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Invariant surfaces and orbital behaviour in dynamical systems of 3 degrees of freedom. II
P h y s i c a 6D (1982) 126--136 North-Holland Publishing C o m p a n y
INVARIANT SURFACES AND ORBITAL BEHAVIOUR IN DYNAMICAL SYSTEMS OF 3 DEGREES OF FREEDOM. II
G. C O N T O P O U L O S European Southern Observatory, Garching bei Miinchen. Germany and
P. M A G N E N A T and L. M A R T I N E T Geneva Observatory, Sauverny, Switzerland Received 3 April 1981 Revised 25 June 1981
Two formal integrals of motion besides the Hamiltonian are f o u n d in a s y s t e m of three degrees of f r e e d o m with three u n p e r t u r b e d frequencies o9~, o92, o93 which are close to, or exactly at, the double r e s o n a n c e 6 : 4 : 3. The integrals are truncated at various orders and they are used to find theoretical invariant surfaces on a surface of section. The theoretical invariant surfaces are then compared with the empirical results. The a g r e e m e n t is improved as the order of the integrals increases from the 5th up to the 1 lth degree. This is surprising because theoretical considerations indicate that one formal integral breaks down b e y o n d order 6, because of the second r e s o n a n c e 4 : 3. ( H o w e v e r it is possible to construct one further integral by a different method). As expected the a g r e e m e n t is worse when the perturbation b e c o m e s large. A check of the c o n s t a n c y of the truncated integrals along various orbits s h o w s that the relative variations b e c o m e smaller as the order increases; they b e c o m e larger as the perturbation increases.
1. Motivation
It is well known that dynamical systems close to integrable ones possess integral surfaces (KAM theorem). Such integral surfaces can be given approximately by asymptotic expansions provided by formal integrals of motion like the "third" integral of galactic dynamics [1]. H o w e v e r there is an important difference between systems of two degrees of freedom and systems of n > 2 degrees of freedom. In the second case the motion in phase space is expected to show Arnold diffusion, making the system practically stochastic, even if the integral surfaces have a large total measure. Such a diffusion does not appear in systems of two degrees of freedom, that have been studied extensively up to now. But the number of studies of Arnold diffusion is limited [2, 3, 4] and its practical applicability has been questioned in certain cases
0167-2789[82/0000-0000/$02.75 0 1982 North-Holland
(see, e.g. [5]). The basic question is whether there exist some restrictions in the effectiveness, or even the appearance, of Arnold diffusion. This question is at the root of the applicability of statistical mechanics in generic dynamical systems. In order to study this problem it would be of interest to study in detail some of the simplest systems in which Arnold diffusion is expected to occur, namely systems of three degrees of freedom. In such systems one can visualize the "invariant surfaces" corresponding to orbits lying on integral surfaces (isolating orbits). T h e r e f o r e one can see empirically whether a given orbit is isolating or ergodic, whether there is diffusion and how fast it goes. Such a visualization is practically impossible in systems of more than three degrees of freedom. Our study uses both numerical experiments and theoretical results to understand the general
G. Contopoulos et al./lnvariant surfaces in 3-D dynamical s y s t e m s
form of the orbits. We are interested especially in finding the onset of unstable and stochastic motion. Then by comparing our results with the already k n o w n results for systems of two degrees of f r e e d o m we hope to understand the onset and the significance of Arnold diffusion.
then we can derive the values of H, @t, qb2, and the form of the integral surface (3) and of the invariant surface (4). We consider the general case of a Hamiltonian with second order lowest order terms
(5)
Ho = (/)10 + (/)20 -[- (~30, 2. Introduction
where
In a recent paper [6] (called h e r e a f t e r p a p e r I) the general characteristics of the orbits in a dynamical system of three degrees of f r e e d o m were studied by constructing empirically their invariant surfaces. In the present paper we find theoretically the invariant surfaces by using the theory of the "third integral" of Contopoulos [1], and c o m p a r e the theoretical with the empirical results. F u r t h e r m o r e we examine how well the formal integrals are c o n s e r v e d along particular orbits. The dynamical s y s t e m is assumed to be described by a time-independent Hamiltonian H = H(x,
(1)
y, z, it, ~, ~).
This is an integral of motion. If we have two more integrals of motion, ~ = qb~(x, y, z, x, 2), z)
(i = 1, 2),
(2)
1
(/)10 = 2 (.~2 ~_A x 2 ) ,
we can eliminate two variables, say 2) and 2, and derive a three-dimensional surface in the fourdimensional space (x, y, z, 2) y, z, ~) = 0
(3)
on which lie the projections of the orbits, that have their initial conditions on it. If we take now a surface of section, say z = 0, we find an "invariant s u r f a c e " in the space
(x, y, ~) [(x,
y, g) = 0,
(4)
which depends on the values of H, ~1 and ~2. If the initial conditions for an orbit are given,
~ 2 0 = ~1( ) ~2 +By2),
1
(/)30 = 2 (~2 _[_ CZ2),
(6)
and A, B, C are positive. In the non-resonant case (i.e. when o91 = A1/2, o92 = B 1/2, and 0) 3 = C 1/2 do not satisfy any linear relation with integer coefficients) we can construct three integrals ~ (i = 1, 2, 3) starting with lowest order terms ~0. The sum of these integrals is the Hamiltonian H = ~ l + ~ 2 + ~3. In the non-resonant integrable case a canonical change of variables brings H to a normal f o r m H = H ( ~ l o , ~20, ~30) in the new variables, where ~io have now the numerical values of qbi. The solutions in the new variables are quasipeiodic with frequencies 05 = o 9 z O H l a d P , o = 6ol + " " " ,
052 = o920H/OdP2o = o92 + " " " ,
053 =
= o ~ 3 + ' " . The higher order terms depend on the values of H, ~1 and qbz. If for a particular orbit there are two resonant relations between 051, 052, 053, this orbit is periodic. The corresponding invariant surface (4) is reduced to one or more invariant points. All orbits lying on the particular surface (3) for the same values of H, (JDI, (~)2, satisfy the same resonance conditions, therefore they are periodic. If there is only one resonance condition between 051,052, 053 the orbit fills a two-dimensional surface in the space (x, y, z, ~) and its invariant surface (4) is reduced to a curve. Up to now we have considered integrable cases. If, however, the Hamiltonian system is o930H/0~3o
F(x,
127
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
128
not integrable, then the periodic orbits are isolated, i.e. there are no three-dimensional surfaces in the 4-dimensional space filled with periodic orbits. F u r t h e r m o r e , near the unstable periodic orbits there are no integral surfaces of the f o r m (3), or invariant surfaces of the f o r m (4). More accurately, we can have integral surfaces, or invariant surfaces, with infinite oscillations, or foldings, which do not close, and the corresponding orbits are not quasi-periodic. Empirically we see that the successive consequents on a surface of section do not f o r m a simple surface, but fill a region of three dimensions in the space ( x , y , ~ ) . Thus we have a " d e s t r u c t i o n " of the invariant surfaces. It is known, f r o m the corresponding problem of two degrees of f r e e d o m , that this destruction is very large if the perturbation is large enough so that we have interaction of several resonances, especially low order resonances [7-10]. This is more evident in s y s t e m s of more than two degrees of freedom, e.g. in s y s t e m s of three degrees of f r e e d o m one can have double r e s o n a n c e s between the frequencies ~1, to2, to3 and these allow a larger diffusion of the consequents of the orbits. If we have r e s o n a n c e s between the unperturbed frequencies to1, oo2, to3, say M I ~ I + n 2 ~ 2 + n 3 ~ 3 = O,
(7)
with n~ integers (not all zero) t h : n we can construct integrals q~, starting with second order terms of the f o r m Ot I
Ol3
~o = - - 4)1o + a2 4)20 + - - 4)30, 0.) 1
£02
(,D3
(8)
where n l o / i + / 1 2 0 : 2 + n30t3 = 0.
