Transient times, resonances and drifts of attractors in dissipative rotational dynamics

0.75 MB·

by  A. Celletti

Transient times, resonances and drifts of
attractors in dissipative rotational dynamics
4
1
0
Alessandra Celletti Christoph Lhotka
2

n DipartimentodiMatematica DipartimentodiMatematica
a
J Universita`diRomaTorVergata Universita`diRomaTorVergata

7
ViadellaRicercaScientifica1 ViadellaRicercaScientifica1
1

I-00133Roma(Italy) I-00133Roma(Italy)
]
h
([email protected]) ([email protected])
p
-
h
January 20, 2014
t
a
m
[

Abstract
1
v
In a dissipative system the time to reach an attractor is often influenced by the pecu-
8
7 liaritiesofthemodelandinparticularbythestrengthofthedissipation. Inparticular, asa
3 dissipativemodelweconsiderthespin–orbitproblemprovidingthedynamicsofatriaxial
4
satellite orbiting around acentral planetandaffected bytidaltorques. Themodelisruled
.
1
bytheoblateness parameterofthesatellite, theorbitaleccentricity, thedissipative param-
0
4 eter and the drift term. We devise a method which provides a reliable indication on the
1 transienttimewhichisneededtoreachanattractorinthespin–orbitmodel;themethodis
:
v based onananalytical result, precisely asuitable normal form construction. Thismethod
i provides also information about the frequency of motion. A variant of such normal form
X
usedtoparametrizeinvariantattractorsprovidesaspecificformulaforthedriftparameter,
r
a
which in turn yields a constraint - which might be of interest in astronomical problems -
betweentheoblateness ofthesatelliteanditsorbitaleccentricity.
Keywords.Spin–orbitproblem,transienttime,dissipativesystem,attractor.
1 Introduction
Weconsideranearly–integrabledissipativesystemdescribingthemotionofanon–rigidsatel-
lite under the gravitational influence of a planet. The motion of the satellite is assumed to be
Keplerian;thespin–axisisperpendiculartotheorbitplaneanditcoincideswiththeaxiswhose
momentofinertiaismaximum.Thenon–rigidityofthesatelliteinducesatidaltorqueprovok-
ingadissipationofthemechanicalenergy.Thedissipationdependsuponadissipativeparame-
terandadrift term.Ifthedissipationwere absent,thesystembecomesnearly–integrablewith
1
the perturbing parameter representing the equatorial oblateness of the satellite. The overall
model depends also on the orbital eccentricity of the Keplerian ellipse. This problem is often
known as thedissipativespin–orbitmodeland it has been extensivelystudied in the literature
(see, e.g., [5], [7], [22]).
The spin–orbit model exhibits different kinds of attractors, e.g. periodic, quasi–periodic
andstrangeattractors(comparewith[1], [2], [9], [15]). Asitoften happensindissipativesys-
tem, the dynamics evolves in such a way that the attractor is reached after an initial transient
regime of motion. The prediction of the transient time to reach the attractor is often quite
difficult (see, e.g., [18], [19]), but it is obviously of pivotal importance to test the reliability
of the result (think, e.g., to the problem of deciding about the convergence of the Lyapunov
exponents). The first goal of this paper is to give a recipe which allows to decide the length
of the transient time, namely the time needed to go over the transient regime and to settle the
system into its typical behavior. Our study is based on the construction of a suitable normal
form fordissipativevectorfields (see [8], compare also with[13], [16], [20], [24]) that gener-
alizes Hamiltonian normal forms that are usually implemented around elliptic equilibria (see
[14]). We compute the frequency in the normalized variables and use it - as well as its back–
transformationto the original variables - for a comparison with a numerical integrationof the
equations of motion. Several experiments are performed as the strength of the dissipation is
varied.Itshouldbekeptinmindthatindissipativesystemsonehastotunethedriftparameter
in order to get specific attractors, since it does not suffice to modify the initial conditions like
intheconservativecase([3],[6]).Adifferentformulationofthenormalform,preciselyasuit-
able parametric representation of invariant attractors, allows to obtain an explicit form for the
drift on the attractor. Taking advantage of the physical definition of the drift term, precisely
as a function of the eccentricity ([23], see also [11]), one can derive interesting conclusions
onalinkbetween theoblatenessparameterand theeccentricityassociatedtoagiveninvariant
attractor.Webelievethatthisconstraintmightbeusefulinconcreteastronomicalapplications.
