical Engineering and Computer Sciences and lhe Eleclronics Research Laboratory, Universily of Calilornia, Berkeley, Califormia94720, USA Received 2 December 1981
We examine the properties of the Fermi mapping with two driving terms, ..+, = u. + sin s(~. - + ~ sin r+,,
~.+I = 6. +
~'~
4+rM
(r ÷ s)u.,,'
where r and s are coprime integers,/L is the amplitude ratio of the driving terms, and M is a constaqt. Linear stability criteria and bifurcation thresholds are derived and confirmed numerically. Global stability limits are obtained by generalizing the cciterion derived for the loss of stability of the last KAM surface between two neighboring island chains to the case of unequal size islands. The analytic estimates are compared with the numericai calculations and found to be in good agreement for ~ - I. The results show that using two frequencies gives considerable dostruction of local stability and an approximately twofold increase in energy (u 2) for the position of the lowest KAM barrier to stochastic heating.
1. Introduction
In this paper we study the effect of two driving frequencies on the stability properties of the Fermi mapping [1, 2]. This two-dimensional area-preserving mapping represents the motion of a ligh* ball colliding elastically with a fixed and an oscillating wall, as depicted in fig. 1. The case of a monochromatic sinusoidal wall velocity has been treated previously [3-6] and its stability properties are now well known. In such systems there exists a lowest (in velocity) invariant phase space curve that separates regions of primarily regular motion from regions of connected stochasticity. A particle initially below such a Kolmogorov-Arnold-Moser (KAM) curve [7,8] cannot penetrate to higher velo":*;""
I~,1 l,,llg, ~I.
As
,~,', :gl il g
kILIII~ ,= mk,,.,,,,,~ 3 l l U ' ~ t ll'll
*l,llglg;, k,.
,~,,IA=,:A,,.
4ll,~ll'Ikllll,,l~+Jrllt
,.~4P a
17l
second wall frequency can significantly increase the velocity at which the first adiabatic barrier appears. *Research sponsored by the Office of Naval Research under Contract N00014.7~,C-0674 and by the Department of Energy Contract DE-ATOE-76ET53059.
0167-2789/82/0000-0000/$02.75 © 1982 North-Holland
Adiabatic limits can be of practical importance in the cyclotron heating of plasmas in magnetic mirror devices [9]. The mapping method has been applied to electron cyclotron resonance heating (ECRH) by Jaeger et al. [10] and by Lieberman and Lichtenberg [11]. Ion ~'!clotron resonance heating (ICRH) has been similarly modelled by Howard and Kesner [12]. These treatments ,-equire that the particle-field interaction be sufficiently localized in order to justify an impulsive mapping model [13]. The theoretical results for ECRH have been verified experimentally for collisionless plasmas [ 14]. Recently it has been suggested that ECRH might be enhanced by using two or more closely-spaced frequencies, with the total RF power held fixed. Recent experiments by Lazar et u,. _I [15] ~ . .v. ., ., .,
.u substantiate t h i s hope, a|-
though the precise reasons for the (approximately factor of two) increase in particle energy are not yet clear. Me'~yuk and Lee [16] have studied the effect of introducing a finite bandwidth of the driving frequency and found that all KAM curves could be effectively destroyed with a bandwidth equal to the orbital
244
J. E. Howard el ai. I Two.frequency Fermi mapping
corresponding to a twofold increase in kinetic energy, in qualitative agreement with experimental results on ECRH [15]. l.J
!
2. Mapping equations I
l f~eO
ql.
v
I wo~i
moving
woll
Fig. I. The Fermi model. A light ball of velocity u makes elastic collisions with a fixed wall and an oscillating wall, ,eparateu b.~ the average distance L.
bounce frequency. However, this technique is difficult to realize experimentally. Here we model the rather comprex ECRH dynamics with the analogous, but far simpler Fermi model, as has previously been done for single frequency heating [5]. In order to produce a periodic mapping., we choose the two frequencies, co, and ~o,. in the ratio of some rational number cod,o, = rls. where r and s are c,~t.rime integers. Taking r and s large, we can closely approximate any irrational number, or, ~ith r = s + 1. obtain closely spaced frequencies. Corresponding to each frequency we find sets of island chains which can overlap, thereby destroying nonlinear stability and increasing the adiat "';~.,....harrier.. In section 2 we derive the basic mapping equations, and in section 3 examine the linear stability of the island chains formed in the phase plane. Bifur;:ation thresholds follow naturally from the linear stability conditions. The principal results are contained in section 4, where the nonlinear stability criterion [I7] is extended to the case ot :mequal size islands. Numerical results are presenteo ,~L;ch substantiate both the linear and nonlinear stability criteria, and the bifurcation thresholds. Tyg,ically we find about a 50% increase in the adia'oatic barrier velocity,
The Fermi-Ulam model [1, 2] consists of a light ball bouncing between a fixed and an oscillating wall, as illustrated in fig. 1. In the simplified model [5], the oscillating wall transfers momentum to the ball instantaneously, without changing its position. The simplified model retains the essential features of the exact model [3, 4] with the advantage of yielding an explicit area-preserving mapping. Here we consider a momentum transfer having two frequency components normalized to maintain the sum of the amplitudes squared constant. (For uncorrelated sources, this gives coustant input power.) The momentum change per collision is then v,,~,- t,, = V, sin ~o,t, + V, sin(0~,t, + 8),
(1)
where V:, + V"~= V:
=
c(,nstant.