(9)
If there is only one resonant relation (7) we can construct two independent integrals, with lowest order terms of the f o r m (8), besides the
Hamiltonian. H o w e v e r if there are two independent relations (7) the quantities o/~ are proportional to o~, therefore ~0 is necessarily proportional to H0. The corresponding integral ~0 differs f r o m H in its higher order terms (see, e.g., [11]), but we cannot construct any further integral by this method. H o w e v e r , the numerical evidence of paper I indicates that, even in the case of double resonance, most of phase space is filled with closed integral surfaces. This indicates that there are other f o r m s of integrals valid in this case.
3. Formal integrals of motion In order to construct formal integrals of motion we solve the Poisson bracket equation {H, 4)} = O,
(10)
where H and 4) are series in the variables x, y, z, ~, ~, ~, starting with second order terms. The successive terms of 4) are calculated by a computer program, once the lowest order terms are given. The program used was constructed by Giorgilli and Galgani [12] (for details see also [13]) and applies to any Hamiltonian in the form of a polynomial, for any number of degrees of f r e e d o m and for any resonances between the unperturbed frequencies to1, to2, to3. The method consists of finding a generating function X, and the corresponding normal form of the Hamiltonian Z. This method does not use any inversions of series, as the v o n Zeipel method, but gives the successive terms of X and Z explicitely, step by step. In this respect it is similar to the methods of Hori [14] and Deprit [151. The main interest of the method of Giorgilli and Galgani is the treatment of resonant and near resonant cases. Suppose that the ratios O91." 602 : ¢.O3 are equal or nearly equal to the ratios of the integers ~1 :/z2 :/z3. Then at each order, s,
G. Contopoulos et al./Invariant surfaces in 3-D dynamical s y s t e m s
the normal form Z (~) and the generating function X(~) are derived from two equations Z (s) = F ~ ) and D X ~) = F ~ ), where D is the operator D = i E3,=] ~n(p~3/3p, - q , 3 / 3 q , ) and the subscripts N and R denote the null space and the range of the function F (~), which is known. (If q]~p ~'q ~p ~ . . . is a term of order s it belongs to the null space if /~1(J~- k0 + / ~ 2 ( h - k2) + . . . . 0, otherwise it belongs to the range). After X is found the construction of integrals of motion, starting with a given lowest order function, is easy. In the present paper we consider the Hamiltonian
H = ½(g2 + ~2 + ~2 +
Ax 2+
-- EXZ 2 -- ~ y z 2,
By2 + C z 2) (11)
where A = 0.9, B = 0.4 or 0.38, C = 0.225. Thus the unperturbed frequencies 091 = A l/z, 092 = B 1/2, 093= C 1/2, have ratios equal, or close to 091 : 092:093 = 6 : 4 : 3. We have always 091 : 093 = 2 : 1 but the other resonance 092:093 = 4 : 3 is only approximate in the near resonance case B = 0.38. Thus in this case we can have two subcases: (a) A single resonance if we set/~1 = 60, ~3 = 30 and/x2 = 39 (because we never go to terms of order t>39); and (b) a double resonance if we set/x~ = 6,/x2 = 4, /x3 = 3. In the first subcase we have the drawback that X contains small divisors of the form (3092-4093). In the second subcase X does not contain any small divisor terms, but we can construct only one integral besides the Hamiltonian. If we have only the resonance 091:093= 2 : 1 we can construct two integrals ~1, ~2 with lowest order terms
129
These integrals do not produce any small divisor terms up to degree 6. H o w e v e r the terms of degree 7 and higher contain terms with divisors (3092-4093). Thus the convergence of these integrals is bad. On the other hand if we use the double resonant form of the generating function )C we can construct only one integral besides the Hamiltonian, starting with the combination 4 q~o= cplo+ TS- C#2o.
(14)
.~ ¢.o2
However, as X and Z can be constructed to any order, we can also continue the construction of the "integrals" ~ , q~2 to any higher order, although they do not satisfy eq. (10) b e y o n d the order 6. To our surprise the invariant surfaces constructed with the help of the functions q~], q~2 gave better and better agreement with the empirical results as we went to higher and higher order, up to order 11 (see section 5). This indicates that the functions q~l, q~2 may approximate two integrals of motion. This possibility is enhanced by the fact that the numerical values of q~l and ~2 are better and better conserved along the calculated orbits as we go to orders b e y o n d 6 up to order 10 (see section 6 and table I). The existence of two formal integrals besides the Hamiltonian, both in the near resonance and in the resonance case, is indicated not only by the empirical calculation of orbits of paper I but also by the following considerations. If we use action-angle variables X = (211/o91) 1/2 COS 0 1 ,
.£ = --(211091) 1/2 s i n Ol,
(15) q~]0= 2 410 + 1 430 091
093
(12) and similar expressions for y, ~, z, ~, we can write the Hamiltonian (11) in the form
and H = ¢otlt + 602/2 + 093/3 q~2o= 420.
(13)
-- ~(211/091)1/2(213/093)
130
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
>( [ l c o s 0.-}-~1 c o s ( 0 . 1 - 20.3) + ~1 COS(0.I +
20.3)]
-- r I ( 2 I ~_/o92) 1/z( 213/ o03)
1 × (lc0s0.2+~c0s(0.2-20.3)
+ 1 c0s(0.2 + 203)].
(16)
By canonical t r a n s f o r m a t i o n s of variables we can eliminate all c o m b i n a t i o n s of angles e x c e p t the r e s o n a n t c o m b i n a t i o n s ( 0 " ! - 20"3) and (30"2403). T h u s we find, in the n e w variables H = ~o111+ oJ212+ 00313+ ~(I~) -- E(211[tol)l/2(13/2o93)
COS(0" 1 -- 2193)
+ ~ cos(30"2- 40"3) + • • ",
(17)
w h e r e rl~ is o f o r d e r r/. Using n o w the generating f u n c t i o n S = Jl(0.t - 2"03) + J2(30"2 - 40"3) + J30"3,
(18)
we derive the final c a n o n i c a l a c t i o n - a n g l e variables (Ji, tOi) such that tOl = Ol -- 20"3, rll = Jl,
tO2 = 30"2 -- 40"3,
12 = 3Jz,
~3 = "03,
13 = - 2 J 1 - 4J2 + J3,
250 times larger than in the case "0 = 0.1. This explains w h y the case of small r/ ( r / = 0.1) can be a p p r o x i m a t e d rather well by the formal integrals derived by the p r o g r a m of Giorgilli and Galgani [12], while the a p p r o x i m a t i o n is considerably w o r s e in the case rl = 0.3. If we w a n t to find a m o r e a c c u r a t e third integral, up to any higher degree, we can use a m e t h o d d e v e l o p e d by K u m m e r [16] for a similar double r e s o n a n c e H a m i l t o n i a n given by F o r d and L u n d s f o r d [17]. This integral exists even in the exact r e s o n a n c e case w h e r e one usually e x p e c t s only one integral besides the Hamiltonian. T h e basic idea is to c o n s i d e r the case w~:w2:o~3=6:4:3, " 0 r = 0 as defining a new "unperturbed" Hamiltonian fflo = (2Jl/o~t)l/2(J3 - 4J2 - 2 J 0 cos q/I.
Then,
(21)
if
we add a small p e r t u r b a t i o n --(2¢O3~/E) COS ~b2 we can find a third formal integral in p o w e r s of (2~o:0~/~). T h e existence of this integral explains the a p p e a r a n c e of g o o d invariant s u r f a c e s as f o u n d in p a p e r I.
