the lissajous transformation ii. normalization

T H E L I S S A J O U S T R A N S F O R M A T I O N

I I . N O R M A L I Z A T I O N

ANDRI~ DEPRIT and ANTONIO ELIPE* National Institute of Standards and Technology,

Gaithersburg, MD 20899, U.S.A.

(lZeceived 10 May, 1990; accepted 15 November, 1990)

Abst ract . Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non isolated equilibria.

Key words: Dynamics - HaIniltonian systems - perturbations - reduction - isotropic oscillators

Imus praecipites per mille pericula rerum Turrigerasque arces, rupes et inhospita saxa.

CASTELLESI de CORNETO, de Sermone Latino, Basel 1518.

1. I n t r o d u c t i o n

The Lissajous t r ans fo rmat ion (Depri t 1990) was invented to handle per- tu rbed elliptic oscillators. These systems are described by Hami l ton ians of the type

7-I = 7~o + eF

the principal par t being the Hamil tonian of the elliptic oscillator

7-lo 1 2 y2 w2x2 = 7 ( X + + + j y 2 )

and )2, a funct ion of the Car tes ian variables (x, y, X, Y) and possibly of a small pa r ame te r e. In Section 2, we show how the Lissajous t r ans fo rma t ion yields a very simple a lgor i thm to nonna l ize pe r tu rbed elliptic oscillators, especially when the pe r tu rba t ion is a power series in e over the real a lgebra of polynomials in the Car tes ian variables (x, y, X , Y).

For a demons t ra t ion problem, we select a two-paramete r subset of per- tu rb ing funct ions in the th ree -paramete r family of quar t ic potent ia ls

= 2o~x2y 2 + ~ x4 "~ 7Y 4 (1)

* Permanent address: Grupo de Mec£nica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain

Celestial Mechanics and Dynamical Astronomy 51: 227-250, 1991.

@ 1991 Kluwer Academic Publishers. Printed in the Netherlands.

228 ANDRI~ DEPRIT AND ANTONIO ELIPE

first introduced in galactic dynamics by Andrle (1966). Those are the most general homogeneous polynomials of degree 4 to admit the following symmetries:

(81) the reflection ( x , y , X , Y ) (S2) the reflection (x, y, X, Y)

) ( x , - y , X , - Y ) in the x-axis; ) ( - x , y , - X , Y ) in the y-axis.

When a = 0, Andrle's galactic systems are separable in the Cartesian coordinates, and one can take advantage of this fact to handle the systems as pairs of Duffing's oscillators subject to a perturbation in x2y 2 assum- ing that the parameter a is sufficiently small (Andrle 1969). This approach leads to very complex developments involving elliptic functions; progress in that direction appears possible now that automated symbolic processors for operating with elliptic functions and integrals are becoming available.

There are more parameters in (1) than one needs to make simple, yet informative models for quasi-spherical potentials in galactic dynamics. Caran- icolas and Barbanis (1982) [see also Davoust (1983), Contopoulos (1989), Caranicolas (1984; 1990) or Caranicolas and Varvoglis (1984)] have shown indeed that much can be accomplished with a fanfily in only one parameter, namely

y = (x2 + ~y2)2.

For our part, we seek for simplifications primarily by imposing additional finite symmetries rather than by cutting on the number of parameters.

On the whole, we depart from Andrle's model on two counts. First, taking 7 =/3 in (1), we restrict ourselves to potentials of the type

]2 : 20~x2y2 71_ /3(X4 .~_ y4) (2)

because, in addition to ($1) and ($2), the potential in (2) admits (S3) the reflection ( x , y , X , Y ) ~ ( y , x , Y , X ) i n the first diagonal as a symmetry. The three symmetries ($1), ($2) and ($3) generate a group of order 8 usually referred to as the symmetries of the square. They introduce into the problem peculiar behaviors not typical of perturbed elfiptic oscillators in general. In particular, contrary to what one should expect from less symmetric perturbed oscillators [see, e.g., Miller (1990)], in the present model the position of all singularities after averaging are independent of the parameters. Nonetheless, we retained this model for the purpose of illustrat- ing a normalization in terms of Lissajous variables because we can extricate its structures without much algebraic complication.

What becomes of these symmetries when the Lissajous variables are introduced in (2) will be explained in a subsequent note by Miller (1990). 1

1 In contrast to what happened in galactic dynamics, Prof. Ferrer pointed to us, quantum

THE LISSAJOUS TRANSFORMATION. II: NORMALIZATION 229

We differ also from Andrle in viewing the systems not as pairs of Duffing's oscillators subject to a small pe r tu rba t ion in x 2 and y2, bu t as pairs of ha rmon ic oscillators in the field of 1;. We obta in t h a t l; be small compared to 7-/o not by imposing tha t the parameters ~ and fl be suffciently small, bu t by res t r ic t ing our analysis to a sufficiently small ne ighborhood of the equil ibr ium at the origin in the phase space.

Although the analysis we make is limited to the neighborhood of the origin, we ought to pay some attention to the global shape of the effective potential

~ e ~ 1 2 / 2 ~03 ~ + u s) + Z(~ + y~) + 2 .~y ~,

especially with a view of determining the regions of the parameter plane (c~,/?) where the equipotential curves are open. We identify these regions by locating the singular points of Y~. (i) Needless to say, the effective potential admits a minimum at the origin (0,0) where it vanishes. (ii) Along the line a = fl, the effective potential is the axisymmetric surface

Ve ----~03 r l 2 2 + j3r4 with r 2 = x2 ÷ y2.

The equation O~;2/i)r = 0 admits real roots if and only if fl < 0, in which case the singular points of ];e make a circle of radius

0.) r - with f l l = _ f l .