The transit time between collisions with the moving wall is
t.+= -
=
t.
2L
~ ,
Dn+l
(2)
where L is the wall separation and (O r
=-
(O~
r S
=
o.
(3)
""
with r and s coprime integers. The phase ~ does not have an appreciable effect on global stability and will be set equal to zero in what follows. Defining the phase & = ,o,t/s and amplitude ratio ~ = %/V,, (I) and (2) can be written in terms
J. E. Howard et al. I Two.frequency Fermi mapping
of the scaled velocity u = v / V as
u.÷,=]u.+sins'~.+~sinr'l',I V ' l + ~,:
'
4~'M 6.÷, = 6. + (r + s)u.+,'
(4)
(5)
where the absolute value has been introduced in (4) to account for negative velocities. In deriving (5) we first symmetrized (2) in r and s by means of the identity
to,
-r
.% =
S
=
203 r+s'
(6)
where 03 = ~(~,, + ,o,)
(7)
and then introduced the parameter M ~ L031~r.
245
which all period-one primary fixed points are unstable, and us, the adiabatic barrier, which bounds the region of connected stochasticity. For ~t = 0 there is a family of s-fold isiand chains at u = 50, 33.33, 25, etc., with smaller chains of secondary islands at the harmonics of tos. When /~ > 0 additional island chains associated with to, appear at u = 37.5, 30, etc. A secondary chain of five islands associated with to, + to, can be clearly seen at u ~ 41 w h e n / z = 1.0. The linear stability limit, u~., is seen to increase slowly with ~t, while us makes large jumps at particular values of/z. W h e n / z = I, the chains of primary islands at u = 50 and 25 have bifurcated from two to three islands, owing to the increasing influence of the co, driving term in (1). Theoretical expressions for the dependence of ut. and us on p. will be derived in sections 3 and 4 and compared with numerical mappings in section 4. Bifurcations are discussed briefly in section 3 and in more detail in ref. 20.
(8)
Eqs. (4) and (5) constitute the desired normalized two-frequency mapping, fully symmetric in the integers r and s. It is straightforward to generalize them to include an arbitrary number of frequency components. In working with this mapping we shall fix L and 03 while varying r, s and/~. The limits/z = 0 and oo yield simple Fermi-sine maps with known stability properties. It is the region /z ~ !, for which no nonlinear stability theory exists, that is of primary interest here. Although one usually requires r ~- s in cyclotron heating applications, we are not limited by this restriction for the Fermi mapping. Fig. 2 shows a series of numerical surface~ of section (u vs. cb) for rls = 312 and M = 125, and several values of ~. As usual, there are three distinct regions; a stochastic region at low velocities, an inter.mediate region with islands embedded in a stochastic sea, and a primarily regular region at high velocities. Two critical velocities delimit these three zones: UL below
3. Linear stability and bifurcation of period one fixed points
From (4) and (5), the period one ( 6 ~ b + 2-irk) fixed points common to both frequencies are given by _
Mc~
Uok- k '
k=
1,2,3 .....
sin s60 + ~ sin r60 = O,
(9) (lO)
where for convenience we have defined 2M
M,~ = ~ .
(ll)
r+s
When ~ = 0 there exists an additional set of fixed points given bv s6 ~ s6 + 2wk~, which includes the common set (9),
UOs =
sMe~ ks '
k~ = I, 2, 3, ....
(12)
al. I Two-lrequency Fermi
J. E. H o w a r d el
24~
rls=3/2
mapping
M=!25
#.l. = 0
6£
k I
k s k~ :~
5¢
3-
I.I
41 4 3-
5-
3~ ~--U
2
4
5- 7 8 5 4 5
6 7 8 iO
B
6-
2~
9 12~5
~-.-- UL
I~
i v
-Tr
0
~r
(a)
G O -,
-----:i
:. . . . . .
- ~~.~.~.~..
. . . . . . . . . . . .
~.' •
--~--.",
i
......
~
". . . . . . . . . .
' ....
•
'~:,~,.~..,.~-.........~-_~
,,'~_'. ~
'
i
-~T
..~;.,'"
'i
0
(b)
.
.
.
.
.
.
.
~..
..,,.~_-iiii.i~"~ . . . . . . . . . . .
• ,.,i~ ' ! i l < ~ ~ .
;~-
~i
,-r
¸247¸
i
i
~P--u L
-Tr
O
7/"
(c)
/.I. -I.O
I
4
-'n"
0
Tr
(d) Fig. 2. Surfaces of section for rls = 312, M = 125 and four values of ~. The locations of several comn~on, r-fold and s-fo~d island chains are given in the first plot. The linear stability limit UL and adiabatic barrier us are indicated in each case.
