Mon. Not. R. Astron. Soc. 364, 253–271 (2005)
doi:10.1111/j.1365-2966.2005.09572.x
Optimized Nekhoroshev stability estimates for the Trojan asteroids
with a symplectic mapping model of co-orbital motion
C. Efthymiopoulos1 and Z. Sándor2
1 Research
Centre for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efessiou 4, GR-11527 Athens, Greece
of Astronomy, Eötvös University, Pázmány Péter stny. 1/A, H-1117 Budapest, Hungary
2 Department
Accepted 2005 August 26. Received 2005 July 26
ABSTRACT
This paper reports analytic estimates of the domain of Nekhoroshev stability for the orbits of
Jupiter’s Trojan asteroids calculated in the space of proper elements (D p , e p ), for a stability
time exceeding the age of the Solar system (t stability = 1010 yr). The model used is a family of
Hadjidemetriou mappings, for different values of the proper eccentricity e p , that represent the
Poincaré sections of co-orbital motion in the Hamiltonian of the planar and circular restricted
three-body problem. These explicit mappings are shown to reproduce accurately the dynamics
that is implicitly induced by the corresponding Hamiltonian model. Optimal Nekhoroshev
estimates are obtained by constructing the Birkhoff normal form for symplectic mappings.
Our optimization is based on an ‘iterated remainder’ criterion. The asymptotic behaviour of
the Birkhoff series is determined by a precise analysis of the accumulation of small divisors
in the series terms at consecutive orders of normalization. About 35 per cent of asteroids from
a recent catalogue (AstDys), with proper inclination I p 5◦ , are shown to be Nekhoroshevstable over the age of the Solar system. By calculating a resonant Birkhoff normal form, this
percentage increases to 48 per cent.
Key words: celestial mechanics – minor planets, asteroids – Solar system: general.
1 INTRODUCTION
A key concept in the study of non-linear dynamical systems is the
concept of exponential stability. This refers to Hamiltonian systems
of n degrees of freedom of the form
H (J , φ) = H0 (J ) + H1 (J , φ),
(1)
where (J , φ) are n-dimensional action–angle variables, H 0 and H 1
are the integrable part and the perturbation, respectively, and the
Hamiltonian (1) satisfies appropriate steepness and analyticity conditions. Then for a sufficiently small the drift of all orbits in an
open domain of the space of actions is exponentially slow, i.e. the
following statement holds true:
|J (t) − J (0)| < O( α ) for all t T , with T = O e(0 /) ,
b
(2)
where α, b and 0 are parameters depending on the number of
degrees of freedom and the particular form of H 0 and H 1 . The
proof of (2) is the content of the theorem of Nekhoroshev (1977) –
see also Benettin et al. (1985), Lochak (1992) and Pöshel (1993).
In the case of motion around an elliptic equilibrium, the effective
perturbation is identified to the amplitude of oscillations, i.e. the
distance of orbits from the equilibrium point. One may thus obtain
E-mail: [email protected] (CE); [email protected] (ZS)
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2005 The Authors. Journal compilation times of stability that are exponentially long in the inverse of the distance from the elliptic equilibrium (Giorgilli 1988; Bazzani, Marmi
& Turchetti 1990; Fassò, Guzzo & Benettin 1998; Guzzo, Fassò &
Benettin 1998; Niederman 1998). Conversely, by fixing the time T,
equation (2) can be used to estimate the size of a domain of stability
around the equilibrium point, i.e. the maximum distance that can be
reached by an orbit that remains within the domain of stability for
a time at least equal to T. In Solar system dynamics, T is equal to
the age of the Solar system.
Before the theorem of Nekhoroshev, particular examples of exponential stability were given in the literature by Moser (1955) and
Littlewood (1959a,b). However, the notion of exponential stability was overshadowed for a number of years by the parallel notion
of KAM stability. The KAM theorem (Kolmogorov 1954; Arnold
1963a,b; Moser 1962) asserts stability for all times of those orbits
with initial conditions belonging to a Cantor set of tori, of non-zero
measure, on which the phase flow is invariant. On the other hand,
exponential stability applies to all the orbits in open domains of
phase space, independently of whether a particular orbit does or
does not lie on an invariant torus. This global character of exponential stability renders it physically relevant to the characterization of
stability in generic Hamiltonian dynamical systems.