Along the rim of equilibria, 1;~ reaches a maximum value w4/(167). (iii) Throughout the half plane/? < 0, there are four singular points:

03 03 0J 03 E1(7--~,01, E3(0,~----~), E~(-~,0), E~(0, 2V~) The potential at these points takes the value w4/(16~'). Furthermore, the Hessian being

03U( l÷~)y2-2032x 2 at E1 and E5,

~ 2 ( l ÷ ~ ) x 2 - 2 0 3 2 y 2 at E3 and ET,

there follows that El, E3, E5 and E7 are all centers when ~ > c~ and all saddles when fl < ~. (iv) In the region of the parameter plane where ~i = -c~ - / ~ > 0, there are four more singularities:

0J (M 02 03 03 03 ~.) ¢d

E~(2v~,2v~), E4(-2-~,2v~), e6( 2v~' 2v~ 1' ES(2v~' 2v~ 1

physicists [see, e.g, (Matinyan et al. 1982; Pullen and Edmonds 1981)] started with the one-parameter model (fl - 7 -- 0); extension to the two-parameter model (7 -- 0) came later with the study of quantum chaos by Vorob'ev and Zaslavk~i (1987).

230 ANDRl~ DEPRIT AND ANTONIO ELIPE

At these points, the potential equals w4/i85). The Hessian is

4 w ~ x y - 2w 2 1 + (x 2 + y2) at E2 and E6,

( ° ) - 4 w 2 ~ x y - 2 ~ 2 1+ (x 2+y2) at E4 and Es

There results that all four singularities are saddles when c~ < ~, and all four are centers when c~ > ft. As ~ tends to co, all four equilibria El, E3, E5 and E7 tend to the origin; so do the equilibria E2, E4, E6 and Es when 5 tends to oo. In which case, the domain in which the normalization at the origin could claim to have any kind of relevance is much affected by the presence of these equilibria.

The normalization, we need hardly mention, is carried out in Lissajous variables as a canonical transformation

~ : ( i ' , g ' , L ' , G ' ) ~ ( ~ , g , L , G )

for the purpose of rendering the new elliptic anomaly ~' ignorable in the transformed Hamiltonian v#7-/, thereby making of L', at least formally, an integral of the normalized system. Any integral manifold L ' = constant, we shall show in Section 3, turns out to be a sphere, and the Lissajous transformation supplies a set of coordinates particularly well suited to follow the flow of the normalized system on orbital spheres.

The demonstrat ion problem is separable in the following three cases (Gralmnaticos et al 1983): (i) It is separable in the Cartesian variables when a = 0, the second inte-

gral being the function

1 2 1 2, 2 ~(X4 y4). ~(X - Y 2 ) + ~ w (x - y 2 ) + _

(ii) Noting that the Poisson bracket of the angular momentum with i2),

(G, ~) = 4 ia - ~ ) x y i x2 - y2),

vanishes identically if and only if a -- fl, we infer that the demonstrat ion oscillator is symmetric with respect to the group of rotations

x = x' cos ¢ - yl sin ¢, X = X ' cos ¢ - Y ' sin ¢, y = x' sin ¢ + y' cos ¢, Y = X ' sin ¢ + Y' cos ¢,

hence it is separable in the polar coordinates. [For a full solution in terms of Jacobian elliptic functions, see Lakshmanan et al. (1980)].

(iii) Finally, after application of the canonical transformation

x = i x ' - y')/Vr2, X = ( X ' - Y ' ) / V / 2 , y = (x' + y ' ) / v % Y = ( x ' +

the Hamiltonian of the demonstration system becomes


"]-/= { l(xl2-b~2xl2)-b l(°~[-l~)Xt4 t2 w2yl2) 1 - -

+ + + + )y,4 + '2,

obviously a separable system when a = 3~. In that case, according to Bountis et al. (1982, p. 1262), the second integral is the function

X Y + w2xy + 4~xy(x 2 + y2) = 2wI2. + 4j3xy(x 2 + y2).

In the normalized problem, we shall also encounter those exceptional values of the parameters, this time, however, as the places where the system becomes degenerate, that is, where it adnfits non isolated equilibria (Section 4). We obtain a complete picture for the behavior of the normalized system by following its evolution along a circle centered at the origin in the parameter plane (a,/3) (Section 6). We shall find, as a rule, that the phase flow on an orbital sphere is identical to that of a rigid body in rotation about a fixed point, and this everywhere in the parameter plane except along the lines of degeneracy. These are precisely the places at which a butterfly bifurcation occurs. Nothing out of the ordinary here: sinfilar events have been mentioned in other domains of nonlinear dynamics, most recently in optical polarization dynamics by David et al. (1989b).

2. Normalization by Averaging

The Lissajous transformation A reduces the Hamiltonian 7-/o to the function A#'Ho = wL, and this changes the Lie derivative Lo associated with 7-/o into the single partial derivative Lo = wO/Oe. One characterizes, therefore, the kernel of L0 as the real algebra of functions independent of the elliptic anomaly e, and the image of L0 as the ker(Lo)-module of functions F for which there exists a function F' such that wOF'/Oe = F. Hence, given a function F that is periodic in e with period T, the average

lfo belongs to ker(L0); with F expressed as a Fourier series

r = ~-~ (Cj cos j i + Sj s i n j t ) , j>__0

it is readily seen that (F)t = Co, and that the periodic function

1 F' = ~ -~ (Sj cos je - Cj sin jg)

j>l

is such that Lo(F') = F - (F)~. In other words, when the perturbation is periodic in the elliptic anomaly g~, normalizing a perturbed elliptic oscillator

232 ANDRt~ DEPRIT AND ANTONIO ELIPE

amounts to averaging the dynamical system over g. Put in this perspective, the normalization responds most adequately to what we sense intuitively should be accomplished. The orbits of the system are conceived as ellipses of an elliptic oscillator slowly changing in size (through variations in the actions L and G) and position (through variations in the angle g) as an effect of the perturbation. The latter modifies also the timing along the ellipses as marked by the elliptic anomaly. Yet fluctuations in the orbital time are of short period, and they are smoothed out by the averaging process.