J. E. Howard el al. ! Two-lrequo.ncy Fermi mapping
248
These fixed points have period P. where P / Q is s/k, reduced to lowest terms. Similarly, w h e n ~t --+ :r. setting r
-"
~r
k, = !, "L 3,
""
"
(13) ....
which also includes (9) as a subset. At intermediate values of/.t the r- and s-fold islands at d~0 = 0 and 7;- are u n p e r t u r b e d , but the other fixed points move according to (10), which locates the angular coordinate ~bo independent of uo. In general, (10) must be solved numerically for cho(/a, r, s). 60 = O, +_ rr are obvious but not exhaustive solutions. We now investigate cases for which (10) may be solved analytically. (a) ~t ~ I. Regarding the /,t term as a perturbation, the unperturbed solut|ons of (10) are &0.=0 +.rr ___2,+r .+ s - - I ) s" s . . . . . . (---~--
zr, _.+rr,
(14)
k!,5. u ) = sin s6 + u sin r ~
&,,--
~ sin r~So s cos stho + k,r cos r~b4(
i
)
t'+!
_"+ ~h,,::::O,
......
-
_
_
.
.
.
.
x.+ I = T • x,,
'20) p r i m a r y fixed
(2 ! )
The stability in the neighborhood of x0 is then determined by the lint:arized m a p p i n g , (16)
Ax,+ ~= ~ A x , ,
(22)
where ~ is the Jacobian matrix of T. The motion near xo is stable if and only if ITr ~el < 2.
(23)
From (4) and (5) we find (17) Tr ,~ = 2 -
4.rrM
(p + I)UoV
which yields tile two families of solutions,
x (cos Sd~o I- g cos rd~o). ± 2w _*~4 n , ,+,"+",+= O, .~'+ . . . . .S . . . .r-t-S ...........
(19)
There are a total of 1 +[~(r + s)] ~b. roots, and [ ~ ( r - s + I)] d,t, roots, where [ ] d e n o t e s the integer part. In addition, there are several cases where (10) is e x a c t l y solvable for all p., namely r - 5 . We shall e n c o u n t e r these explicit solutions in the c o n t e x t of linear stability and bifurcation. The linear stability theory of fixed points of arbitrary order is given in ref. 5. H e r e we calculate the stability limits of the p r i m a r y periodone fixed points of the t w o - f r e q u e n c y Fermi mapping. As we shall see, these limits enable us to delineate the bifurcations the p r i m a r y island chains must m a k e with increasing tz as they change from s-fold to r-fold s y m m e t r y . Writing (4) and (5) in matrix f o r m ,
xo = T • xo.
A similar application of N e w t o n ' s method is of course possible in the limit p.--, ~c. (b) la = !. For equal a m p l i t u d e s (10) transforms |o sin r 2+ ~ +,,cos
r-
(15)
then gives ,+-8= +bo- FJFt'+, or th(~t) ~
+-3rr s' r - s '
__.zr -
with x - - ( u , cb), the period-one points are given by
a total of + _s + ! rot, Is in [-rr, rr]. Writing
F( +b,,. ~ ) -+ F'(~Su. ~t ) ( + - ~bo) = O,
ebb =
(24)
(18)
If 0 < cos S4+o+ ~ cos r4>o< I + tip, a sufficient
J. E. Howard et al. ! Two-frequency Fermi mapping stability condition is Tr .T > - 2 , uo2> u~ = V ~
or
,"
(25)
It should b e noted th~ stable secondary fixed points, or higher period primary fixed points may exist below UL. Also note that UL has m a x i m u m at ~ = p. For T r . ~ > 2, a sufficient condition for instability is clearly cos S4,o+ t~P cos r4,o < O,
(26)
which is to be solved for ~ and 4,0 simultaneously with (10). Since (26) is independent of u0, we see that if u0> ut., a stability threshold exists when cos sOo + UP cos r4,o = O,
(27)
sin s4,o + tx sin r4,o = O.
(10)
249
for 4,0 = 0 and we conclude that this point never bifurcates for u > u L. For 4,0- m however, (27) yields (-1)" + ~ * p ( - l ) ' =0, which has the solution t~*= lip, provided that r - s is odd. In particular, if r - s 1, this is the only bifurcation, giving birth to two new fixed points, exactly the required number to change from s-fold to (s + l)-fold symmetry. This bifurcation is clearly seen in fig. 2d, Another broad class of bifurcations is revealed by setting 4,0 = -+¢d2 in (10) and (27). Setting r=-s+2m, m = 1 , 2 , 3 , . . . . (27) is satisfied, since r and s must be odd in order to be coprime. Eq. (10) becomes ! +/x cos wm = 0, which is satisfied by Ix* = 1 and m odd. Thus, a pair of tangent bifurcations occurs at ¢b* = _-t_rd2 when t ~ * = l for r = s + 2 , s + 6 , . . . . Again, when r - s = 2 , these are the only Vossible bifurcations. For s < r _<5, complete algebraic solutions of (10) and (27) may be obtained by means of the factorization
These equations admit a simple geometrical interpretation if we observe that they are of the form F(~,4,0) = aF/04,o=O. As this is just the condition for a double root of (10), it follows that (10) and (27) together determine the bifurcations of the common fixed points as the mapping changes (with increasing ~) from s-fold to r-fold symmetry. That is, for fixed r and s, one or more common fixed points 4,* bifurcate at critical values ~*. In general there are three types of bifurcation: the pitchfork bifurcation (P) in which a stable fixed point destabilizes and gives birth to two nearby stable fixed points; the anti-pitchfork bifurcation(P') in which an unstable fixed
The critical values of ~ are then given by the vanishing of the discriminant of (29), except ~n a few degenerate cases. The bifurcation ang!e 4,* usually follows b : inspection or, if necessary, by numerical solution of (10). The calculated values of 4,~ and ~* and the bifurcation type are listed in table I. Formal solutions of (29) are
l.~UiliS,
~lV~li
al.l~Uillil:.~,.,9
Oilt.i
i~'191dlg:3
l l J I I Ikll
LWUI
Ulll~itillJll~
fixed points; and lastly, the tangent bifurcation (T), in which a stable-unstable hair of fixed points is created. Although all this information is in principle contained ~n (10) alone, (27) is often useful in practice. For example, (10) yields the immediate fixed points ,ko = 0, 11".Eq. (27) gives I +/~p = 0
sin nd, = sin 4,P, ,(cos ~),
(28)
where P, l(cos 6) is a polynomial of degree n - 1. Thus, (10) becomes
P, 1(cos 4,)+ u.P, 1(cos 6)=0.