The example given by Littlewood (1959a,b) deals with a model
of great interest for celestial mechanics: the Lagrangian equilateral configuration in the restricted three-body problem (RTBP). The
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C. Efthymiopoulos and Z. Sándor
Birkhoff normal form, and (b) the use of better variables as coordinates (polar instead of rectangular). The use of a computer program
that performs the perturbative expansions is essential, as it allows
one to calculate explicitly the size of the most important terms in
the remainder series, which, in turn, determine the size of the domain of Nekhoroshev stability. Supplemented with a lemma on the
accumulation of round-off errors, a computer-assisted estimate obtained in the above way acquires the status of a real theorem [see, for
example, the computer-assisted proof of KAM stability in a model
Hamiltonian system by Celletti, Giorgilli & Locatelli (2000)]. As
explained below, the obtained estimates are quite precise even without a rigorous treatment of the round-off error.
We should point out that the use of the circular RTBP as a model
for Trojan dynamics is already a great simplification. This neglects
all the phenomena caused by the overlapping of resonances in the
secularly changing elliptic RTBP. In particular, the common practice
of averaging over short-period terms with a period equal to Jupiter’s
mean motion renders the averaged Hamiltonian of the planar and
circular problem integrable. However, in the case of co-orbital motion, the influence of the short-period terms in determining the size
of the stability region is non-negligible. In the present paper we
demonstrate how the small divisors introduced by the circular RTBP
model alone affect the asymptotic behaviour of the Birkhoff series
and, thereby, Nekhoroshev stability. We may conclude that using
a Hamiltonian model averaged over the short-period terms should
be avoided in the study of Nekhoroshev stability, although it is
harmless in the determination of proper elements (Beaugé & Roig
2001), which, by definition, refer to nearly integrable orbits. This
implies that, besides its mathematical challenge, the optimization
of Nekhoroshev estimates in the circular case already represents
a physically interesting problem. The generalization of these estimates in the elliptic and secularly changing elliptic RTBP is an
obvious next step that would bring the theoretical estimates closer
to the physical reality.
In a recent paper (Efthymiopoulos 2005), improved estimates
of the size of the domain of Nekhoroshev stability of the Trojan
asteroids were obtained by using a mapping model of co-orbital
motion (Sándor, Érdi & Murray 2002) in the framework of the planar and circular RTBP. These estimates were expressed in terms
of the asteroids’ proper element values. The region shown to be
Nekhoroshev-stable corresponds to asteroids with proper eccentricity e p = 0 and proper inclination I p = 0 (with respect to Jupiter’s
orbit). Asteroids with libration amplitudes up to Dp = 10.◦ 6 were
shown to be Nekhoroshev-stable for a time of at least 1010 yr. The
theoretically calculated libration amplitude is equal to about onethird of the value of the maximum observed libration amplitude
for real asteroids (35◦ ). A comparison with real asteroids with
e p 0.07 and I p 10◦ , from the catalogue of proper elements of
Milani (1993), showed that about 28 per cent are included in the
theoretically calculated region of stability. However, from the 667
named objects in the recent AstDys catalogue,1 only 11 per cent with
I p 5◦ and e p 0.03 are included in the so-determined stability
region.