In most instances, the perturbation acting on the elliptic oscillator belongs to the real algebra P in the Cartesian variables (x, y, X, Y). In that context, specialists in automated algebraic calculations will note with inter- est that the mapping

A # : F ( x , y , X , V ) ) A#F(e ,g ,L ,G)

is an isomorphism of the algebra P onto the real algebra .4 of Fourier series of the type

I COS } P = ~ Cj,k sin (Jg +kg) (3)

j_>o, Ikl>_o

with coefficients in the real algebra Q of polynonfials in

and

More precisely, if F is a homogeneous polynomial of degree n in the Carte- sian variables, then the coefficients in the Fourier series A#F are themselves homogeneous polynonfials of degree n in s and d. For instance, in the Lis- sajous variables, the perturbation (2) becomes

/ 1(o~ ..-}- 3/~)(d 4 + 4d2s 2 + s 4) - 3 ( a _ fl)d2s2 cos 4g

- ( a + 3fl)[sd(d 2 + s2) cos2t{ + ½d2s2cos 4g]

A#P = + (a - /3) [sd (s 2 cos(4g + 2g) + d 2 cos(4g - 2e))

184 COS(rig "-}- 4~) -- -14d4 cos(rig - 4~)]

(4)

The algebra ker(L0lA) is identical to the Q-algebra of Fourier series in g. Hence the rule--most readily implemented by machine--for decomposing any series (3) into its components

ro=z 0{cos} sin Jg

j>o


in ker(L01A) and

{co } r ~ = ~ Cj,k sin (Jg + ke),

j>_o,M>o

in im(L0lA).

Normalization in the neighborhood of the equilibrium at the origin in phase space is performed by the classical algorithm of Lie transformations (Deprit 1969). The calculation consists essentially in evaluating sequences of Poisson brackets. It is therefore of importance, especially when the analyti- cal development is carried out by machine in object oriented languages, to note that the calculations take place entirely within the algebra .A. Indeed, without loss of generality, it can be assumed that the perturbation is the sum of homogeneous polynomials in (x, y, X, Y), each of them of degree > 2. Hence A#7 ) belongs to the ideal B of A generated by s '2, sd and d 2. Now, it is readily checked that

0.S n 1 ~ n - 2

OL = anwa ' 0 8 n

Od n 1 _ . . a n - 2

OL = -i'1~°~* '

1 . . . . n - 2 1 . . . A n - 2 - - ~ "1 ~L~J (~ . OG -g'nco ~ , OG

In consequence, for any G in B, the partial derivatives O G / O L and OG/OG are at least of degree 0 in s and d, hence for any F C B and G E B, the Poisson bracket (F, G) belongs to that ideal.

For the demonstration problem, we carried out the normalization only to the first order. One infers at once from (4) that the reduced Hamiltonian is

3 cos 4g'. (5) ~ ' = w L ' + ¼(a + 3/3)(d '4 + 4d'2s '2 + s '4) - $ ( a - f l )s '2d '2

3. Orbital Spheres and Their Flow

The most salient feature of the reduced phase space is that each manifold L' = constant is a two-dimensional sphere, a fact most easily recognized if one adopts the Hopf coordinates (Hopf 1931, p. 654):

,2 r ,2 r 1 = - = L - sin2g', = G' (6) 1'1 ~ L cos 2g', ~ ~ 3

since those coordinates are tied by the relation

1 1/2 S(L ' ) : 1 ' 2 + 1 ' . ~ + 1 ' 1 = a _ • (7)

The points I'3 > 0 (northern hemisphere) on S ( L ' ) correspond to ellipses travelled in the direct sense while those for which I'3 is < 0 (southern hemisphere) represent ellipses travelled in the retrograde sense. Any point


on the equator It3 = 0 corresponds to a segment along a strMght line passing through the origin which is its midpoint . The nor th pole (I~1 = It2 = 0, II3 = U / 2 ) stands for a circle travelled in the direct sense, whereas the south pole (It1 = I~2 = 0, I~3 = - U / 2 ) stands for the same circle bu t travelled in the retrograde sense.

With the Cartesian coordinates (6), one covers an entire manifold L ~ = constant including the circular orbits at the poles. That much cannot be achieved with the normalized Lissajous variables: G' and g~ rather serve as cylindrical coordinates in the reduced phase space; as such, they present pole-like singularities at the classes of circular orbits.

In the Hopf variables, the second order Hamil tonian of our demonst ra t ion problem turns out to be a diagonalized quadrat ic form,

H I = 0aL' + 3(a + 3fl) L,2 + 3(a - / 3 ) ( I '2 I'~) a + 3fl 1,32 (8) 8~02 ~L-=2 2 - 2~02 •

The trajectories as the level contours of 7/~ on the sphere S ( U ) are precisely the curves along which the quadric in (8) intersects the sphere S(L ' ) .

Accordingly, we obtain them numerically by elimination of an unknown in the system made of Equation (7) and of equation 7/~ = constant. The procedure is definitely faster than numerical integration of the reduced differential equations. Besides, whereas tracing a homoclinic orbit by numerical integration is a very del- icate affair, one obtains it as the level curve at the energy level of the equilibrium from which it emanates, and this as easily as any other level curve.