4. Theoretical invariant surfaces (19) T h e variables u s e d in the p r o g r a m of Giorgilli and Galgani are slightly different f r o m ours. N a m e l y t h e y use the H a m i l t o n i a n
and the H a m i l t o n i a n takes the f o r m H = w3J3 + (3~o2 - 4w3)J2 + ~(J~) - E ( 2 J l [ w l ) 1/2 × [(-2J~ - 4J 2 + J3)/2to3] cos tOl + -~ cos ~2 + • • •
(20)
n
T + ecxGz~ + ~GyGz~.
As the angle 4/3 is ignorable, the action J3 is a s e c o n d integral of m o t i o n besides the Hamiltonian. T h e p e r t u r b a t i o n "0~ cos q~2 is of degree 7 in j~/2. If we ignore this term we find also an a p p r o x i m a t e third integral of m o t i o n , n a m e l y J2 = ~I2. In the p r e s e n t p a p e r we take the total e n e r g y equal to h = 0.00765, • = 0.5 and rl = 0.1 or rt = 0.3. T h e r e f o r e the p e r t u r b a t i o n term is really small. H o w e v e r in the case r / = 0.3, r/~ is a b o u t
(22)
T h e relations b e t w e e n our variables and theirs are
x s = w[/Zx,
~s = ~/oJl/2,
(23)
and similar relations f o r y, ~, z, 2. F u r t h e r m o r e EG:--e/(wVZt.D3) and rls = - r l / ( w ~ / 2 w 3 ) . We calculate the invariant s u r f a c e s in the GiorgilliGalgani variables and then t r a n s f o r m to our
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems variables for c o m p a r i s o n with the empirical results. The two integrals with lowest order terms (12) and (13) are given in the f o r m @i = Z
Cabcdef X ay bz c2 el2)e~/.
(24)
We take the surface of section z = 0, which is a s y m m e t r y plane of the Hamiltonian (11). Thus we have q~i = Z CabOde[xayb2d2) efr':'
(25)
and d is now always even. Using eq. (22) we find 22 = [2h - ~oj(x2 + 22) - o)2(y 2 + 2)2)]/w3,
(26)
where h is the numerical value of the Hamiltonian (h = 0.00765) and we have omitted the subscripts G. We keep now only the coefficients Cab0d~: up to a given degree, N, in the two integrals. N e x t we eliminate 2) f r o m the two integrals (25) and thus derive the equation of the invariant surface (4). One point of the invariant surface is given, namely the initial point (x0, Y0, z0 = 0, 20, 2)0) while z0 is derived f r o m the energy integral). Thus we have also the numerical values of the integrals q~. We find the values of 2 and 2) for a grid of values of x and y with steps Ax and Ay respectively. N a m e l y we calculate 2 and 2) for all points (x +-max, y +-nay) within certain maximum limits. Each solution (2, 2)) at a given point (x, y) is used as an a p p r o x i m a t e initial condition for the next point ( x ± A x , y) or (x, y _+Ay). In general if 20, 2)0 are a p p r o x i m a t e solutions of the equations (25) at a point (x, y) we write q~ as Taylor expansions and truncate them after the first order terms:
q~(2, 2)) = q~d2o, 2)0)+ \ 05c /o 82 + \ 02) Io 62) = ~, (27) where q~ mean the numerical values of q~. Thus
131
we solve the system (27) and iterate until the corrections 82, 62) are smaller than a fixed accuracy, given a priori. After some preliminary exploration with various values of /~x, Ay and of the accuracy, we found that the solution of the eqs. (27) converged very fast, after a few iterations. If no c o n v e r g e n c e was found after 30 iterations, a special message was printed. It was found that lack of c o n v e r g e n c e meant, in general, nonexistence of the relevant solution. Similarly a special message was printed if the calculated 22 was smaller than zero. It was checked that if (2, 2)) is a solution then ( - ~ , - 2 ) ) is also a solution. In fact the expressions (25) are even in (22)). Thus essentially only two solutions (2, 2)) were needed for each point (x, y). It was checked also that for e = ~/= 0 the solutions are (+2, _-+2)), with 2 = (2ci91o-Ax2) 112, 2) = (2qb20- By2) 1/2. The numerical solutions (2, 2)) for the whole grid of points (x +_max, y _+ n a y ) were printed and stored in the computer, and then were used for plotting the invariant surfaces as described in the next section.
5. Comparison between the theoretical and the
empirical invariant surfaces In paper I a systematic numerical investigation of the conditions of existence of good invariant surfaces was initiated for the Hamiltonian (11) with A = 0 . 9 , B = 0 . 4 , C = 0.225, • = 0.5, the total energy h = 0.00765, and various values of r/, for given initial conditions x0, Y0, 20, 2)0 (~0 is inferred f r o m the equation of energy and z0 = 0 by the definition of the invariant surfaces). Stereographic views allow to visualize the shape of the invariant surfaces when existing. A series of cases with B = 0.38 instead of 0.4 were also calculated but not published. F o r the same fixed initial conditions the numerical invariant surfaces were very similar to those corresponding to B = 0.4.
132
G. Contopoulos et aL[Invariant surfaces in 3-D dynamical systems
At present we will c o m p a r e some of such empirical invariant surfaces with theoretical surfaces obtained by the method described in the previous section for the same initial conditions x0, Y0, ~0, ~0 and different values of 7. (We systematically use the variables x* = A~/2x, y* = Bl/2y, z* = CII2z and omit the symbol *). In all the present e x p e r i m e n t s x0 = - 0 . 0 3 7 8 , Y0 = 0.025, ~ 0 = 0 , ~ 0 = 0 . The order in which the coordinate axes are given in figs. 2--4 corresponds to the order xs, ys, Zs indicated in fig. 1, where we also show the angles 0 and ¢ which define the direction of the o b s e r v e r E with respect to the origin of the axes of representation. We shall always use 0 = 30 ° and ¢ = 0 ° whereas the angle of c o n v e r g e n c e of the two " e y e s " of E towards the origin is 15 °. We begin with case B = 0.38. Fig. 2a shows the empirical invariant surface in the space ( x , y , ~ ) obtained for -q = 0 . 1 , which suggests that 2 isolating integrals exist besides the energy in this case. Figs. 2b,c,d show the corresponding theoretical surfaces c o m p u t e d by means of the integrals (25) truncated after the 5th, 8th and l lth degree terms respectively. The scanning steps /Ix and Ay as well as the a c c u r a c y were set equal to 0.001. It is important to e m p h a s i z e that we are able to c o m p a r e the shape only of the numerical and theoretical surfaces and not the detailed distribution of points on these surfaces. The " n u m e r i c a l " surfaces a p p e a r as sets of points Zs /v
8
Ys
I
X
Fig. 1. Definition of the observer's direction E (0,~0) with respect to the origin of the axes.
which are successive consequents produced by the method of surface of section due to Poincar6, as explained in paper I. The "theoretical" surfaces are sets of points resulting f r o m the scanning method described in the previous section. It stands to reason that, in general, some kth " n u m e r i c a l " point does not coincide with the kth "theoretical" point as the sets are obtained through entirely different procedures. It appears f r o m the figures that the shape of the theoretical surfaces is progressively changing and, as more and more higher-order terms are included, it a p p r o a c h e s closer and closer the " n u m e r i c a l " surface. At the l lth stage there is quite good agreement, in the sense that the numerical and the theoretical surfaces have the same extent. In a similar way we have calculated invariant surfaces in the (y, x, ~) space. Such surfaces are not given here, but they show also a quite satisfactory agreement with the empirical surfaces. In both representations no i m p r o v e m e n t could be o b s e r v e d b e t w e e n orders 10 and 11. Fig. 3 shows the invariant surfaces obtained in the case -q = 0.3 represented in the subspace (x, y, ~). H e r e we o b s e r v e discordance between the empirical (fig. 3a) and theoretical surfaces, even in case (d) (order 11). Similar c o m p u t a t i o n s have been p e r f o r m e d with other values of ft. We conclude that for small enough values of 7, let us say smaller than 0.1, the theoretical invariant surfaces obtained by means of the integrals c o m p u t e d with the Giorgilli-Galgani program are good approximations of the real invariant surfaces, provided the formal series representing these integrals are c o m p u t e d to a sufficiently high order. For larger "O, discordances appear. T h e y have been anticipated, taking account of the discussion in section 3. Let us now consider the case B = 0.4. Fig. 4 displays the numerical invariant surface and the theoretical ones corresponding to "q = 0.1 and calculated f r o m the same initial conditions as before. Taking into consideration the discussion of the previous section we think that
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
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Fig. 2. S t e r e o g r a p h i c v i e w s of i n v a r i a n t s u r f a c e s o n the (x, y,.~) s p a c e for the c a s e B = 0.38, ~ = 0.1; (a) c o r r e s p o n d s to the n u m e r i c a l s u r f a c e , a n d the t h e o r e t i c a l s u r f a c e s are s h o w n at o r d e r N = 5 (b), N = 8 (c) a n d N = 11 (d).