Now, our recently obtained improvement with respect to previous estimates was partly due to the choice of better canonical variables, that is, modified Delaunay-like variables. Nevertheless, the
use of a mapping model means that these estimates are not strictly
comparable to previous estimates on the same problem which were
obtained with the Hamiltonian model of the circular RTBP. This
following proposition is proved: If a massless particle is placed in
the vicinity of either one of the two Lagrangian equidistant equilibrium points (L 4 or L 5 ), the particle will remain in the neighbourhood
of the same points for a time at least as long as
Tstability O
exp
1
log µ 1/2
,
(3)
where µ is the mass parameter (ratio of mass of less massive primary
to total mass of primaries), and is the amplitude of oscillations
around L 4 or L 5 in a linearized approximation of the equations of
motion. The proof of equation (3) is based on an analytic calculation
of formal integrals, i.e. asymptotic series of the canonical variables
corresponding to approximate integrals of motion valid over exponentially long times. The production of such integrals in an indirect
way, i.e. by the normal form of Birkhoff (1920), lies also at the core
of the so-called ‘analytical part’ of Nekhoroshev’s theorem. The
logarithm of in the estimate (3) can be removed by a better treatment of the number theoretical properties of the ratio of fundamental
frequencies of oscillation around L 4 and L 5 . As a result, one is led
from equation (3) to equation (2). Furthermore, in the framework of
the elliptic restricted three-body problem, it is possible to show that
the necessary conditions for applying Nekhoroshev’s theorem are
fulfilled in a phase space open neighbourhood around the triangular
points (Benettin, Fassò & Guzzo 1998).
The best-studied example of a Lagrangian equilateral configuration in our Solar system is the population of Trojan asteroids at the
L 4 and L 5 equilibrium points of the Sun–Jupiter system (see Érdi
1997, for a review). The dynamics of the Trojan asteroids has been
the subject of intense research in recent years, mostly by numerical
(e.g. Levison, Shoemaker & Shoemaker 1997; Tsiganis, Dvorak &
Pilat-Lohinger 2000; Tsiganis, Varvoglis & Dvorak 2005; Dvorak
& Schwarz 2005; Robutel, Gabern & Jorba 2005) or semi-analytical
(Milani 1993; Beaugé & Roig 2001; Sándor & Érdi 2003) methods.
Such investigations have revealed a rich resonant structure at the
boundary of the domain of stability around L 4 or L 5 . Unravelling
this structure seems hardly tractable by current implementations of
Nekhoroshev theory. The latter, instead, seeks to determine a rigorous estimate of the size of the domain of practical stability around
L 4 or L 5 for a time equal to the age of the Solar system, which is
obtained by analytical means.
The discovery of exoplanetary systems offers new grounds for
the implementation of Nekhoroshev theory. In particular, if there
is a giant planet in the habitable zone of an exoplanetary system,
a habitable terrestrial planet can exist near the stable Lagrangian
points of the star–giant planet system (Érdi & Sándor 2005). Thus
the techniques of Nekhoroshev theory offer an analytical way to
prove the existence of such stable regions around the equilateral
Lagrangian points.
The present paper presents optimized estimates of the size of the
domain of stability of Jupiter’s Trojan asteroids in our Solar system,
in the framework of the planar and circular RTBP.
The first realistic estimates of the size of the region of Nekhoroshev stability around L 4 or L 5 , using the circular RTBP, were given
by Giorgilli & Skokos (1997) and Skokos & Dokoumetzidis (2001).
The size of the region of stability found by these authors, for a
Nekhoroshev time T = 1010 yr, was about one-tenth of the size
of the domain within which real asteroids are observed. Four real
objects were included in the determined region of stability. This
was a serious improvement compared to previous estimates on the
same problem, which gave a size of only about 104 km (Simó 1989;
Celletti & Giorgilli 1991). The improvement was due to (a) the use
of a computer program to perform the algebra of calculation of the
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remark notwithstanding, there are a number of reasons for which a
mapping model is clearly preferable over a Hamiltonian model in
the context of the study of Nekhoroshev stability. We mention here
the two main reasons:
(a) The natural choice of origin for perturbative expansions in the
Trojan problem is the family of short-period orbits that surround the
triangular equilibrium points. One such orbit corresponds to one elliptic fixed point in a Poincaré surface of section of the Hamiltonian
model of the circular RTBP, for a value of the Jacobi constant equal
to that of the periodic orbit. Similarly, this orbit corresponds to an
elliptic fixed point in the mapping model of co-orbital motion, for
a constant value of the proper eccentricity. However, in the mapping case the perturbative expansion can be explicitly carried out in
the neighbourhood of the elliptic fixed point, since its position, as
well as the mapping equations, are explicitly known. On the other
hand, in the Hamiltonian case the short-period orbit can only be calculated numerically. Thus, the only analytically known centre for
a Birkhoff expansion is the Lagrangian equilibrium point, i.e. the
limit of the short-period orbits for e p = 0. This fact poses a severe
restriction in the study of Nekhoroshev stability. Let us note that,
in the case of the elliptic RTBP, there are no triangular equilibrium
points, and therefore there is no analytically known starting point for
a Birkhoff construction in the Hamiltonian case. On the other hand,
the corresponding 4D symplectic mapping has elliptic fixed points
(Sándor & Érdi 2003), which correspond again to bifurcations of
the short-period family. Therefore, a Birkhoff construction around
these points is always possible in the mapping model, while it is not
possible in the Hamiltonian model.