On account of the Poisson brackets

( I '1 , I '2 ) = I '3, (I '2, I ' 3 ) = I l l , (I '3, I '1) = I '2,

the equations of motion in the Cartesian variables (6) are derived immedi- ately from (8) by application of the Liouville-Jacobi theorem:

• 4 o e I~ = (I'1,7-l') = --~ I '2I '3,

]~ = (I'2,7-/') - 2 ( a - 3 / 3 ) I ' I ' (9) ¢d 2 3 1,

]~ : (i,3,7_~/) _ 6 ( a --/3) i,11,2" ~2

4. Equilibria

We begin the analysis of this system by searching for its equilibria. These are in fact the local ex t rema of 7-/~ on the sphere S ( U ) . They are the roots of a system in the four unknowns I~1, I~2, II3 and a Lagrange mult ipl ier #; the system is made of the equations

0 12 I'32)] 0. (i 1 ,2 ,3) [n ' + #(I '~ + I 2 + = =

T H E LISSAJOUS T R A N S F O R M A T I O N . Ih N O R M A L I Z A T I O N

and the constraint (7).

~ = 3

235

Fig. 1. Partitions in the parameter plane.

Solutions fall into two classes. The first is made of non isolated equilibria in the following three cases:

(i) when ~ = 0, the meridian circle (I'1 = 0, I'~ + I '2 = 1L'2);

(ii) when a =/3, the equator (I'3 = 0, I'~ + I'~ = I f, t2].

(iii) when c~ = 3/3, the meridian circle (I'2 = 0, I'~ + I'32 = ~L1 ,2).

The degeneracy cases, it is worth noting, correspond to the separability cases present in the original problem. An explanation as to how the first order normalized system passes through its degeneracies nlight possibly shed some light about the way the original system goes through a case of separability.

The second class of equilibria is made of isolated points. Relevant infor- Ination about them is given in Table I. Here, as for the class of non isolated equilibria, we are not surprised to see the equilibria appearing in pairs of diametrically opposite points. This is due to the central synlmetry

(I'a,II2 1'3) ( - I 'a , I' I' ~ , > - - 2 , - - 3 ) ,

236 ANDRIS DEPRIT AND ANTONIO ELIPE

as a m a t e r of fact , a remote consequence of the symmet r ies of the square we a l ready observed in the original problem. The 'modif ied ' energy ment ioned in the th i rd column of Table I is the normalized Hami l ton ian s t r ipped of its cons tan t terms, or

7-[ ° = 7-i' - w L ' 3(a + 3fl) L,2. 860 2

TABLE I Isolated equilibria

Characteristic 7_/0 equation Type

(1L~ 1 , E1 at ~7 , 0, 0) and E3 at ( - T L , 0, 0)

A2 + 3L,2(~ _ 3fl) (. - # ) = o

Maximum in c and a ~ Minimum in a and c' Saddle in b and b'

3(o~ - fl) L,2 8w 2

I l E2 at ( 0 , ~ L , 0 ) and E4 at (0,-I~L, ' 0)

A2 + 6 L , 2 a ( a _ fl) = 0

Maximum in a and b Minimum in a' and b' Saddle in c and c'

3 ( a - fl) L,2 8w 2

E0 at (0,0, 1 ' (0 ,0 , -½L' ) ~ L ) a n d E s a t

A 2 - 2L'2c~(c~ - 3fl) = 0 Maximum in W and d Minimum in b and c Saddle in a and a'

a + 3fl L ,2 8w 2

Discussion of existence and stabi l i ty takes on an intui t ive character when we consider the pair of parameters c~ and/3 as the Car tes ian coordinates of a point . The equilibria, isolated as well as non-isolated, cause us to divide the pa rame te r plane into six sectors using the lines c~ = 0, c~ = [3 and c~ = 3/3 as boundar ies . Figure 1 shows how we have labelled these regions. To the

THE LISSAJOUS TRANSFORMATION. Ih NORMALIZATION 237

variational equations associated with the system (9),

4a (I'352 + I'2f3), =

~2 = 2(a~23/3)(I'153 + I'3/fl), (10)

~3=. 6(a_w -~-/3) (i,2~1 + I'1~2),

corresponds the characteristic equation

A [A ~ + 12(a - 3/3)(a - /3)I '~ + 24a(a - /3 ) I ' ] - 8a(a - 3/3)1'32] (11)

+ 96a(a - 3 / 3 ) ( a - fl)I'm I '2I '3 = O.

Strictly speaking, the unknown A in (11) represents not the characteristic exponent X but the quantity w2X. The eigenvalue A = 0 is due to the fact that the variations are tied by the identity

I'151 + , 1 2~2 + 0 1 353 =

obtained from varying the constraint (7). By reason of this constraint, the variations must re;nain in the plane tangent to the sphere at the point (I '1,I '2,~3). Inserting the coordinates of the equilibria into (11) produces the non trivial part of the characteristic equation mentioned in the first column of Table I.

What Table I tells about the way the system behaves as its para~neters are varied is rendered more vividly by graphics. According to Figure 1, it is sufficient to follow the evolution on the unit circle (F) centered at the origin in the parameter plane. Thus we take

a : c o s ¢ and / 3 = s i n e with 0 < ¢ < 2 ~ .

In Figure 2, we plot the value of 7l ° at the equilibria when the parameter point (a,/3) travels along (F). A quick glance at this plot reveals that the sector boundaries in the parameter plane are lines of bifurcations. Two things are happening on these lines, (i) an exchange of equilibria where the integral 7-/° takes its absolute maximum or mininmnl, and (ii) a swapping of stability. Take for instance the boundary a = 3t~. In region (a), 7_/o reaches its maximum at the equilibria E2 and E4, and its mininmnl at E1 and E3; across the boundary a = 3fl, in region (b), the same integral is still maximum at E2 and E4 but now becomes minimum at E0 and Es. Furthermore, whereas El, E3 are stable and E0, E5 unstable in region (a), stability is reversed in region (b).

El 2

-I

-2


E2

I

I I I

I

3~ ~ 0

b' c'

3Z ~ 0 a b c a ' a

- . . . . . . . . _ _ _ r

C~

Fig. 2. The integral 7-I o at the equilibria as the point (a, fl) travels a circle centered at the origin of coordinates, starting on the boundary line (a = 0, fl < 0).