Thispaperisorganizedasfollows.InSection2wepresenttheequationsofmotionofthe
spin–orbit problem in the conservative and dissipative cases. The construction of the normal
form is developed in Section 3, while the parametric representation of invariant attractors is
providedinSection 4.Theinvestigationofthetransienttimeandtheanalysisofthedriftterm
areperformed in Section5. Someconclusionsare drawninSection 6.
2 The spin–orbit problem with tidal torque
In this Section we describe the spin–orbit model, providing the equation of motion in the
conservativecase (Section 2.1) and under the effect of a tidal torque, due to the internal non–
rigidityofthesatellite(Section 2.2).
2.1 The conservative spin–orbit problem
The spin–orbit model describes the dynamics of a rigid body with mass m, say , that we
S
assumetohaveatriaxialstructurewithprincipalmomentsofinertiaI I I .Thesatellite
1 2 3
≤ ≤
2
moves under the gravitational effect of a perturbing body with mass M. Moreover, we
S P
makethefollowingassumptions:
i) thebody orbitsonaKeplerianellipsearound ;wedenotebyaandethecorrespond-
S P
ingsemimajoraxis and eccentricity;
ii) the rotation axis of is assumed to coincide with the direction of the largest principal
S
axisofinertia;
iii) thespin–axisisassumedtobealignedwith theorbitnormal;
iv) allotherperturbations,includingdissipativeeffects, areneglected.
In order to simplify the notation, we normalize the units of measure; precisely, the mean
motion GM (where is the gravitational constant) is normalized to one. An important role is
a3 G
playedby thefollowingquantity,which isnamedtheequatorialellipticity:
3I I
2 1
ε − .
≡ 2 I
3
Todescribetherotationof withrespectto ,weintroducetheanglexspannedbythelargest
S P
physicalaxis(thatweassumetolieintheorbitalplane)withtheperihelionline(seeFigure1).
The Hamiltonian function describing the spin–orbit model under the assumptions i)-iv)
is(see[5])
y2 ε a
(y,x,t) = ( )3cos(2x 2f), (1)
H 2 − 2 r −
where y is the momentum conjugated to x, r is the orbital radius and f is the true anomaly.
Hamilton’sequationsassociated to(1)are givenby
a
y˙ = ε( )3sin(2x 2f)
− r −
x˙ = y ,
whichare equivalenttothesecond–orderdifferentialequation
a
x¨+ε( )3sin(2x 2f) = 0. (2)
r −
Remark1 a) The parameter ε = 3I2−I1 plays the role of the perturbing parameter: the
2 I3
Hamiltonian (1) is integrable whenever ε = 0, which corresponds to the equatorial
symmetry I = I . For almost spherical bodies, like the Moon or Mercury, the value of
1 2
εis oftheorderof 10−4.
b) It is important to stress that (1) is a non–autonomous Hamiltonian function, due to the
factthatr andf areKeplerianfunctionsof thetime.Introducingtheeccentricanomaly
u, defined in terms of the mean anomaly ℓ (which is a linear function of time) through
0
3
Figure 1: The geometry of the spin–orbit problem: orbital radius r, semi-major axis a, true
anomalyf,rotationanglex.
the well–known Kepler’s equation ℓ = u esinu, the orbital radius and the true
0
−
anomalycanbedeterminedbymeans ofthefollowingKeplerianexpressions:
r = a(1 ecosu)
−
1+e u
f = 2arctan tan . (3)
r1 e 2
(cid:16) (cid:17)
−
c) TheHamiltonian(1)dependsparametricallyontheorbitaleccentricityethroughr and
f provided by (3). We remark that in the case of circular orbits, equation (2) becomes
integrable,sincer isconstantandf coincideswithtime(upto a shift).