G. C o n t o p o u l o s et a l . l l n v a r i a n t s u r f a c e s in 3 - D d y n a m i c a l s y s t e m s
34
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I#,.;
_ , o~. ,. ~ " ; " o °,~'o ' % ' ° : ~
.'.:.':.:.;~.~.:'.
• . ........
.~.
.. .,
° ..... °o',
..
.. ° °..o . ...... ~..:..~
o- •
•
•" •
OoO'q-_~'~-" -% o, o o o -
o Oo ; o . O
"
;.
.- ..'~ .:- . : ' . : . . , i .
o.~., o-.,o;.,,
.o , o . . °
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.; - , . . . . - ' - : - ' . - " ~, . . " .- . ' . . . . , , . . . . - . . .
.o-
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.
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~Oo.-.
..~...,.'...:.
~-..-"-.....:-.,....-" ;..'..'.~, • '."
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~
•
:'~. -
o...
o....Oo.o%
°'~
~
•~.. ;-.,-.~,'~.'.
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--...:--
.~.-, . .. - . . . . . : "....'~ ~ . . . . . . ~ . • • ~, , • .o o . o ° O . ~oOoO.O, | ..:.'.. ~...... : ; . ' . : .~o
"~."
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,,.
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.
.." ...~.. - - ,.:~...,r~.,, ~ •
o" o
,- ,,
~ ~ ~
- ~ ~ , L P '
b
C
d F i g . 3. S a m e
a s fig. 2 f o r ~ = 0.3.
o-~I
"
G. Contopoulos et al./lnvariant surfaces in 3-D dynamical systems
.~{+r.~"-:....:::'.
:'~'~'-~7"':.:."":~+.:.".~.~.J~ o~ .:
:.-..-.~.:
.%
,..
°
• ~.
%° " "
.,,,~.....:..
o"
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-~o" q.o
,..:.
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.++~ •
+'..
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"
r.- -..-.~./..;.y..-/(l~,,~t:.,,.~-~.~9, .,:.,.,,..,:.. " ",.: ' ~~ • "'~" " "" "" . . . . . . ~"~ " -..---..::.~'~:-~.~'~...v..,..:... ~ +
~
.-I...,~.}%°.-...:*,:
a
b
C
Cl F i g . 4. S a m e a s fig. 2 f o r t h e c a s e B = 0.4, "O = 0.1.
~
,,
. : . " I+*,,~- Oo. + l r . "
.~-t¢,~.~.. ,..:....-.....;.~,~..~y~.~,.
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135
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•
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G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
136
Table I Relative errors of the integrals &t and ~bz truncated after the terms of Nth degree for various (A = 0.9, B = 0.38, C = 0.225). N
5
6
7
8
9
10
r/ = 0.1 7=0.1
d)l &2
1.84x 10 3 1.53x 10 I
1.38x 10 3 9.46 x 10 2
9.45 x 10-4 7.19x 10-2
7.75 x 10-4 7.19x 10 2
6.86x 10 4 7.19x 10 2
5.61 x 10 4 3.74x 10 2
= 0.3 r/ = 0.3
(6t ,~2
1.03x 10 2 3.72
6.34 x 10 3 5.11 x 10 -~
6.24 x 10 3 7.94x 10-I
3.56 X 10 3 2.15x 10-1
4.41 x 10 3 3.30x 10 -t
2.65 × 1 0 -3 1.27x 10-l
the satisfactory the
present
because
results
case
which
are
even
are obtained more
in t h e e x a c t r e s o n a n c e
integral, besides structed.
surprising,
case only one
the Hamiltonian
(See however,
in
can be con-
the discussion
at t h e
ved when
we reach order
10. A n a l o g o u s
con-
c l u s i o n s c a n b e d r a w n f r o m t h e c a s e B = 0.4. We
conclude
that
the functions
¢1 a n d
¢2
b e h a v e as f o r m a l i n t e g r a l s o f m o t i o n , e s p e c i a l l y w h e n t h e p e r t u r b a t i o n ~ is small.
e n d o f s e c t i o n 3.) F i n a l l y f o r ~ = 0.3 w e o b s e r v e t h e s a m e d i s c o r d a n c e as in t h e c a s e B = 0.38 between the empirical and theoretical surfaces.
Acknowledgement W e w i s h to t h a n k D r . G i o r g i l l i f o r p r o v i d i n g us w i t h an i m p r o v e d v e r s i o n o f his p r o g r a m .
6. The "constancy" of the formal integrals References In a d d i t i o n to t h e c o m p a r i s o n d e s c r i b e d in t h e previous
section we can obtain supplementary
information integrals
on the applicability of the formal
by
periodically
checking
their
con-
stancy along a chosen phase trajectory. Starting from
the same
initial c o n d i t i o n s
have performed
as b e f o r e
we
the integration of the equations
o f m o t i o n f o r 1000 t i m e u n i t s a n d c a l c u l a t e d t h e values every
of the ten
integrals
time
units
o s c i l l a t i o n is p e r f o r m e d
u p to d i f f e r e n t (A
complete
orders
typical
z-
in a b o u t 15 t i m e units).
We then deduce an estimate of the fluctuations o f t h e s e i n t e g r a l s w i t h t i m e . In t a b l e I w e g i v e f o r B = 0.38 t h e r a t i o o f t h e s t a n d a r d d e v i a t i o n to t h e m e a n
v a l u e s o f t h e i n t e g r a l s (~1 o r W2)
c a l c u l a t e d f r o m o r d e r 5 to o r d e r 10, f o r "9 = 0.1 a n d ~ = 0.3. T h e r e s u l t s c o n f i r m t h e c o n c l u s i o n s o f t h e p r e v i o u s s e c t i o n . T h e r e is a s m a l l r e l a t i v e e r r o r f o r b o t h i n t e g r a l s in t h e c a s e ~ = 0.1 a n d t h e f l u c t u a t i o n s d e c r e a s e f r o m o r d e r 5 to o r d e r 10. F o r ~ = 0.3, o n l y qh p r e s e n t s r e l a t i v e l y s m a l l fluctuations.