5. S o l u t i o n s

In order to characterize the bifurcations on the boundary lines, we must determine what is the global general flow on the orbital spheres inside the various regions mentioned in Figure 1. Fortunately, the demonstrat ion problem is simple enough that the general solutions can be expressed in Jacobian elliptic functions.

Because the equations are left unchanged by the transformation

(I'1 I'2,I'3,t ,a,fl), ~ ( - I ' 1 , I ' I ' , - 2 , - 3 , - t , - a , - / 3 ) ,

any solution obtained for a _> 0 can be turned into a solution for a _< 0, and conversely. Therefore we restrict the analysis to the half parameter plane

> 0. Furthermore, for the sake of simpler notations, we introduce the dimensionless quantities:

a' = L'a/~ 3, ~' = L'~/w 3, H = 47-1°/L'w,

the dimensionless Hopf variables

T H E LISSAJOUS T R A N S F O R M A T I O N . II: NORMALI ZATI ON 239

(~ = 2 ~ i / L ' , (i = 1,2,3)

and an independent variable 7---also dimensionless--such that dT = w d t . In those notations, the equations of the motion become:

d(1 d r - 4 a (2(3,

d(2 _ 2 ( 0 / - 3~t)(,'3(1, (12) d7

d(3 _ - 6 ( a ' -/~')(1(2; d r

their solutions belong to the unit sphere

(~ + (2 + (2 1, (13) 3 ~--

and they admit the integral

H 3 t - - c , - ' ' = ~(a + 3/3')(~. (14)

Not only to make the discussion more concise but also to keep the contact between the mathematical formulas and the geometry of the dynamics, we denote by

H1,3 3 ! 1 t : - - ~ ( o ~ - - at), H2,4 = 3 ( a ' - fit), H0,5 = - ~ ( a - 3/T)

the values of the integral H at the corresponding equilibria. In those notations,

H = Ha,3(~ + H2,4(~ + H0,5( 2. (15)

One gets a firmer grasp on how the solutions evolves as the parameter point moves in its plane by imagining that the level curves of H over the unit sphere (13) constitute lines of equal heights for a landscape on the unit sphere. The relative altitudes of the equilibria are changing from one region to another; the hill, for instance at a certain equilibrium flattens, and eventually turns into a depression, and the corresponding tectonic evolution on the unit sphere forces compensations transforming other equilibria into mountain passes. As a guide to the analysis, the reader will find in the table below a summary of these changes as the point (a,/3) moves along a circle of equations a = p sin X,/3 = P cos X when X goes from 0 to 7r.


B o u n d a r y /3' > ~' = 0 H2,4 = Ho,5

Sec tor (c) fl' > a ' > 0 Ho,5 < H2,4

B o u n d a r y fl' : ~' > 0 Ho,5 < H2,4

Sec tor (b) 3/3' > a ' >/3' > 0 Ho,5 < H1,3

B o u n d a r y 3/3' = oz' > 0 Ho,5 : H1,3

/ %

Sector (a) a ' > max(3/3',0) H1,3 < H0,s

< H1,3

< H1,3

= HI ,3

< H2,4

< H2,4

< H2,4

B o u n d a r y a ' : 0 > /3' H1,3 < H0,5 = H2,4

We now proceed to discuss in detail the solutions of system (12) in the various regions mentioned in Figure 1.

B o u n d a r y / 3 ' > a ' = 0.

Clearly, the Equations (12) represent a differential rotation about the (1- axis at an angular velocity 6/3'~1. Because/3 is > 0, the rotation is direct in the hemisphere ~1 > 0, and retrograde in the opposite hemisphere. All points on the great circle (1 = 0 are stationary; they are all at the energy level H = _3/3, = H2,4 = H0,s.

Sector (c): fl' > a ' > 0.

As the parameter point moves away from the positive side of the/3-axis, a pothole develops at Eo and Es. Indeed, in region (c), according to Table I, Ho,s <: H2,4 < H1,3. Moreover, Ho,s __ H < H1,3. Under these conditions, by elinfinating (2 and (1 in turn from (13) and (15), we find that

(2 2 2 .),1 (a2 (2) and (2 2 2 = - = (16)


the coefficients in these expressions being the real quantities

~/ 2c~1 71 = 3/31 _ c~1,

/3(,8' - o?) ~ 3 : V 3 ~ - ~ ' ,

The difference

• ( H L 3 - Ho ,5) (H- H2,4)

~ H - H0,5 a2 = H2,4 - Ho,5 '

[ H1,3 - H :

is positive above the energy H2,4, and negative below. Thus, we are led to distinguish three regimes of phase flows: around the hills at E1 and E3, in the depressions around Eo and Eh, and along the divide lines joining E2 and E4.

H2,4 < H < H1,3 In this interval, the nlodulus

a2 3(fi 7 : c~') ~ t I lies between 0 and 1. Upon substituting (16) into the differential Equa- tions (12), thereafter introducing the phase ¢ such that (2 = b2 sin ¢, we find that the solution reduces to the quadrature

o = r ( ¢ , k3), de

¢2 : n 2 ( r - to) = ~/1 _ k~ sin2 ~

where n2 is the frequency

n2 = 2 ( f l ' - 3a')7173a2 = 2ff(a ' - /~ ' ) (H - H0,5).