(b) Linked to the above remark, the study of stability in the neighbourhood of the short-period family is closer in spirit to Nekhoroshev’s theorem itself. In fact, this approach is equivalent to studying
Nekhoroshev stability in the 1:1 ‘single resonant’ domain of the action space (Morbidelli & Guzzo 1997). A point at the centre of this
domain, i.e. a short-period orbit, is transformed to an elliptic fixed
point of the symplectic mapping for a specific value of e p .
In view of the above remarks, our choice was to repeat the calculation of Nekhoroshev stability by constructing a mapping model
for co-orbital motion, via the Hadjidemetriou (1991) method, in a
way that reproduces as accurately as possible the dynamics of the
Hamiltonian model of the planar and circular RTBP. At the same
time, we extended the calculation to a much wider class of mapping models representing the Poincaré surface of section of the
Hamiltonian model for a number of values of the proper eccentricity up to e p 0.21. This allowed us to obtain estimates covering
almost the entire domain of proper elements e p and D p where real
asteroids are observed. The construction of the mapping models,
as well as a number of comparison tests with the corresponding
Poincaré mappings in the exact Hamiltonian model, are described in
Section 2.
In the previous paper (Efthymiopoulos 2005), the Nekhoroshev
estimates were obtained by a direct construction of formal integrals,
i.e. without use of a normal form. This is the discrete analogue of
the Contopoulos (1960) ‘third integral’ (see Contopoulos 2002 for
a review of formal integrals). In the present paper the integrals are
constructed by the Birkhoff normal form, as explained in Section 3.
There are a number of advantages to this approach. The crucial quantity is the remainder of the normal-form series at a particular order of
truncation r. Fixing r, the remainder is a convergent series starting
with terms of order r + 1, the size of which determines the size of the
domain of Nekhoroshev stability for any fixed time of stability. Our
analytical treatment is based on an optimized analytical calculation
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2005 The Authors. Journal compilation 255
called ‘iterated remainder’. This is an improvement of the iterated
remainder used by Bazzani et al. (1990) in their analytical proof of
exponential stability in isochronous symplectic mappings.
Section 4 presents the outcome of our new optimized analytical treatment. In summary, about 35 per cent of asteroids with
proper elements taken from AstDys, with I p 5◦ , are shown to be
Nekhoroshev-stable over the age of the Solar system. The size of
the estimated domain of Nekhoroshev stability is about half the size
of the domain occupied by real asteroids in the space of proper elements. Besides being, to our knowledge, the best available Nekhoroshev estimates in the literature, these estimates are quite realistic,
rendering the present analytical method competitive to other numerical methods of determination of the domain of stability. The
main resonances (small divisors) responsible for the asymptotic behaviour and, eventually, divergence of the formal series are analysed. A non-rigorous extension of our results by using a resonant
Birkhoff normal-form construction suggests a still higher percentage
(48 per cent) of Nekhoroshev-stable asteroids. Section 5 summarizes
the main conclusions from the present study.