!!1 I ' l l .
(29)
ZU.
Additional insight into the structure of the bifurcations may be gleaned from ttle averaged Hamiltonian (derived in the text section) /# = ~C;u 2 + v ( 4 , ) ,
(30)
where G is a constant and, for a common island
250
3.
Howard et ai. I Two-frequency F e r n i mapping
E
r/s:3/2
Table I
rls
~b~
ix*
Type
2/1 371 312 4/!
180 ° 90 ° 180 ° 180 ° 65.91 ° 180 ° 52.24°'1, 127.76 = J 180 ° 55.80 ° 90 o 180 °
1/2 ! 213 I/4
P'
413
3t' 5/2 5/3 5!4 (n ~- I)ln (n + 21!n
T
415
P P' T P+ T T P T T P
180 °
nl(n + I)
P o r P'
90 °
I
T
3V618 314 ,t/5 2/5 0.9414
I
-
2 -'n"
o
~ 0
--/'N t,
-~
-rr
(el ~r
.
I
3"rr
ju.: I
7,- ,.,-t
.
.
,I
0
Tr
2
~
O
27r
rr
.
.
i
2~
,
I
37r
4,
(d) 2rr
3Tr
F:5
r/s:3/2 V Ok_:~/,~- - - / ' ' k ~ : - ~ ' ~ : :'7"~--'7'~"----~
21 -
t -~
,,t
0
i
rr
.t
2~r
/
3."
to)
u ~ I.
I. .
0
.
.
.
1.
....
~
i .
.
2u
.
.
.
.
3rr
H.=O
Fig. 3. Theoretical level c u r v e , and effective potential for r/s = 3/2 and four values of ,. Note the p i t c h f o r k bifurcation of the elliptic fixed po!,,t at ~b = ~. w h e n tt = 2/3, and the corresponding change in ' , ' ( ~
|
-v
...........
~
.......
0
~
........
u
chain,
~ ..................... _j
2rr
3Tr
v(~,) =
i- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b)
~ r I.
- ""
.
.
.
.
.
L. . . . . . . . . . .
0
1 ........................
~
l
....................
2Tr
3~r
0
~*:2/3
, (,
Vl-g~
c o s s,~ + r c o s r,~
)
(31)
is the effective potential. At a pitchfork or antipitchfork bifurcation, V ' - V " - V " - O, while V'# 0 for a tangent bifurcation. Thus, tangent bifurcations arise from inflection points in V(~), while pitchfork h i f l l r o ~ t ; ~ n © e t , ~ m f~.e,m tk:~.,t order critical points. More exotic bifurcations may be produced from potentials having higher order singular points [19]. A set of calculated levels curves and their associated effective potential~ are shown in fig. 3 for the case r l s = 3/2. Note the pitchfork bifurcation at 0 = 180~ when ~t - 2/3, as predicted.
I. E. H o w a r d e l a|. /Two.[requencyFermimapping
251
where O= s~ and 4.1. Derivation of averaged equations
Uo~ - sM,~ik~,
Oo, = 2¢tk:~,
(33)
terms o f n as locally approximates the Fermi map The destruction of KAM curves may be a,,t.erstood intuitively as arising from the interaction of neighboring island chains. Each island possesses a stochastic layer about its separatrix, so that for sufficiently large island widths these layers overlap, permitting diffusion over the region of velocity space occupied by the two islands. The island width is easily calculated from an approximate Hamiltonian in which only the dominant resonant term is retained, with all other terms averaged out. Analy:ic and numerical calculations show that the last KAM curve lying between two contiguous island~ is destroyed when the sum of the island h~df-widths equals approximately two-thirds the distance between the island centers measured in action space (velocity in our problem). This criterion has been shown to be quite sharp for equalwidth island [18] and works reasonably well if the widths are not too disparate [5]. Escande and Doveil [21] have derived a criterion valid for two islands with unequal widths, but their method is difficult to apply owing to the influence of neighboring large islands. We m©r©[or© © m p m y here m e simpler --twothirds rule," which agrees well with numerical calculations for comparable-sized islands. We begin by defining slowly changing variables near the s-fold elliptic fixed points, u, = u - uo~,
O~ ffi 0 - 0o,,
(32)
dd, = sin O + ~ sin(eOis) ~ 8(n - m), d. .o_. dO, 2wsM~ dn = u0~ u.