The
behavior
o f q~2 is n o t
factory although a small improvement
satis-
is o b s e r -
[1] G. Contopoulos, Z. Astrophys. 49 (1%0) 273. [2] C. Froeschl6, Astrophys. Space Sci. 14 (1971) 110; C. Froeschl6 and J.P. Scheidecker, Astrophys. Space Sci. 25 (1973) 373. [3] J.L. Tennyson, M.A. Lieberman and A.J. Lichtenberg, A.I.P. Conf. Proc. No. 57, p. 272. [4] B.V. Chirikov, J. Ford and F. Vivaldi, A.I.P. Conf. Proc. No. 57, p. 323. [5] G. Contopoulos, in N.R. Lebovitz, W.H. Reid and P.O. Vandervoort (eds.), Theoretical Principles in Astrophysics and Relativity (Univ. Chicago Press, Chicago, 1978), p. 93. [6] L. Martinet and P. Magnenat, Astron. Astrophys. 96 (1981) 68 (paper I). [7] G. Contopoulos, Bull. Astron. Ser. 3, 2 (1%7) 223. [8] M.N. Rosenbluth, R.A. Sagdeev and J.B. Taylor, Nuclear Fusion 6 (1%6) 297. [9] B.V. Chirikov, Nucl. Phys. Inst. Siberian Section USSR Acad. Sci. Rep. 267; CERN Trans. 71-40 (1%9). [10] L. Martinet, Astron. Astrophys. 32 (1974) 329. [11] F. Gustavson, Astron. J. 71 (1966) 670. [12] A. Giorgilli and L. Galgani, Celest. Mech. 17 (1978) 267. [13] A. Giorgilli, Computer Phys. Commun. 16 (1979) 331. [14] G. Hori, Publ. Astron. Soc. Japan 18 (1%6) 287; 19 (1%7) 729. [15] A. Deprit, Celest. Mech. 1 (1%9) 12. [16] M. Kummer, J. Math. Anal. Appl. 52 (1975) 64. [17] J. Ford and H.G. Lundsford, Phys. Rev. AI (1970) 59.
INVARIANT SURFACES AND ORBITAL BEHAVIOUR IN DYNAMICAL SYSTEMS OF 3 DEGREES OF FREEDOM. II
G. C O N T O P O U L O S European Southern Observatory, Garching bei Miinchen. Germany and
P. M A G N E N A T and L. M A R T I N E T Geneva Observatory, Sauverny, Switzerland Received 3 April 1981 Revised 25 June 1981
Two formal integrals of motion besides the Hamiltonian are f o u n d in a s y s t e m of three degrees of f r e e d o m with three u n p e r t u r b e d frequencies o9~, o92, o93 which are close to, or exactly at, the double r e s o n a n c e 6 : 4 : 3. The integrals are truncated at various orders and they are used to find theoretical invariant surfaces on a surface of section. The theoretical invariant surfaces are then compared with the empirical results. The a g r e e m e n t is improved as the order of the integrals increases from the 5th up to the 1 lth degree. This is surprising because theoretical considerations indicate that one formal integral breaks down b e y o n d order 6, because of the second r e s o n a n c e 4 : 3. ( H o w e v e r it is possible to construct one further integral by a different method). As expected the a g r e e m e n t is worse when the perturbation b e c o m e s large. A check of the c o n s t a n c y of the truncated integrals along various orbits s h o w s that the relative variations b e c o m e smaller as the order increases; they b e c o m e larger as the perturbation increases.
1. Motivation
It is well known that dynamical systems close to integrable ones possess integral surfaces (KAM theorem). Such integral surfaces can be given approximately by asymptotic expansions provided by formal integrals of motion like the "third" integral of galactic dynamics [1]. H o w e v e r there is an important difference between systems of two degrees of freedom and systems of n > 2 degrees of freedom. In the second case the motion in phase space is expected to show Arnold diffusion, making the system practically stochastic, even if the integral surfaces have a large total measure. Such a diffusion does not appear in systems of two degrees of freedom, that have been studied extensively up to now. But the number of studies of Arnold diffusion is limited [2, 3, 4] and its practical applicability has been questioned in certain cases
0167-2789[82/0000-0000/$02.75 0 1982 North-Holland
(see, e.g. [5]). The basic question is whether there exist some restrictions in the effectiveness, or even the appearance, of Arnold diffusion. This question is at the root of the applicability of statistical mechanics in generic dynamical systems. In order to study this problem it would be of interest to study in detail some of the simplest systems in which Arnold diffusion is expected to occur, namely systems of three degrees of freedom. In such systems one can visualize the "invariant surfaces" corresponding to orbits lying on integral surfaces (isolating orbits). T h e r e f o r e one can see empirically whether a given orbit is isolating or ergodic, whether there is diffusion and how fast it goes. Such a visualization is practically impossible in systems of more than three degrees of freedom. Our study uses both numerical experiments and theoretical results to understand the general
G. Contopoulos et al./lnvariant surfaces in 3-D dynamical s y s t e m s
form of the orbits. We are interested especially in finding the onset of unstable and stochastic motion. Then by comparing our results with the already k n o w n results for systems of two degrees of f r e e d o m we hope to understand the onset and the significance of Arnold diffusion.
then we can derive the values of H, @t, qb2, and the form of the integral surface (3) and of the invariant surface (4). We consider the general case of a Hamiltonian with second order lowest order terms
(5)
Ho = (/)10 + (/)20 -[- (~30, 2. Introduction
where
In a recent paper [6] (called h e r e a f t e r p a p e r I) the general characteristics of the orbits in a dynamical system of three degrees of f r e e d o m were studied by constructing empirically their invariant surfaces. In the present paper we find theoretically the invariant surfaces by using the theory of the "third integral" of Contopoulos [1], and c o m p a r e the theoretical with the empirical results. F u r t h e r m o r e we examine how well the formal integrals are c o n s e r v e d along particular orbits. The dynamical s y s t e m is assumed to be described by a time-independent Hamiltonian H = H(x,
(1)
y, z, it, ~, ~).
This is an integral of motion. If we have two more integrals of motion, ~ = qb~(x, y, z, x, 2), z)
(i = 1, 2),
(2)
1
(/)10 = 2 (.~2 ~_A x 2 ) ,
we can eliminate two variables, say 2) and 2, and derive a three-dimensional surface in the fourdimensional space (x, y, z, 2) y, z, ~) = 0
(3)
on which lie the projections of the orbits, that have their initial conditions on it. If we take now a surface of section, say z = 0, we find an "invariant s u r f a c e " in the space
(x, y, ~) [(x,
y, g) = 0,
(4)
which depends on the values of H, ~1 and ~2. If the initial conditions for an orbit are given,
~ 2 0 = ~1( ) ~2 +By2),
1
(/)30 = 2 (~2 _[_ CZ2),
(6)
and A, B, C are positive. In the non-resonant case (i.e. when o91 = A1/2, o92 = B 1/2, and 0) 3 = C 1/2 do not satisfy any linear relation with integer coefficients) we can construct three integrals ~ (i = 1, 2, 3) starting with lowest order terms ~0. The sum of these integrals is the Hamiltonian H = ~ l + ~ 2 + ~3. In the non-resonant integrable case a canonical change of variables brings H to a normal f o r m H = H ( ~ l o , ~20, ~30) in the new variables, where ~io have now the numerical values of qbi. The solutions in the new variables are quasipeiodic with frequencies 05 = o 9 z O H l a d P , o = 6ol + " " " ,
052 = o920H/OdP2o = o92 + " " " ,
053 =
= o ~ 3 + ' " . The higher order terms depend on the values of H, ~1 and qbz. If for a particular orbit there are two resonant relations between 051, 052, 053, this orbit is periodic. The corresponding invariant surface (4) is reduced to one or more invariant points. All orbits lying on the particular surface (3) for the same values of H, (JDI, (~)2, satisfy the same resonance conditions, therefore they are periodic. If there is only one resonance condition between 051,052, 053 the orbit fills a two-dimensional surface in the space (x, y, z, ~) and its invariant surface (4) is reduced to a curve. Up to now we have considered integrable cases. If, however, the Hamiltonian system is o930H/0~3o
F(x,
127
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
128
not integrable, then the periodic orbits are isolated, i.e. there are no three-dimensional surfaces in the 4-dimensional space filled with periodic orbits. F u r t h e r m o r e , near the unstable periodic orbits there are no integral surfaces of the f o r m (3), or invariant surfaces of the f o r m (4). More accurately, we can have integral surfaces, or invariant surfaces, with infinite oscillations, or foldings, which do not close, and the corresponding orbits are not quasi-periodic. Empirically we see that the successive consequents on a surface of section do not f o r m a simple surface, but fill a region of three dimensions in the space ( x , y , ~ ) . Thus we have a " d e s t r u c t i o n " of the invariant surfaces. It is known, f r o m the corresponding problem of two degrees of f r e e d o m , that this destruction is very large if the perturbation is large enough so that we have interaction of several resonances, especially low order resonances [7-10]. This is more evident in s y s t e m s of more than two degrees of freedom, e.g. in s y s t e m s of three degrees of f r e e d o m one can have double r e s o n a n c e s between the frequencies ~1, to2, to3 and these allow a larger diffusion of the consequents of the orbits. If we have r e s o n a n c e s between the unperturbed frequencies to1, oo2, to3, say M I ~ I + n 2 ~ 2 + n 3 ~ 3 = O,
(7)
with n~ integers (not all zero) t h : n we can construct integrals q~, starting with second order terms of the f o r m Ot I
Ol3
~o = - - 4)1o + a2 4)20 + - - 4)30, 0.) 1
£02
(,D3
(8)
where n l o / i + / 1 2 0 : 2 + n30t3 = 0.