By inversion, there results that ¢ is the Jacobian amplitude am (¢2, k2), hence

~1 : :=[=~fla2 dn(¢2,k2), ~2 = b2sn(¢2,k2), ~3 = T73b2cn(¢2,k2). (17)

Thus, around the hills at E1 and E3, the flow circulates about the (1- axis in the direct sense about the equilibrimn El, in the opposite sense about E3. At the very top where H = H1,3, the solution collapses onto the equilibrium, E1 or E3 since there b 2 = 0, 71a2 = 1, and k2 = 0. i

I H0,5 < H < H2,4 [ The flow in this energy interval is obtained from i

I I

the previous one by the Reciprocal Modulus transformation (Han- cock 1909, p. 249). Thus, putting

242 ANDRI~ D E P R I T AND A N T O N I O E L I P E

n~2 = k2n2 = 2~ /2~ ' (H1 ,3 - H) ,

k: = 1 / k 2 =

¢5 =

in place of n2, k2 and ¢2 in the Equat ions (17), we obtain the solution

~: = 3=71a2cn(¢~,k~), (2 = a2sn(¢~,k~), ~3 =-4-73b2dn(¢~,k~). (18)

On the boundary H = Ho,5, it reduces to the equil ibrium, either Eo or E5. Elsewhere, it defines a circulation of the phase flow, in the retrograde sense around Eo and in the direct sense around Es.

I

Homoclinic orbits at H = H2,4. I We find here the trajectories which r

::lost likely will be called to play a critical role when the pe r tu rbed elliptic oscillator, as is done in Caranicolas (1989; 1990), is analyzed at energy levels far above the min imum at the origin in the Cartesian phase space. Very likely they will prove to be the furrows in which stochasticity sets in and grows, eventually to invade the entire landscape. In tha t regard, it would be very interesting to follow by numerical integrat ion in a Poincar6 section diagram what becomes of the homoclinics we are about to deternfine for increasing values of the energy in the original problem. At the energy level H = H2,4, the roots a2 and b2 are both equal to 1, and the quadrature for (2 reduces to

d~2 2 ~- +2(3fl ' - ol t ) '~l '~3 dr.

Taking (2 = tanh ¢.~, we obtain that

¢.~ = 4-2~/6a ' ( /3 ' - o~ ' ) ( r - TO).

To the upper sign in ¢.~ correspond the solutions

(: - cosh ¢~ (2 = tanh ¢~, (3 - ¢~ ' cosh "

They are made of two half great circles going from E5 at T ---- - - ~ to Eo at 7 = c~, on both sides of the diameter (1 = (3 = 0 in the same nleridian plane (M+). On the other hand, to the lower sign in ¢~r per tain the solutions


-4-~1 -4-"}, 3 C1 - cosh ¢-----~2" ~'2 = tanh ¢7, ~'3 - cosh ¢7

going from E0 at 7 = - o o to E5 at r = oo. Like the previous ones, they belong to the same meridian (M_) where each of them is just the half circle joining E0 and Es. To unders tand the kind of bifurcations taking place at the boundaries between sectors in the pa ramete r plane, one should observe that , as a ~ ~ 0, both planes (M+) and M_ tend to the meridian plane ~'1 = 0 whereas they come to coincide with the equatorial plane C3 = 0 when a ~ --+ /3 ~. More will be said later about the transit ions at the boundaries.

Now that we have outl ined the steps one should take to get the solutions expressed in elliptic functions, we propose to shorten our exposition by simply stat ing what the solutions are in the other cases.

B o u n d a r y / 3 1 = ~' > O.

All along tha t half line, H1,3 = H2,4 > Ho,5. The flow is due to a differential rota t ion about the ~'3-axis at the angular velocity -4a1~'3, thus retrograde in the nor thern hemisphere around E0 and direct in the southern hemisphere around Es.

S e c t o r (b): 3/3' > a' > [3' > O.

At the same t ime that depressions were developing at Eo and Es, the ground was swelling under E2 and E4 so much so that eventually, when the paramete r point crosses the boundary line, H0,s < H1,3 < H2,4. Throughou t Sector (b), we consider the functions

/313'- ~' / . - -o,~ ~2= V ~ , al=VH--(,3:.[-~o,5,

/3(~'- z')' / "~,~ - "

= < l T,,o is o, the <o,,,,

(:'1 : al s n ( ¢ l , k , ) , G = :[:(~2a, c n ( ¢ l , k , ) , ~3 = -t-(~3b1 d n ( ¢ l , k l ) .

with the modulus kl and the frequency n] defined as follows:

I / ,,

244 ANDRt~ D E P R I T AND ANTONIO E L I P E

and the phase ¢1 = ?~1( T - TO). The flow circulates about Eo in the retrograde sense at the same t ime that it circulates about E5 in the direct sense.

I H1,3

ceding solution, we find that

¢1 = b l s n ( ¢ ~ , k 4 ) , ¢2 = -at-(~2al dn(¢4,k~),

with

k4 = 3(~' /3,) - ~ - YI~ '

I < H _< H2,4 I By Reciprocal Modulus t ransformat ion in the pre-

¢3 = +53b1 cn(¢ 4, kl)

n 4 = 2~/3(a ' - / T ) ( H - Ho,5),

and ¢~ = n~ (r - To). The phase flow is in direct circulation about the equilibrium E2, and in retrograde circulation about E4.

I Homoclinic orbits at H = H1,3. [ The argument of the hyperbol ic J

[

functions being

¢4' = 2 ~ / 3 ( . ' - Z')(3Z' - . ' ) ( T - T0),

we obtain

+52 T53 ¢1 = tanh ¢]1, ¢2 - cosh ¢4 I' ¢3 - cosh ¢-------~11

as the homoclinics going from E3 at r = - o o to E1 at 7 = oc whereas

+52 ±53 ¢1 = - tanh ¢]1, ¢2 - cosh ¢4 I' ¢3 - cosh ¢-----~{

s tand for the two homoclinics passing from E1 to E3 as ~- goes from - o o to + ~ .

B o u n d a r y 3/'3' = a ' > 0. [ I

The meridian plane ¢2 = 0 is the energy level curve at which H = H1,3 = Ho,s. The flow consists of a differential rotat ion about the ¢2-axis at the angular velocity 4a~¢2, thus in the direct sense in the henfisphere ¢2 > 0, and retrograde in the other hemisphere.

S e c t o r (a) : ~ ' > max(3Z' , 0).