(34)
(35)
where (5) has been expanded about Uo~. Fourier-expanding the train of delta functions, 8(n-m)= m~-x
~ e 2'~i'",
(36)
I=-x
and averaging (34) over n, we find that the first term survives for I - -+k, while the second term is slowly varying only if ! ~ +_rkJs, which is only true near the common fixed points, not considered here. Thus, the averaged form of (34; 's simply du__z~= sin O, dn X/1--~'"
(37)
Eqs. (35) and (37) are derivable from the averaged Hamiltonian ~ , = 21rsM~ d~ cos 0"-~, uo., 2 VI--+~"
(38)
which yields the island amplitude A = (l +
2uL \~rSMe~]
•
(39)
Similarly, expanding u and 0 about an r-fold
J. E. Howard el at. I Two-~nrquency Fermi mapping
252 chain gives ,~
.x, = \ / ~ 1
+ t~
,,~:
,
I,:2
.
(40)
Eqs. (39) and (40) are generally valid only for 0t <~ I and ~ ~ 1, respectively, when the fixed r~oints are well approximated by (10). Neverthe',¢ss. a useful overlap criterion may be obtained r-v focusing on the 0o, = 0 (rood 2,-r) fixed point, which i~ unperturbed for all ~ and elliptic for u ~ u~. The widths A, and A, are valid for these central islands for arbitrary ~,. The motion near the common fixed points is obtained in the same manner by keeping the terms in the 8-functior, satisfying I = +--k, and I = -'-rk,/s, ~dmultaneously: d,i _- sin s4; + g sin rd, dn \I1 + g
(41)
separatrices, the o u t e r m o s t of which first overlaps with the separatrices of neighboring island chains. In terms of the angle d~ the central island width is ~iven by
A:= G~/~4
[1` s i n "4\( s26 ¢/ 5) + Ers i)n " ( - ~s ) ] "
Taking ris = 3/2 as an example we find A2 = 81.t 3 + 18/.t2 ÷ (1 + 4 ~ " ) 3/: -- I .
.
.
12G~"X/1-'~"
.
.
.
.
.
.
(46)
.
'
which varies by only 30% as t.t varies from 0 to ~c and by only 2% in the range 0 < ~ < I. In general, for r > s, A(/x) is either monotonic decreasing or has a gentle maximum near Ix = 0.5. In all cases (45) gives m
A(0) = x f r dn =
ac ---i
uii
Integrating thc~,~ ,,quations then gives the averaged Hamiltonian for the c o m m o n fixed points
!-
,
(,
)
(47,
;
Thus, for closely spaced frequencies the variation of A(t.t~ is entirely negligible so that wP may take 4,..
.... ~cossd~+ ....... t:/ . . ._C;~i \ / t ~:-......... ~t ,~ r ~osr~b , (43)
A ~ A(0) = k/G-~s-
',,,'here
In extreme cases and use (45); for from (18). More may be found in
2 rr,',Ik.~ (; ...... ~. .... .
(44)
Calculation of the island width in this case is complicated by the sequence of bifurcations as ~t increases from 0 to x. As discussed in section 3. this calculation can be carried out explicitly ,,nlv for the particular values of r and s listed in l:thle I. A v,idth can be calculated explicitly for the central island at cb, = O, which is elliptic for all g. F,~r ~t ~ 1 a single separatrix crosses the , ::: u,, line at d~ = ± rr/s (mod 2rr), but at larger g: there i~ in general a complex sy~,,em of nested
(48) one must calculate the angle cb~ # = 1 one may easily obtain d~,. information on island widths ref. 20.
4.2. O v e r l a p c r i t e r i o n Having outlined a procedure for calculating approximate island widths, we now turn to the choice of an appropriate overlap criterion for estimating the loss of globai stability with increasing/~. A useful rule of thumb is the "twothirds rule" [18],
a, + a: >
luo,- Uo:l,
(49)
$.:E. l.lo,ward:et:aLI Two,frequency, F~i mapping where-uot and uo2 a r e the two resonance positions and A t and &2 are the r e s o n a n c e halfwidths & , As or &, as the case may be. In deriving (49) it is implicitly assumed that the islands are of comparable size. s o t h a t the secon-
agreement for p. = I b u t only qualitative predictions for ~ ~ 1 or tt ~' 1. In the latter cases, although (49) still predicts KAM destruction quite accurately, the precise value of ;t at which this occurs is predicted less accurately, ~t(or N -t) being a perturbation. We consider three controlled numerical experiments in examining the effect of two frequencies on stochasticity. In the first, M is held fixed, while ~ is varied from 0 to oo. As ~t is "turned on," the r-fold islands grow rapidly and interact with s-fold and common islands, large jumps in the adiabatic barrier us generally occurring at fairly small values of ~t. In the second numerical experiment, we hold ~ fixed and vary M. Here (49) will be seen to give accurate predictions of the jump in the adiabatic barrier. In the third experiment we hold M and tt fixed and vary the frequency ratio r/s. On the several possible combinations of overlapping r-fold, s-fold and common islands, we choose for illustration only two. For overlap of the k, and ks + 1 fold chains, using (9), (11), and (48) in (49) gives M < M*, with M* = ~t-~
(2ks + I ) : V l - ' ~ ,
(501
which is equivalent to the single frequency overlap criteria for p ,~ 1. For overlap of at', r and an s-fold chain, using (12), (13), (39) and (40) in (49) yields
9 (r+s~I
k,
l+rk~
skr
(51)
253
However, care must be exercised in using this result f o r ~ v e r y small, as overlap is then determined by neighboring s..ch,Jns and their perturbation by interspersed r-chains.