Expandingr andf givenin(3)inpowerseriesofe,theFourierexpansionofequation(2)
can bewrittenas
+∞
m
x¨ + ε W( ,e) sin(2x mt) = 0 , (4)
2 −
m6=0X,m=−∞
where we introduced the coefficients W(m,e), decaying as powers of e (see, e.g., [5]). Using
2
acompactnotation,wewrite(4)as
x¨+εV (x,t) = 0 , (5)
x
where V = V(x,t) is a time–dependent periodic function (the subscript x denotes derivative
withrespect to theargument).In particular, weconsidera trigonometricfunction by retaining
in (5) just the most important harmonics (see [5]). Precisely, keeping the same notation V for
thetrigonometricapproximation,wedefine
1 5 13
V(x,t) ( e2 + e4)cos(2x 2t)
≡ − 2 − 4 32 −
h
7 123 17 115
+( e e3)cos(2x 3t)+( e2 e4)cos(2x 4t) . (6)
4 − 32 − 4 − 12 −
i
4
We now introduce the definition of a p : q spin–orbitresonancefor p,q Z with q > 0
as aperiodicsolutionof(4), say t R x = x(t) R,such thatit satisfies ∈
∈ → ∈
x(t+2πq) = x(t)+2πp forany t R .
∈
Theaboveexpressionimpliesthattheratio betweentheperiodofrevolutionandtheperiodof
rotationisequaltop/q.ItiswidelyknownthattheMoon,alikemostoftheevolvedsatellitesof
the Solar system, move in a 1:1 spin–orbit resonance (usually referred to as thesynchronous
resonance); within the Solar system only Mercury moves in a non–synchronous resonance
([10], [12]), precisely in a 3:2 spin–orbit resonance1, since twice the orbital period is equal to
thricetherotationalperiodwithinan erroroftheorderof10−4 (see[5]).
2.2 The dissipative spin–orbit problem
Duetoassumptioniv)ofSection2.1,dissipativeeffectshavebeendiscardedand inparticular
weneglectedtheeffectofthenon–rigidityofthesatellite.Thiscontribution,whichturnsoutto
bethemostrelevantdissipativeeffect,inducesatidaltorque([21],[23]),whichcanbewritten
as afunction dependinglinearlyon theangularvelocityx˙:
T
(x˙;t) = K L(e,t)x˙ N(e,t) . (7)
d
T − −
h i
In theaboveexpressionwehaveintroducedthefunctionsLand N as
a6 a6
˙
L(e,t) = , N(e,t) = f
r6 r6
(recall that r and f are known functions of the time). Moreover, the coefficient K is the
d
dissipativeconstant,whoseexplicitexpressionisgivenby
k R M
K 3n 2 ( e)3 ,
d
≡ ξQ a m
where n denotes the mean motion (that we have normalized to one), k is the so–calledLove
2
number(see [17]), the constant ξ is defined through I = ξmR2 with R denoting the equa-
3 e e
torialradius,Qiscalled thequalityfactor(providingthefrequencyofoscillationwithrespect
to therate ofdissipationof energy, [17]). In order tocompare thesizeof thedissipativeeffect
with that of the conservative part, we notice that astronomical measurements provide a value
forK oftheorderof10−8 for theMoonorMercury.
d
In the following we reduce the tidal torque by considering (as in [11]) its average over
oneorbitalperiod.Inparticular,takingtheaverageof(7)withrespecttotimeoneobtains(see
[23])
¯ ¯(x˙) = K L¯(e)x˙ N¯(e) (8)
d
T ≡ T − −
h i
1Theastronomicalconsequenceofa1:1resonanceisthatthesatellitealwayspointsthesamefacetothehost
planet.Mercury’s3:2spin–orbitresonancemeansthat,almostexactly,duringtwoorbitalrevolutionsaroundthe
Sun,Mercurymakesthreerotationsaboutitsspin–axis.
5
with
1 3
L¯(e) (1+3e2 + e4)
≡ (1 e2)9/2 8
−
1 15 45 5
N¯(e) (1+ e2 + e4 + e6) . (9)
≡ (1 e2)6 2 8 16
−
In conclusion, the following differential equation describes the spin–orbit problem under the
dissipativeeffect duetothetidaltorque:
a 3
x¨ +ε sin(2x 2f) = K L¯(e)x˙ N¯(e) . (10)
d
r − − −
(cid:16) (cid:17) h i
Asin (5), weusea compactnotationre-writing(10)as
x¨ +εV (x,t) = µ(x˙ η), (11)
x
− −
where we have introduced µ and η as follows: µ K L¯(e), η N¯(e)/L¯(e). As we can see,
d
≡ ≡
µ depends on the dissipative constant (as well as on e), and therefore we call it dissipative
parameter,whileη isjustafunctionoftheeccentricity,andwe callit thedriftparameter.