For the ampl i tudes in tha t par t of the parameter plane, we must take

THE LISSAJOUS TRANSFORMATION. II" NORMALIZATION 245

2oJ ~1 : V 3 ( j : ~t ) ,

/ <~' - 3 / ~ ' =

/ H2, 4 -- H

a3 = VH-~,4 :j-H--~, 5 ,

/. H__H__~ H_I, 3 b3 : VL,2,4_

I H1,3 < H < Ho,5 ] The modulus and frequency being ]

2a' ( H - _//H) 2 q ( a ' 3fl')(H2,4 H), k 3 = O/~-~/,j, ~-2~ 4 ' n 3 = -- _

the solution is given by the equations

~1 = - I - e l a 3 d n ( ¢ 3 , k3), ~2 = :Fe2b3cn(¢3 ,k3) , 6 = b 3 s n ( g g 3 , k3)

for the phase (/)3 : 7"/'3(7" - - TO) . There is direct circulation of flow about the equihbrium/773 and retrograde circulation around Ea.

Ho,5 < H < H2,4 The Reciprocal Modulus transformation changes

frequency and modulus into the quantities

I 0~1-- 3flt (H- HH) ~i/'t3 : 2q2o~t(y- H1,3 ) k~ = 2--a7 H---:2,~ '

and yields the solution

6 = Te, a3cn(¢Lk~) , ¢2 = +e2b3dn(¢~,k~) ¢3 = a3sn(¢~,k~) ,

the phase argument being ¢I = n~(r-7-0). The flow rotates in the direct sense about E2 and, in the opposite sense around E4.

Homoclinic orbits at H = Ho,5. [ The two solutions going fi'om E5 to i

I

E0 are "at- el Tel

¢1 - cosh ¢~' ¢2 - cosh ¢~' ~'3 = tanh ¢,~,

whereas the two solutions going from Eo to Es are + q -4-el

¢ 1 - cosh ¢~' ¢2 = cosh ¢~' ¢3 = - tanh ¢~.

cases, the phase argument is ¢~ = 2 ~ / 2 a ' ( a ' - 3/3'). In both

B o u n d a r y a ' = 0 >/7'.

We have returned to a situation analogous to the one we found on the boundary at the top of our analysis, with one difference though, namely that the differential rotation has changed sense. It is retrograde in the hemisphere ¢1 > 0, and direct in the opposite hemisphere.


G

\ ~>~=o

# < c ~ = 0 , G

Fig. 3. Evolution of the phase flow through the butterfly bifurcations.


6. Butterf ly Bi furcat ions

It is time to encapsulate all the details we belabored into a global picture to give a sense of the evolution in the averaged phase flow in dependence to its parameters. By numerical integration, for L' = 1, we obtained various solutions of the differential system (9) at several points along the unit circle n the parameter plane, in particular at those points where the circle crosses a boundary line. These trajectories are plotted on a orthographic projection of the orbital sphere. By electronic collage, the diverse spheres have been assembled in Figure 3. The resulting picture summarizes most vividly the salient points of the analysis we made case by case in Section 5. It is most informative, for instance, to find out in the diagram how the flow reverses its sense of circulation as the parameter point, initially somewhere on the positive side of the/%axis journeys through regions (c), (b) and (a) to end on the negative side of the same axis.

On the sphere at the top of the figure, the numerical integration confirms that the flow incurs a rotation around the ~l-axis. We learned from the analysis--not from the numerical in tegrat ion-- that all points in the meridian C1 = 0 are equilibria. From watching the flow on the two spheres just below, we observe how the system got into that state of degeneracy: through a b u t t e r f l y bifurcation. 2 The meridian of stationary points opens up with the (2 serving as an hinge (the butterfly opens its wings!). On the edge of the wings are inscribed the homoclinic orbits emanating from the equilibria E2 and E4, the only two equilibria to survive the coming out of the bifurcation in the meridian plane (1 = 0.

As the parameter point clears the next boundary line (a =/3) , the butterfly spreads its wings more and more openly until they come to coincide with the equatorial plane C3 = 0. At that moment, all points, and not only E0 and Ej, become stationary, and the system enters a state of degeneracy. Yet no sooner has the butterfly folded its wings than it reopens them, this time using the Cl-axis as the unfolding hinge.

The process repeats itself on the next boundary: the wings folding in the meridian plane (2 to reopen along the (3-axis, until they are fully spread in the meridian ~'1 = 0. Notice how, in this state of degeneracy, the circulation is now retrograde on the front side of the sphere, and direct on the back side.

In sum, it has taken three butterfly bifurcations for the system to revert its sense of circulation in the flow.

2 The name of butterfly bifurcation is borrowed from David et al. (1989a) who discovered this type transition while studying a single laser pulse propagating as a single travelling wave in an anisotropic lossless medium.

248 ANDRt~ D E P R I T AND A N T O N I O E L I P E

7. Conclusions

We demonstrated on an example how well suited is the Lissajous transformation for handling a wide class of elementary and not so elementary non linear systems. In conjunction with an algebraic processor, it leads to an- swers of an algebraic nature in a mat ter of minutes. Furthermore, most of the significant results can be turned without much effort into informative graphics, a facility most physicists appreciate very much. Of course, the technique is limited to low energies, in the neighborhood of a central equilibrium. Above a certain threshold, one should not expect that the formal series developed by Lie transformations would stay close for a very long time to the solutions of the original problems. Nevertheless, comparing the situ- ations we encountered at low energies with the Poincar6 section diagrams, we find that a number of features we identified at low energies persist at the higher levels analyzed by Caranicolas and others. It seems, for instance, that the zones of stochasticity located in the original problem first emanate h'om the neighborhood of the unstable equilibria identified in the normalized version, owing to the fact, probably, that the full perturbation canses the homoclinic solutions to intersect.