(9)
If there is only one resonant relation (7) we can construct two independent integrals, with lowest order terms of the f o r m (8), besides the
Hamiltonian. H o w e v e r if there are two independent relations (7) the quantities o/~ are proportional to o~, therefore ~0 is necessarily proportional to H0. The corresponding integral ~0 differs f r o m H in its higher order terms (see, e.g., [11]), but we cannot construct any further integral by this method. H o w e v e r , the numerical evidence of paper I indicates that, even in the case of double resonance, most of phase space is filled with closed integral surfaces. This indicates that there are other f o r m s of integrals valid in this case.
3. Formal integrals of motion In order to construct formal integrals of motion we solve the Poisson bracket equation {H, 4)} = O,
(10)
where H and 4) are series in the variables x, y, z, ~, ~, ~, starting with second order terms. The successive terms of 4) are calculated by a computer program, once the lowest order terms are given. The program used was constructed by Giorgilli and Galgani [12] (for details see also [13]) and applies to any Hamiltonian in the form of a polynomial, for any number of degrees of f r e e d o m and for any resonances between the unperturbed frequencies to1, to2, to3. The method consists of finding a generating function X, and the corresponding normal form of the Hamiltonian Z. This method does not use any inversions of series, as the v o n Zeipel method, but gives the successive terms of X and Z explicitely, step by step. In this respect it is similar to the methods of Hori [14] and Deprit [151. The main interest of the method of Giorgilli and Galgani is the treatment of resonant and near resonant cases. Suppose that the ratios O91." 602 : ¢.O3 are equal or nearly equal to the ratios of the integers ~1 :/z2 :/z3. Then at each order, s,
G. Contopoulos et al./Invariant surfaces in 3-D dynamical s y s t e m s
the normal form Z (~) and the generating function X(~) are derived from two equations Z (s) = F ~ ) and D X ~) = F ~ ), where D is the operator D = i E3,=] ~n(p~3/3p, - q , 3 / 3 q , ) and the subscripts N and R denote the null space and the range of the function F (~), which is known. (If q]~p ~'q ~p ~ . . . is a term of order s it belongs to the null space if /~1(J~- k0 + / ~ 2 ( h - k2) + . . . . 0, otherwise it belongs to the range). After X is found the construction of integrals of motion, starting with a given lowest order function, is easy. In the present paper we consider the Hamiltonian
H = ½(g2 + ~2 + ~2 +
Ax 2+
-- EXZ 2 -- ~ y z 2,
By2 + C z 2) (11)
where A = 0.9, B = 0.4 or 0.38, C = 0.225. Thus the unperturbed frequencies 091 = A l/z, 092 = B 1/2, 093= C 1/2, have ratios equal, or close to 091 : 092:093 = 6 : 4 : 3. We have always 091 : 093 = 2 : 1 but the other resonance 092:093 = 4 : 3 is only approximate in the near resonance case B = 0.38. Thus in this case we can have two subcases: (a) A single resonance if we set/~1 = 60, ~3 = 30 and/x2 = 39 (because we never go to terms of order t>39); and (b) a double resonance if we set/x~ = 6,/x2 = 4, /x3 = 3. In the first subcase we have the drawback that X contains small divisors of the form (3092-4093). In the second subcase X does not contain any small divisor terms, but we can construct only one integral besides the Hamiltonian. If we have only the resonance 091:093= 2 : 1 we can construct two integrals ~1, ~2 with lowest order terms
129
These integrals do not produce any small divisor terms up to degree 6. H o w e v e r the terms of degree 7 and higher contain terms with divisors (3092-4093). Thus the convergence of these integrals is bad. On the other hand if we use the double resonant form of the generating function )C we can construct only one integral besides the Hamiltonian, starting with the combination 4 q~o= cplo+ TS- C#2o.
(14)
.~ ¢.o2
However, as X and Z can be constructed to any order, we can also continue the construction of the "integrals" ~ , q~2 to any higher order, although they do not satisfy eq. (10) b e y o n d the order 6. To our surprise the invariant surfaces constructed with the help of the functions q~], q~2 gave better and better agreement with the empirical results as we went to higher and higher order, up to order 11 (see section 5). This indicates that the functions q~l, q~2 may approximate two integrals of motion. This possibility is enhanced by the fact that the numerical values of q~l and ~2 are better and better conserved along the calculated orbits as we go to orders b e y o n d 6 up to order 10 (see section 6 and table I). The existence of two formal integrals besides the Hamiltonian, both in the near resonance and in the resonance case, is indicated not only by the empirical calculation of orbits of paper I but also by the following considerations. If we use action-angle variables X = (211/o91) 1/2 COS 0 1 ,
.£ = --(211091) 1/2 s i n Ol,
(15) q~]0= 2 410 + 1 430 091
093
(12) and similar expressions for y, ~, z, ~, we can write the Hamiltonian (11) in the form
and H = ¢otlt + 602/2 + 093/3 q~2o= 420.
(13)
-- ~(211/091)1/2(213/093)
130
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
>( [ l c o s 0.-}-~1 c o s ( 0 . 1 - 20.3) + ~1 COS(0.I +
20.3)]
-- r I ( 2 I ~_/o92) 1/z( 213/ o03)
1 × (lc0s0.2+~c0s(0.2-20.3)
+ 1 c0s(0.2 + 203)].
(16)
By canonical t r a n s f o r m a t i o n s of variables we can eliminate all c o m b i n a t i o n s of angles e x c e p t the r e s o n a n t c o m b i n a t i o n s ( 0 " ! - 20"3) and (30"2403). T h u s we find, in the n e w variables H = ~o111+ oJ212+ 00313+ ~(I~) -- E(211[tol)l/2(13/2o93)
COS(0" 1 -- 2193)
+ ~ cos(30"2- 40"3) + • • ",
(17)
w h e r e rl~ is o f o r d e r r/. Using n o w the generating f u n c t i o n S = Jl(0.t - 2"03) + J2(30"2 - 40"3) + J30"3,
(18)
we derive the final c a n o n i c a l a c t i o n - a n g l e variables (Ji, tOi) such that tOl = Ol -- 20"3, rll = Jl,
tO2 = 30"2 -- 40"3,
12 = 3Jz,
~3 = "03,
13 = - 2 J 1 - 4J2 + J3,
250 times larger than in the case "0 = 0.1. This explains w h y the case of small r/ ( r / = 0.1) can be a p p r o x i m a t e d rather well by the formal integrals derived by the p r o g r a m of Giorgilli and Galgani [12], while the a p p r o x i m a t i o n is considerably w o r s e in the case rl = 0.3. If we w a n t to find a m o r e a c c u r a t e third integral, up to any higher degree, we can use a m e t h o d d e v e l o p e d by K u m m e r [16] for a similar double r e s o n a n c e H a m i l t o n i a n given by F o r d and L u n d s f o r d [17]. This integral exists even in the exact r e s o n a n c e case w h e r e one usually e x p e c t s only one integral besides the Hamiltonian. T h e basic idea is to c o n s i d e r the case w~:w2:o~3=6:4:3, " 0 r = 0 as defining a new "unperturbed" Hamiltonian fflo = (2Jl/o~t)l/2(J3 - 4J2 - 2 J 0 cos q/I.