Remark2 Thetidaltorquein(11)vanishesforx˙ = η;inviewof(8),thetidaltorquevanishes
asfaras
N¯(e) 1+ 15e2 + 45e4 + 5 e6
x˙ = 2 8 16 . (12)
≡ L¯(e) (1 e2)3(1+3e2 + 3e4)
2
− 8
When e = 0 equation (12) implies that x˙ = 1, which corresponds to the synchronous reso-
nance.FortheactualMercury’seccentricityamountingtoe = 0.2056,(12)providesthevalue
x˙ = 1.256, while for future use we notice that e = 0.285 corresponds to x˙ = 1.5, namely the
3:2resonance.
3 A normal form construction
Our next task is to develop a normal form which transforms (11) into a system of equations
which is normalized up to a given order (see [8]). This allows us to compute a normalized
frequency, which will provide useful information on the dynamical behavior of the model
describedby (11).
Letus write(11)as thefirst–orderdifferentialsystem
x˙ = y
y˙ = εV (x,t) µ(y η) . (13)
x
− − −
Let us denote the frequency vector of motion associated to the one–dimensional, time depen-
dent equation (13) as ω(y) = (ω (y),1). Assume that the vector field (13) is defined on a set
0
6
A T2,whereA Risanopenset.Lety Abeaninitialconditionsuchthatthefrequency
0
× ⊂ ∈
ω = ω (y ) satisfiesthefollowingnon–resonancecondition:
0 0 0
ω m+n > 0 for any (m,n) Z2 , n = 0 .
0
| | ∈ 6
Welookforatransformationofcoordinatesdefined uptoasuitableorderN Z inε,µ,say
+
∈
Ξ : A T2 R T2, suchthatthenewvariables are(Y,X)with
N
× → ×
(Y,X,t) = Ξ (y,x,t), Y R , (X,t) T2 . (14)
N
∈ ∈
In thetransformed setofcoordinateswerequirethat theequationsbecome:
˙
X = Ω(Y;ε)+O (ε,µ)
N+1
Y˙ = µ(Y η)+O (ε,µ), (15)
N+1
− −
where O (ε,µ) denotes a function whose Taylor series expansion in ε, µ contains only
N+1
monomialsεjµm withj +m N +1.
≥
According to [8], the transformation (14) is obtained as the composition of two transfor-
mations.Thefirstonebringstheoriginalvariables(x,y,t)intointermediatevariables(x˜,y˜,t),
soto removeterms dependingonε;then, from(x˜,y˜,t)we implementanotherchangeofvari-
ablesto (X,Y,t) insuch away toobtain(15).
Thenormalform(15)isparticularlyuseful,sinceneglectingO (ε,µ)onecanintegrate
N+1
thesecond equationas
Y(t) = η +(Y η)e−µ(t−t0) , (16)
0
−
wherewedenoteby(X ,Y )theinitialconditionsattimet = t inthenormalformvariables.
0 0 0
The expression (16) shows that, in the approximationobtained neglecting higher order terms,
the solution tends to Y = η as time tends to infinity. Inserting (16) into the first of (15) we
obtainthedependenceof X˙ ontime, whoseintegrationprovidesX = X(t) withX(0) = X .
0
Indeed the solution (16) provides the natural attractor, which can be found in the original
coordinatesbyintegratingequations(13)withε = 0.
The local behavior near quasi–periodic attractors of some dissipative systems, precisely
conformallysymplecticsystems2, has been studied in [4]. The main result of [4] is that there
exists a transformation of coordinates such that the time evolution becomes a rotation in the
angles and a contraction in the actions. The normal form (15) is consistent with such result:
indeed, neglecting higher order terms, the expression (16) shows that the normalized action
contracts exponentiallyin thedissipativeparameter, whilethe first of(15) showsthat the lim-
itingbehaviorofthenormalized angleis alinearrotationwithfrequency Ω(η).