of the island chains as given by (9), (11) and (13) for several values of k, k, and ks. The k~ = 2-8 islands are clearly visible, but the ks ffi 9 chain (uo = 11.1) falls beneath the linear stability limit and has disappeared. Numerically, we find the last KAM curve at about us = 27, as indicated in the figure, in agreement with the overlap criterion (50), which predicts that KAM surfaces exist between the ks = 3 and 4 island chains, but not between the ks = 4 and 5 chains. As we increase ~t to 0.1, the adiabatic barrier increases dramatically to uB = 37. This jump is due to the intercession of the kr = 5 islands at u0, = 30, whose stochastic layer overlaps those of the ks = 3 and 4 islands at uo~ = 33.33 and 25. Applying the overlap criterion, we find that the ks = 3 and k, = 5 islands overlap for arbitrarily small ~, but that overlap of the k, = 5 and k~ = 4 islands should not occur until ~t >0.5. The underlying reason for this premature overlap is the chain of four secondary islands at u0 ~-28, which may b~. seen in fig. 2b. There is also a chain of six secondary islands at u0 ~ 27, barely visible in fig. 2b. These two secondary chains effectively bridge the gap between the kr = 5 and ks = 4 islands, so that Or""~,rlap occurs at ~ ~ 0.065. As tt is increased from 0.1 to 0.25, uB once again suddenly increases, as the k, = 4 islands at u0,= 37.5 grow and overlap with the k: = 3 islands at u0s = 33.33. According to theory this jump should occur at tt = 0.0025, whereas it does not actually occur until ,t •0.15. The underlying reason for .this discrel~ancy is the great difference in size of the two islands, the smaller being distorted with no significant in-
J. E. Howard et ai. I Two.Irequency Fermi mapping
254
50
. . . . . . . . . -r.....
-
ADIABATIC
......... ~ . . . . . . . . .
; .......
BARRIER
,.'s=3 2
L
M.:t25
UB
.q----
~,,
: 4
NUM[RICAL
DATA
~ - - ~s--3 ~.
~
~
k,:5
L ..............
.9 M'--- s e c o n d o r ~ e s
!.
-,. ~-
--
k :2
! 2,." . . . . . . . . . . 0 0 0.~
'-
02
/.,t
- -~ . . . . . . . . . . . . . . . . 0.3 0.4
Ftg. 4 A d i a b a t i c barrio; as a f u n t t i o n o f ~. f o r rls = 3/2 and .%! = !25. T h e j u m p s in uB a r e a s , , o c i a t e d w i t h o v e r l a p o f the t,,land c h a i n s i n d i c a t e d in the right-hand ,'rmrgin.
termediate island chain to mediate the overlap. To complete this series of r/s = 3/2 cases we include the m a p p i n g for vt = !. Here the twothird~, rule successfully predicts that the k = 1 ch~,,in ..~t u,, = 50 cannot o~ erlap the k, = 4 chain at 37~ (Note that the c o m m o n island chain at u , = 5~) has bifurcated at d, = 180 ° and ~hat the r-fold chain at uo, = 37.5 has u n d e r g o n e a more complex division (trifurcation) into three is!ands.) The o b s e r v e d variation of uB with Ix is shown in fig. 4. The iimbility of the two-thirds rule to predict the exact value of ~ at which two I:,and chains overlap, for small ~, it is not surprising. Since ~ enters the o v e r l a p condition in (51!. as a perturbation, ~ significant change in t.t corresponds to only a small c h a n g e in the overlap criterion itself. l-or t.,: ~ 1 we expect much better agreement, ~ hich is in fat:t roun,! numerically. To illustrate thi,,, fig. 5 presents un as a function of M for t.L = 0 and i (eq. (49) is always solvable for M). To follow a given K A M curve with changing M, we note that in the absence of overlap, the island centers, and therefore ut~, are proportional to M. H o w e v e r . the island width A-.-
•
x_. .......
t, .....
,t.....~_,.~__x_,.t._,l,~d
50
....................
tOO
H,,O
.k. ......... , _ , t . ~ _ a . _ - _ . k - - - . L
500
x=~
_
IO00
M
Fig. 5. Adiabatic barrier as a function of M for ~ = 0 and I. The solid "'zig-zag'" curves were calculated using the twothirds rule, while the circles were obtained by numerically iterating the mapping I0" times.