Then,
(21)
if
we add a small p e r t u r b a t i o n --(2¢O3~/E) COS ~b2 we can find a third formal integral in p o w e r s of (2~o:0~/~). T h e existence of this integral explains the a p p e a r a n c e of g o o d invariant s u r f a c e s as f o u n d in p a p e r I.
4. Theoretical invariant surfaces (19) T h e variables u s e d in the p r o g r a m of Giorgilli and Galgani are slightly different f r o m ours. N a m e l y t h e y use the H a m i l t o n i a n
and the H a m i l t o n i a n takes the f o r m H = w3J3 + (3~o2 - 4w3)J2 + ~(J~) - E ( 2 J l [ w l ) 1/2 × [(-2J~ - 4J 2 + J3)/2to3] cos tOl + -~ cos ~2 + • • •
(20)
n
T + ecxGz~ + ~GyGz~.
As the angle 4/3 is ignorable, the action J3 is a s e c o n d integral of m o t i o n besides the Hamiltonian. T h e p e r t u r b a t i o n "0~ cos q~2 is of degree 7 in j~/2. If we ignore this term we find also an a p p r o x i m a t e third integral of m o t i o n , n a m e l y J2 = ~I2. In the p r e s e n t p a p e r we take the total e n e r g y equal to h = 0.00765, • = 0.5 and rl = 0.1 or rt = 0.3. T h e r e f o r e the p e r t u r b a t i o n term is really small. H o w e v e r in the case r / = 0.3, r/~ is a b o u t
(22)
T h e relations b e t w e e n our variables and theirs are
x s = w[/Zx,
~s = ~/oJl/2,
(23)
and similar relations f o r y, ~, z, 2. F u r t h e r m o r e EG:--e/(wVZt.D3) and rls = - r l / ( w ~ / 2 w 3 ) . We calculate the invariant s u r f a c e s in the GiorgilliGalgani variables and then t r a n s f o r m to our
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems variables for c o m p a r i s o n with the empirical results. The two integrals with lowest order terms (12) and (13) are given in the f o r m @i = Z
Cabcdef X ay bz c2 el2)e~/.
(24)
We take the surface of section z = 0, which is a s y m m e t r y plane of the Hamiltonian (11). Thus we have q~i = Z CabOde[xayb2d2) efr':'
(25)
and d is now always even. Using eq. (22) we find 22 = [2h - ~oj(x2 + 22) - o)2(y 2 + 2)2)]/w3,
(26)
where h is the numerical value of the Hamiltonian (h = 0.00765) and we have omitted the subscripts G. We keep now only the coefficients Cab0d~: up to a given degree, N, in the two integrals. N e x t we eliminate 2) f r o m the two integrals (25) and thus derive the equation of the invariant surface (4). One point of the invariant surface is given, namely the initial point (x0, Y0, z0 = 0, 20, 2)0) while z0 is derived f r o m the energy integral). Thus we have also the numerical values of the integrals q~. We find the values of 2 and 2) for a grid of values of x and y with steps Ax and Ay respectively. N a m e l y we calculate 2 and 2) for all points (x +-max, y +-nay) within certain maximum limits. Each solution (2, 2)) at a given point (x, y) is used as an a p p r o x i m a t e initial condition for the next point ( x ± A x , y) or (x, y _+Ay). In general if 20, 2)0 are a p p r o x i m a t e solutions of the equations (25) at a point (x, y) we write q~ as Taylor expansions and truncate them after the first order terms:
q~(2, 2)) = q~d2o, 2)0)+ \ 05c /o 82 + \ 02) Io 62) = ~, (27) where q~ mean the numerical values of q~. Thus
131
we solve the system (27) and iterate until the corrections 82, 62) are smaller than a fixed accuracy, given a priori. After some preliminary exploration with various values of /~x, Ay and of the accuracy, we found that the solution of the eqs. (27) converged very fast, after a few iterations. If no c o n v e r g e n c e was found after 30 iterations, a special message was printed. It was found that lack of c o n v e r g e n c e meant, in general, nonexistence of the relevant solution. Similarly a special message was printed if the calculated 22 was smaller than zero. It was checked that if (2, 2)) is a solution then ( - ~ , - 2 ) ) is also a solution. In fact the expressions (25) are even in (22)). Thus essentially only two solutions (2, 2)) were needed for each point (x, y). It was checked also that for e = ~/= 0 the solutions are (+2, _-+2)), with 2 = (2ci91o-Ax2) 112, 2) = (2qb20- By2) 1/2. The numerical solutions (2, 2)) for the whole grid of points (x +_max, y _+ n a y ) were printed and stored in the computer, and then were used for plotting the invariant surfaces as described in the next section.
5. Comparison between the theoretical and the
empirical invariant surfaces In paper I a systematic numerical investigation of the conditions of existence of good invariant surfaces was initiated for the Hamiltonian (11) with A = 0 . 9 , B = 0 . 4 , C = 0.225, • = 0.5, the total energy h = 0.00765, and various values of r/, for given initial conditions x0, Y0, 20, 2)0 (~0 is inferred f r o m the equation of energy and z0 = 0 by the definition of the invariant surfaces). Stereographic views allow to visualize the shape of the invariant surfaces when existing. A series of cases with B = 0.38 instead of 0.4 were also calculated but not published. F o r the same fixed initial conditions the numerical invariant surfaces were very similar to those corresponding to B = 0.4.
132
G. Contopoulos et aL[Invariant surfaces in 3-D dynamical systems
At present we will c o m p a r e some of such empirical invariant surfaces with theoretical surfaces obtained by the method described in the previous section for the same initial conditions x0, Y0, ~0, ~0 and different values of 7. (We systematically use the variables x* = A~/2x, y* = Bl/2y, z* = CII2z and omit the symbol *). In all the present e x p e r i m e n t s x0 = - 0 . 0 3 7 8 , Y0 = 0.025, ~ 0 = 0 , ~ 0 = 0 . The order in which the coordinate axes are given in figs. 2--4 corresponds to the order xs, ys, Zs indicated in fig. 1, where we also show the angles 0 and ¢ which define the direction of the o b s e r v e r E with respect to the origin of the axes of representation. We shall always use 0 = 30 ° and ¢ = 0 ° whereas the angle of c o n v e r g e n c e of the two " e y e s " of E towards the origin is 15 °. We begin with case B = 0.38. Fig. 2a shows the empirical invariant surface in the space ( x , y , ~ ) obtained for -q = 0 . 1 , which suggests that 2 isolating integrals exist besides the energy in this case. Figs. 2b,c,d show the corresponding theoretical surfaces c o m p u t e d by means of the integrals (25) truncated after the 5th, 8th and l lth degree terms respectively. The scanning steps /Ix and Ay as well as the a c c u r a c y were set equal to 0.001. It is important to e m p h a s i z e that we are able to c o m p a r e the shape only of the numerical and theoretical surfaces and not the detailed distribution of points on these surfaces. The " n u m e r i c a l " surfaces a p p e a r as sets of points Zs /v
8
Ys
I
X
Fig. 1. Definition of the observer's direction E (0,~0) with respect to the origin of the axes.