The method we have used to uncover the phase structure of the normalized system points to a general approach to this class of problems. The Lissajous transformation played an essential role in that it yielded 'pseudo- coordinates' in which to express the averaged Hamiltonian, much like the components of the angular velocity for a solid body free to rotate about a fixed point. The behavior of the elliptic oscillator at low energies, we have seen, is commanded by a Hamiltonian that is a quadratic form in the pseudo-

T, I ' ,2± ,2 ,2 L2/4 . We propose coordinates ~ 1, 2 and I'3 over the sphere 1 1 T I 2 + 1 3 - -

that such quadratic Hamiltonians be studied for themselves, and that the phase flow they deternfine be exanfined for its dependence on the coefficients in the quadratic form taken as parameters, irrespective of the particular dynamical system whose normalization might be reduced to that quadratic form. This note gives reasons to believe that the general analysis of Ha- miltonian systems represented by a quadratic form in pseudo-coordinates promises to be quite useful in studying wide classes of perturbed integrable systems.

Acknowledgements

Support for this research came in part to the first author (A. D.) from the Applied and Computational Mathematics Program of the D.A.R.P.A. (Washington). The second author (A. E.) gratefully acknowledges having been awarded a fellowship by the European Space Agency to spend the academic year 1988-1989 at the U. S. National Insti tute of Standards and


Technology.

We are g ra te fu l to our col league Dr. Bruce Miller for hav ing appl ied his

g raph ics sof tware in r e d r a w i n g all our f igures, in p a r t i c u l a r for the collage

in the las t f igure d la M a u r i t s Cornel ius Escher .

R e f e r e n c e s

systems. Bountis, T.,

Painlev~ Caranicolas,

209-215.

Andrle, P.: 1966, "A third integral of motion in a system with a potential of the fourth degree," Bulletin of the Astronomical Institutes of Czechoslovakia 17, 169-175.

Andrle, P.: 1969, "The stabihty problem of oscillations along the axis of symmetry in a galaxy," Bulltein of the Astronomical Institutes of Czechoslovakia "I. The first order perturbation in non-resonant case," 20 317-322; "II. The first order perturbation in a general resonance case," 21 132-139 (1970); "III. Lyapunov's linear theory and a generaluzation of the results," 23 40-147 (1972).

Caranicolas, N. and Barbanis, B.: 1982, "Periodic orbits in nearly axisymlnetric stellar "Astron. and Astroph. 114, 360-366. Segur, H. and Vivaldi, F.: 1982, "Integrable Hamiltonian systems and the property," Phys. Rev. A 25, 1257-1264. N.: 1984, "Periodic orbits in a resonant dynamical system." Cel. Mech. 33,

Caranicolas, N. and Varvoglis H.: 1984, "Families of periodic orbits in a quartic potential." Astron. and Astroph. 141,383-388.

Caranicolas, N.: 1989, "Some properties of an intrinsically degenerate Hamiltonian system," J. Astrophys. Astr. 10, 197-202.

Caranicolas, N.: 1990, "A mapping for the study of the 1/1 resonance in a galactic type Hamiltonian," Celes. Mech. 47, 87-96.

Cayley, A.: 1895, An elementary treatise on elliptic functions, Constable and Company, Ltd, London; reprinted by Dover Pubhcations, Inc., New York, 1961.

Contopoulos, G.: 1989, "Asymptotic curves and escapes in Hamiltonian systems," Eu- ropean Southern Observatory Preprint # 677, accepted for publication in Astron. Astroph..

Davoust, E.: 1983, "Periodic orbits in elliptical galaxies," Astron. Astroph. 125, 101-108. David, D., Hohn, D. D., and Tratnik, M. V.: 1989, "Integrable and chaotic polarization

dynamics in nonlinear optical beans," Physics Letters A 137, 355-364. David, D., Hohn, D. D., and Tratnik, M. V.: 1989, "Horseshoe chaos in a periodically

perturbed polarized optical beam." Physics Letters A 138, 29-36. Deprit, A.: 1969, "Canonical transformations depending on a small parameter," Celes.

Mech. 1, 12-30 . . . . Deprit, A.: 1991, "The Lissajous transformation. I: Basics," Celest. Mech. 51, 203-224

(this issue). Grammaticos, B., Dorizzi, B., and Ramani, A.: 1983, "Integrability of Hamiltonians with

third- and fourth-degree polynomial potentials," J. Math. Phys., 24, 2289-2295. Hancock, H.: 1909, Lectures on the Theory of Elliptic Functions, John Wiley and Sons,

reprinted by Dover Publications, Inc., New York, 1958. Hopf, E.: 1931, "Uber die Abbildungen der dreidimensionalen Sph£re auf die Kugelfl/~che."

Math. Ann. 104, 637-665. Lakshmanan, M. and Kaliappan, P.: 1980, "On the energy levels of an isotropie anharmonic

oscillator," J. Phys. A: Math. Gen. 13, L299-L302, (1980). Lakshmanan, M. and Sahadevan, R.: 1985, "Coupled quartic anharmonic oscillators,

Painlev6 analysis, and integrability," Phys. Rev. A 31,861-876. Matinyan, S. G., Savvidi, G.K. , and Ter-rutyunyan-Savvidi, N. G.: 1982, "Stochasticity

of classical Yang-Mills mechanics and its elimination by using the Higgs mechanism," JETP Lett. 34, 590-593.


Miller, B. R.: 1991, "The Lissajous transformation. III: Parametric bifurcations." Celest. Mech. 51,251-270 (in this issue).

Pullen, R. A. and Edmonds, A. R.: 1981, "Comparison of classical and quantal spectra for a totally bound potential," J. Phys. A: Math. Gen. 14, L477-L484.

Vorob'ev, P. A. and Zaslavskff, G. M.: 1987, "Quantum chaos and level distribution in the model of two coupled oscillators," Soy. Phys. JETP 65,877-881.

the lissajous transformation ii. normalization

Documents