V ' M , so that an o v e r l a p must inevitably occur for a given island c h a i n as M is d e c r e a s e d , with the chain contint.,mg to sink into the stochastic sea as M decreases further. For t~ = 0 w e follow only the s-fold islands, while for ~ = I we m u s t examine all possible c o m b i n a t i o n s of rfold, s-fold and c o m m o n islands. The solid zigzag curves r e p r e s e n t the theoretical formula (49), with the numerical results s h o w n as individual pcint~. The dashed line indicates the average pn oportionality u, --- V ' M . The a g r e e m e n t b e t w e e n t h e o r y and n,~,merical values is seen to be quite good. We have also e x a m i n e d a , u m b e r of other cases, varying the value ol r and s to d e t e r m i n e first if the two-thirds rule for overlap holds and also to investigate the fractional j u m p in the action as r and s b e c o m e larger. Specifically, we take r = s + 1 and c o n s i d e r s = !, 2, 3, 4, . . . . For fixed M, this c o r r e s p o n d s to taking the two frequencies s u c c e s s i v e l y closer together. Of particular interest here is the case s = I, both theoretically b e c a u s e of the simplicity of the interspersed island structure, and practically b e c a u s e a second harmo.qic may be g e n e r a t e d naturally in a physical system. B e c a u s e the r. fold islands are interspersed h a l f w a y b e t w e e n the s-fold chains and reinforce the second h a r m o n i c s of the s - s y m m e t r y , we might expect
J. E. Howard et aLI Two-[requency :Fermi mapping 28 . . . . . . . . t "'
26
~'
I ....
~ ' •
For large s(lls <0.05) the r-fold and s-fold islands nearly cohlcide near us, so that we would expect no increase in us with increasing /z.
i
decreases at first with increasing s, but does not fall t o zero as e x p e c t e d , This residual change is apparently due to t h e mutual destruction of closely spaced r-fold and s-fold island chains and a small increase in island width due to the normalization. The oscillations in ~q at smaller s can be understood in terms of the resonance interspersal, the maxima occurring when the resonances are evenly spaced near the adiabatic barrier. From the fixed point conditions (12) and (13) the r-fold and s-fold islands are evenly interspersed where
ADIABATIC BARRIER r/s=2/I M=50
24
UB 22
20
.~4
18
L., , 16 0.0
0.~
I 0.2
i p. 0.3
*
secondaries
EL~_ks= 2
0.4
Fig. 6. Adiabatic barrier as a function of/~ for rls - 211 and M =50, The agreement between the theory and the numerical data is much better in this case, since the jumps occur at larger p..
particularly good agreement with the two thirds overlap criteria. This is indeed the case as seen in fig. 6, which graphs ua against/~ for rls = 2/i. The arrow at /~ = 0.39 indicates the predicted overlap of the k, = 3 and k~ = 2 island chains, in excellent agreement with the value of /z = 0.35 at which the last KAM surface between these islands disappears. The smaller jumps in the adiabatic ba, rier between these two major chains involve the interaction of secondary islands and are therefore not predicted by the first order theory. Generally the theory predicts whether, for given p, the last K A M surface between any two island chains is destroyed. To demonstrate this, we calculate the maximum change in Ua over a large ~'ange of !~, for r = s + I and fixed M. The results are shown in ,fig. 7, which plots tl,e relative change in ua,
2rM=ff = s M ~ + sMeff k, k, k~+l "
k,, = ~(2m
-
(54)
which implies s >> 1 for moderate m. If we evaluate u0, at the adiabatic barrier u = a ~ / 2 ~ - M , where Meif~ MIs and a is approximatery equal to the increase in u, clue to two O.
7 . . . . . . . . . . .
!
m': I
m'= 3
0.6 0.5 0.4 0.3
•
.-
'
j
m': 2 O.
0.301
as a function of 1Is, where us(I) ~ u,(/~ = I) WF~max.
I)s,
o.if- ---J--. .. (52)
(53)
Taking r = s + I and k, = k, + m then gives, for k , ~ 1,
1
u . ( I ) - u.(O) = us(O)
255
1
t
i
l l l J |
i
i
O.Ol
i
i
i i i i ~L...
O. i
~L
I
l ~ 1 =:J_
l.O
lls
Fig. 7. Fractional increase in u, as a function of lls for M = 500. According to theory the increase should vanish for s ~ 100. The residual change is probably due to the strong interaction of overlapping r- and s-fold island chains.
]. 17.. Howard a al. i Th'o-[reaaency Fermi mapping
256
frequencies, then (55) becomes
$
The maximum enhancement thus occurs at a set of odd integer values, m' = 2m - 1. Conversely, the s and r resonances are cemered on top of one another for even integers m ' = 2m, where we expect ~ to be a minimum, i.e.. ! 4,-r - = a 2m
s
(56)
2M"
From these results and the position of the s and r island chains from (12) and (13). we expect the last KAM curves to be destroyed at a = 1.4to give the peaks, and at a --- I. ! to give the minimum. These values of I/s are shown on fig. 7 as arrows, which indicme resonable agreement with the predictions of (55) and (56).