which are successive consequents produced by the method of surface of section due to Poincar6, as explained in paper I. The "theoretical" surfaces are sets of points resulting f r o m the scanning method described in the previous section. It stands to reason that, in general, some kth " n u m e r i c a l " point does not coincide with the kth "theoretical" point as the sets are obtained through entirely different procedures. It appears f r o m the figures that the shape of the theoretical surfaces is progressively changing and, as more and more higher-order terms are included, it a p p r o a c h e s closer and closer the " n u m e r i c a l " surface. At the l lth stage there is quite good agreement, in the sense that the numerical and the theoretical surfaces have the same extent. In a similar way we have calculated invariant surfaces in the (y, x, ~) space. Such surfaces are not given here, but they show also a quite satisfactory agreement with the empirical surfaces. In both representations no i m p r o v e m e n t could be o b s e r v e d b e t w e e n orders 10 and 11. Fig. 3 shows the invariant surfaces obtained in the case -q = 0.3 represented in the subspace (x, y, ~). H e r e we o b s e r v e discordance between the empirical (fig. 3a) and theoretical surfaces, even in case (d) (order 11). Similar c o m p u t a t i o n s have been p e r f o r m e d with other values of ft. We conclude that for small enough values of 7, let us say smaller than 0.1, the theoretical invariant surfaces obtained by means of the integrals c o m p u t e d with the Giorgilli-Galgani program are good approximations of the real invariant surfaces, provided the formal series representing these integrals are c o m p u t e d to a sufficiently high order. For larger "O, discordances appear. T h e y have been anticipated, taking account of the discussion in section 3. Let us now consider the case B = 0.4. Fig. 4 displays the numerical invariant surface and the theoretical ones corresponding to "q = 0.1 and calculated f r o m the same initial conditions as before. Taking into consideration the discussion of the previous section we think that
G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
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G. C o n t o p o u l o s et a l . l l n v a r i a n t s u r f a c e s in 3 - D d y n a m i c a l s y s t e m s
34
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G. Contopoulos et al./lnvariant surfaces in 3-D dynamical systems
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G. Contopoulos et al./Invariant surfaces in 3-D dynamical systems
136
Table I Relative errors of the integrals &t and ~bz truncated after the terms of Nth degree for various (A = 0.9, B = 0.38, C = 0.225). N
5
6
7
8
9
10
r/ = 0.1 7=0.1
d)l &2
1.84x 10 3 1.53x 10 I
1.38x 10 3 9.46 x 10 2
9.45 x 10-4 7.19x 10-2
7.75 x 10-4 7.19x 10 2
6.86x 10 4 7.19x 10 2
5.61 x 10 4 3.74x 10 2
= 0.3 r/ = 0.3
(6t ,~2
1.03x 10 2 3.72
6.34 x 10 3 5.11 x 10 -~
6.24 x 10 3 7.94x 10-I
3.56 X 10 3 2.15x 10-1
4.41 x 10 3 3.30x 10 -t
2.65 × 1 0 -3 1.27x 10-l
the satisfactory the
present
because
results
case
which
are
even
are obtained more
in t h e e x a c t r e s o n a n c e
integral, besides structed.
surprising,
case only one
the Hamiltonian
(See however,
in
can be con-
the discussion
at t h e
ved when
we reach order
10. A n a l o g o u s
con-
c l u s i o n s c a n b e d r a w n f r o m t h e c a s e B = 0.4. We
conclude
that
the functions
¢1 a n d
¢2
b e h a v e as f o r m a l i n t e g r a l s o f m o t i o n , e s p e c i a l l y w h e n t h e p e r t u r b a t i o n ~ is small.
e n d o f s e c t i o n 3.) F i n a l l y f o r ~ = 0.3 w e o b s e r v e t h e s a m e d i s c o r d a n c e as in t h e c a s e B = 0.38 between the empirical and theoretical surfaces.
Acknowledgement W e w i s h to t h a n k D r . G i o r g i l l i f o r p r o v i d i n g us w i t h an i m p r o v e d v e r s i o n o f his p r o g r a m .
6. The "constancy" of the formal integrals References In a d d i t i o n to t h e c o m p a r i s o n d e s c r i b e d in t h e previous
section we can obtain supplementary
information integrals
on the applicability of the formal
by
periodically
checking
their
con-
stancy along a chosen phase trajectory. Starting from
the same
initial c o n d i t i o n s
have performed
as b e f o r e
we
the integration of the equations
o f m o t i o n f o r 1000 t i m e u n i t s a n d c a l c u l a t e d t h e values every
of the ten
integrals
time
units
o s c i l l a t i o n is p e r f o r m e d
u p to d i f f e r e n t (A
complete
orders
typical
z-
in a b o u t 15 t i m e units).
We then deduce an estimate of the fluctuations o f t h e s e i n t e g r a l s w i t h t i m e . In t a b l e I w e g i v e f o r B = 0.38 t h e r a t i o o f t h e s t a n d a r d d e v i a t i o n to t h e m e a n
v a l u e s o f t h e i n t e g r a l s (~1 o r W2)
c a l c u l a t e d f r o m o r d e r 5 to o r d e r 10, f o r "9 = 0.1 a n d ~ = 0.3. T h e r e s u l t s c o n f i r m t h e c o n c l u s i o n s o f t h e p r e v i o u s s e c t i o n . T h e r e is a s m a l l r e l a t i v e e r r o r f o r b o t h i n t e g r a l s in t h e c a s e ~ = 0.1 a n d t h e f l u c t u a t i o n s d e c r e a s e f r o m o r d e r 5 to o r d e r 10. F o r ~ = 0.3, o n l y qh p r e s e n t s r e l a t i v e l y s m a l l fluctuations.
The
behavior
o f q~2 is n o t
factory although a small improvement
satis-
is o b s e r -
[1] G. Contopoulos, Z. Astrophys. 49 (1%0) 273. [2] C. Froeschl6, Astrophys. Space Sci. 14 (1971) 110; C. Froeschl6 and J.P. Scheidecker, Astrophys. Space Sci. 25 (1973) 373. [3] J.L. Tennyson, M.A. Lieberman and A.J. Lichtenberg, A.I.P. Conf. Proc. No. 57, p. 272. [4] B.V. Chirikov, J. Ford and F. Vivaldi, A.I.P. Conf. Proc. No. 57, p. 323. [5] G. Contopoulos, in N.R. Lebovitz, W.H. Reid and P.O. Vandervoort (eds.), Theoretical Principles in Astrophysics and Relativity (Univ. Chicago Press, Chicago, 1978), p. 93. [6] L. Martinet and P. Magnenat, Astron. Astrophys. 96 (1981) 68 (paper I). [7] G. Contopoulos, Bull. Astron. Ser. 3, 2 (1%7) 223. [8] M.N. Rosenbluth, R.A. Sagdeev and J.B. Taylor, Nuclear Fusion 6 (1%6) 297. [9] B.V. Chirikov, Nucl. Phys. Inst. Siberian Section USSR Acad. Sci. Rep. 267; CERN Trans. 71-40 (1%9). [10] L. Martinet, Astron. Astrophys. 32 (1974) 329. [11] F. Gustavson, Astron. J. 71 (1966) 670. [12] A. Giorgilli and L. Galgani, Celest. Mech. 17 (1978) 267. [13] A. Giorgilli, Computer Phys. Commun. 16 (1979) 331. [14] G. Hori, Publ. Astron. Soc. Japan 18 (1%6) 287; 19 (1%7) 729. [15] A. Deprit, Celest. Mech. 1 (1%9) 12. [16] M. Kummer, J. Math. Anal. Appl. 52 (1975) 64. [17] J. Ford and H.G. Lundsford, Phys. Rev. AI (1970) 59.