Fordetailson thenormal form algorithmused to obtain(15)we refer to [8]; here wejust
state the final result. At the normalization order N = 3 the normalized frequency Ω(Y;ε) in
2Aflowf : definedonasymplecticmanifold isconformallysymplectic,whenf∗Ω = eµtΩ
forsomeµretalwMith→ΩtMhesymplecticform.Noticethat(13)iMsaconformallysymplecticflowaccordtingtosuch
definition(see[3]).
7
(15)turnsouttobe
ε2 1 5 98
Ω(Y;ε) = Y +ε2 e2 + + (17)
− 8(Y 1)3 8 (cid:18)(Y 1)3 (3 2Y)3(cid:19)
h
− − −
1 63 3444 578
e4 + +
64 (cid:18)−(Y 1)3 (2Y 3)3 − (Y 2)3(cid:19)
− − −
1 31280 390 45387
e6 + + ,
768 (cid:18)(Y 2)3 (Y 1)3 (3 2Y)3(cid:19)
i
− − −
where we expanded the coefficients up to the 6th order in the eccentricity. The expansion of
thedrift η = N¯(e)/L¯(e)(with N¯, L¯ as in (9))tothesameorderisgivenby:
3e4 173e6
η = 1+6e2 + + . (18)
8 8
Equations(15)togetherwith(17),(18)willbeusedinSection5.2foradynamicalinvestigation
based on a normal form approach; precisely, we will backtransform the frequency Ω in the
originalvariablesandcomparetheresultwithanumerical integration.
4 A parametric representation of invariant attractors
Withreferencetoequation(11),weintroduceinthisSectionaparametricrepresentationofan
invariant KAM attractor with Diophantine frequency; as it is well known ([3], [6]), the equa-
tionsfortheembeddingcanbesolvedundersuitablecompatibilityconditions,whichprovidea
relationbetweenthefrequencyandthedrift.Inparticular,suchcompatibilityconditionsallow
us to provide an explicit computation of the drift that we perform up to the 4-th order in the
series developmentin the perturbing parameter. These results are used in Section 5.3 in order
to investigate in some specific cases the relation between the drift and the frequency, as well
as thedependenceon theotherparameters (mostnotablytheoblatenessand theeccentricity).
Let us recall that the frequency vector of motion associated to (11) is written as ω =
(ω ,1).Wesay thatω satisfiestheDiophantinecondition,whenevertheinequality
0
ω m+n −1 C m forall (m,n) Z2 , m = 0 (19)
0
| | ≤ | | ∈ 6
is satisfied for some positive real constant C. Next we provide the following definition of a
KAMattractorfor(11).
Definition3 A KAM attractor for (11) with rotation number ω = (ω ,1) satisfying (19) is a
0
two–dimensionalinvariantsurface,describedparametricallyby
x = θ +u(θ,t), (θ,t) T2 , (20)
∈
where the flow in the parametric coordinate is linear, namely θ˙ = ω , and where u = u(θ,t)
0
isa suitableanalytic,periodicfunction,suchthat3
1+u (θ,t) = 0 forall (θ,t) T2 . (21)
θ
6 ∈
3Therequirement(21)guaranteesthat(20)isadiffeomorphism.
8
LetD bethepartialderivativeoperatordefined as
∂ ∂
D ω + . (22)
0
≡ ∂θ ∂t
Inserting (20) in (11) and using the definition (22), it is readily seen that the function u must
satisfythedifferentialequation
D2u(θ,t)+εV (θ+u(θ,t),t) = µ ω +Du(θ,t) η . (23)
x 0
− −
(cid:16) (cid:17)
NoticethattheinversionoftheoperatorD2 provokestheappearanceofthewell–knownprob-
lem of the small divisors ([5]). An approximate solution of (23) can be found as follows. Let
usexpandu and η in Taylorseries ofεas
∞ ∞
u(θ,t) = u (θ,t)εk , η = η εk . (24)
k k
X X
k=1 k=0
Inserting(24)in(23)and equatingsameorders in ε,oneobtainstheiterativeequations
D2u (θ,t)+µDu (θ,t) = V (θ,t)+µη
1 1 x 1
−
D2u (θ,t)+µDu (θ,t) = V (θ,t) u (θ,t)+µη
2 2 xx 1 2
−
...