5. Discussion
The addition of a second driving frequency to the Fermi mapping produces a considerable enrichment of the phase plane structure. This c:m be seen, for example, by comparison of figs. 2a and 2d. A new se~ of islands appear, corn~sponding to the fixed points of the second frequency. The islands associated with the fixed points common to both frequencies become considerably more involved, with bifurcations changir~g the island topology. Stochasticity near the separatrices of the bifurcated i~;lands decreases the area occupied by local KAM curves (islands). Interactions also occur belho ]n~ct
tw,~:'n ........................
csrcl+ar ;e|,~nrle •.,~.,
,a~,a,~uo
,~¢
~t
one
.......
which destroys the KAM surfaces between islands at lower driving amplitude. The twothirds rule, which was developed to predict when the last K A M surface between neighboring equal width island chains is destroyed, is applied to the two-freque.ncy Fermi map with its islands of unequal width. It is found that the rule works well unless the islands are very different in size. Comparing the results of figs. 2a and 2d it is s e e n that the last adiabatic barrier
increases from the single-frequency value u B 27, above the k, = 4 island centered at u - 25, to us = 37, above the k, = 3 island ,:entered at u = 33.3. These results are in agreement with the two-thirds rule. Considering the Fermi acceleration model as an analog to electron cyclotron resonance heating, our results have implications for the heating problem. If the resonances are fairly evenly interspersed near the adiabatic barrier, then for a given input power it is possible to increase the energy obtained by stochastic heating by a factor of two, by employing two heating frequencies rather than one. We can also relate the requirement for interspersed resonances found for the Fermi acceleration model to ECRH. From the definition of phase slip sA~ in (5) we see that
sMe~ = Uo,
2~oa'
(57)
where o , s : ¢ru/! is the "bounce frt.qeency" corresponding to t w o mapping periods and o', is the driving frequency. Substituting Me,/Uo~ from (12) into (57) and noting that 8,0 : o ' , - ,o, : o',/s, we have for interspersed resonances
....
~ynuti~'.tu y
and higher order islands of the second symmetry, giving rise to ve-y interesting island
svructures. The addition of a second set of islands, with their own stochastic layers interpersed among the first set of islands, leads to an interaction
8¢o
= (2m - l)o'..
(58)
This result is consistent with a numerical calculation by Rognlien [22], which indicates peaks in the ECRH induced heati:.g rate for frequency separations satisfying (57). A peak in ECRH
$.E. Howard c¢ al.i Two,Srequency Femi mopping Q heating Wa s also found experimentally to occur when ~ =cob [15]. There are additional constraints that must be satisfied in an ECRH experiment, such as the
257
Systems, Ann. ef Math. 77 (Princeton Univ. Press, Princeton, N.J., 1973). [9] T, H. ~:~x, Nucl. Fusion 15 (1975) 737. [10] E.F. ~Jaeger. A.J~ Lichtenberg and M. A. Lieberman, Plasma Phys. 14 (1972) 1073. sma Phys. in a Tanngineering
L,S,..vj.
:. l i ~ o i ;
t,,t;i|~titil;IiiLIVll3
UA
UUUI
LIICUi'CUI;til
and practical importance leave considerable scope for future work on this problem.
References [1] E. Fermi, Phys. Rev. 15 (1949) 1169. [2] S. Ulam, Proc. 4th Berkeley Syrup. on Math. Stat. e~nd Prob. 3 (Univ. of Calif Press, Berkeley, 1961), p. 315. [3J G. M. Zaslavskii and B. V. Chirikov, Soviet Physics Doklady 9 (1965) 989. [4] A. Brahic, Astr. and Astrophys. 12 (1971) 98. [5] M. A. Lieberman and A. J. Lichtenberg, Phy.~. Rev. A5 (1972) 1852. [6] A. J. Lichtenber8, M. A. Lieberman and R. H. Cohen, Physica ID (1980~ 291. [7] V. I. Arnold, Russian Math. ~urvey 18 (1963) 9. [8] J. Moser, Stable and Random Motions in Dynamical
,ieberman. Plasma Phys. 17 (~975) 679. [15l B. H. Quon et al., Bull. Am. Phys. Soc. 26 (1981) 893. [16] C, R. Menyuk and Y. C. Lee, Phys. Fluids 23 (1980) 2225. [171 B. V. Chirikov, Physics Reports 52 (1979) 263 [lS] A. J. Lichtenberg, in Intrinsic Stochasticity in Plasmas, C. Laval and D. Gresillon, eds. (Editions de Physique, Orsay, 1979), p. 13. [19] I. Stewart, Physica D 2 (1981) 245. [20] J. E. Howard, M. A. Lieberman and A. J. Lichtenberg, Bifurcations of the Two-Frequency Fermi Map, UCB/ERL Memorandum M81/56 (1981), University of California, Electronics Research Lab. [21] D, F. Escande and F. Doveil, Physics Lett. 83A (1981) 307. [22] T. D. Rognlien, Bull. Am. Phys. Soc. 26 (1981) 1035. [23] J. L. Tennyson, M. A. Lieberman and A. J. Lichter~berg, AIP Conf, Proc. No. 57, Nonlinear Dynamics avd the Beam-Beam Interaction, M. Month and J. C. Herrera, eds., (1979) 272.