D2u (θ,t)+µDu (θ,t) = S (θ,t)+µη , (25)
k k k k
whereattheorderk thefunctionS isknownanditdependsonthederivativesofV aswellas
k
on u , ..., u . At each order one determines first η so that the right hand sides of (25) have
1 k−1 k
zero average. After having determined η , the k–th equation in (25) can be used to find u as
k k
thesolutionofthefollowingequation:
D2u (θ,t)+µDu (θ,t) = S (θ,t) , (26)
k k k
where S has zero average (in fact, S = S S¯ , wheree the bar denotes the average over θ
k k k k
−
andt). Tosolve(26), letus expandu inFourierseries as
e ek
u (θ,t) = uˆ(k) ei(mθ+nt) ,
k mn
(mX,n)∈Z2
whereuˆ(k) denotethe(unknown)Fouriercoefficientsofu .InsertingtheFourierseriesin(26)
mn k
andexpandingalsotheleft handsideinFourierseries,we obtain
(ω m+n)2 +iµ(ω m+n) uˆ(k) ei(mθ+nt) = Sˆ(k) ei(mθ+nt) , (27)
− 0 0 mn mn
(mX,n)∈Z2h i (m,Xn)∈IS(k)
where (k) denotes the set of the Fourier indexes of S . Equation (27) allows to determine u
IS k k
as
ˆ(ke)
S
u (θ,t) = mn ei(mθ+nt) . (28)
k
(ω m+n)2 +iµ(ω m+n)
(m,Xn)∈IS(k) − 0 0
9
Notice that the assumption (19) on ω guarantees that u is well defined (no zero divisors
0 k
appear in (28)). An alternative (weaker) assumption would be that ω m + n > 0 for all
0
| |
(m,n) (k).
∈ IS
Due to the fact that the function V is assumed to be a trigonometric function (see (6)),
alsoS istrigonometricand itis convenientto writeit as
k
S (θ,t) = Sˆ(k,c)cos(mθ +nt)+Sˆ(k,s)sin(mθ +nt)
k mn mn
(m,Xn)∈IS(k)h i
for suitable real coefficients Sˆ(k,c) and Sˆ(k,s). Then in place of (26) we can write the solution
mn mn
inareal formwhichis suitablefornumericalcomputationsas
Sˆ(k,c)
mn
u (θ,t) = (ω m+n)cos(mθ+nt) µsin(mθ +nt)
k − (ω m+n)[(ω m+n)2 +µ2] 0 −
mX,n∈ISh 0 0 (cid:16) (cid:17)
Sˆ(k,s)
mn
+ (ω m+n)sin(mθ+nt)+µcos(mθ+nt) .
(ω m+n)[(ω m+n)2 +µ2] 0
0 0 (cid:16) (cid:17)i
In conclusion,thealgorithm to computethedrift consistsin solving iterativelyequations (25)
to obtain the functions u ; at each order, by imposing that the right hand sides have zero
k
average, weobtaintheterms η oftheseries expansion(24)ofthedrift.
k
We provide here the η expanded up to the third order (the fourth order can be obtained
k
through a reasonable computer time, but the expression becomes too long to be displayed
here):
η = ω
0 0
η = η = 0
1 3
a2 b2
η =
2 −2(ω 1)(µ2 +4(ω 1)2) − (2ω 3)(µ2 +(3 2ω)2)
− − − −
c2
(29)
− (2ω 1)(µ2 +4(1 2ω)2)
− −
with
5e2 13e4 7e 123e3 1
a = 1 + , b = , c = e2 115e2 51 .
− 2 16 2 − 16 −6 −
(cid:0) (cid:1)
The above solution for η defines the drift parameter in (24) up to finite order; this expression
will be used in Section 5.3 to obtain in particular a constraint between the parameters of the
model(precisely,theoblatenessparameter andtheeccentricity).
5 Transient time and attractor’s drift
We devote this Section to the discussion of some dynamical features of the attractors associ-
ated to the model described by equation (11). In particular, we provide a numerical method
10