2 THE MAPPING MODEL
2.1 Derivation of the mapping model
The Hamiltonian of the circular and planar restricted three-body
problem is most conveniently expressed in terms of modified
Delaunay-like variables (Brown & Shook 1964):
x=
a
− 1,
a
x2 =
τ = λ − λ ,
a ( 1 − e2 − 1),
a
(4)
,
(5)
where a is semimajor axis, e is the eccentricity, λ is the mean longitude, is the longitude of the pericentre of the asteroid, and the
primed variables refer to Jupiter. The Hamiltonian is
H = H0 − µR,
(6)
where
H0 = −
1
− (1 + x)
2(1 + x)2
(7)
is the integrable part (Kepler’s problem in a frame rotating with the
mean motion of Jupiter),
µ = m /(m 0 + m ) = 0.000 953 875 36
is the mass parameter (m 0 is the mass of the Sun, m is the mass of
Jupiter), and R is the perturbing function. The latter can be expressed
as a function of the above canonical variables and of the eccentric
anomaly E:
R=
1
1
1
− − r cos φ + ,
r
2
= (r 2 − 2r cos φ + 1)1/2 ,
r = a(1 − e cos E),
φ = τ + v − M,
v − M = 2 arctan
√
(1 + e)(1 − e) tan (E/2) − E + e sin E.
A mapping model for Trojan asteroids can be derived by using
the method proposed by Hadjidemetriou (1991), which is based on
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C. Efthymiopoulos and Z. Sándor
√
(x − 1)( 1 − e2 − 1) is a first integral of the mapping (14), which
represents the connection between x (or practically a) and e. In fact,
a constant value of x 2 refers to an almost constant value of e because
the variation (a − a )/a is quite small. Thus, instead of x 2 we rather
use e (when x = 0) as the parameter of the mapping. Furthermore,
on the mapping plane, one has for the proper eccentricity of the
asteroid e p = e + e, so that e p = e in the circular case. Therefore,
the adopted values of e are practically equal to proper eccentricity
values e p . In the sequel we shall use e p to denote both the value of
the eccentricity on the Poincaré section and the value of the proper
eccentricity.
In order to calculate a normal form for the mapping (14), the
latter should be expressed in polynomial form. This is achieved by a
two-variable Taylor expansion around the period-one elliptic fixed
point (τ ∗ , x ∗ ), which is located in the neighbourhood of L 4 . This
fixed point is determined by finding the corresponding root of the
mapping (14). Introducing the new variables u and v through τ =
τ ∗ + u and x = x ∗ + v, the expansion for a fixed initial e p has the
form
the averaged Hamiltonian of the problem. According to Sándor et al.
(2002), the averaged Hamiltonian reads
H̄ = H0 − µ
1
2π
2π
R dM,
(8)
0
that is, the averaging is performed with respect to the mean anomaly
M of the asteroid. However, it is convenient to transform the averaging process to one according to the eccentric anomaly E:
1
R̄ =
2π
2π
0
1
R dM =
2π
2π
R(1 − e cos E) dE.
(9)
0
We note that the averaged perturbing function depends only on the
canonical variables x, τ and x2 : R̄ = R̄(x, τ, x2 ). According to
Schubart (1964), during the averaging E is varied between 0 and
2π, and the other variables are kept fixed.
Now, a symplectic mapping is obtained by the following mixedvariables generating function:
W = xn+1 τn − Tshort [H0 (xn+1 ) + µ R̄(xn+1 , τn , x2 )],
(10)
where, in our case, the short period is T short = 2π. The mapping corresponds to the canonical transformation via the generating function
(10), namely
∂W
∂W
xn =
,
τn+1 =
.
(11)
∂τn
∂xn+1
∂ R̄1 (x, τ, x2 )
∂x
+ O(x 2 ).
x=0
z n+1 = eiω0 z n ,
(13)
τn+1 = τn − 2π
∂ R̄0 (τn , x2 )
∂τn
1 − 2πµ
∂ R̄1 (τn , x2 )
∂τn
dH0 (xn+1 )
− 2πµ R̄1 (τn , x2 ).
dxn+1
−1
z − z∗
z + z∗
√ ,
Y = √ ,
(18)
2
2i
where X and Y are the normal coordinates of the linearized mapping. (In the variables u and v the invariant curves of the linearized
mapping are ellipses, whose semimajor axes are rotated by an angle ϕ 0 . By a standard canonical transformation, these ellipses are
transformed to concentric circles – see Appendix.)
,
X=
(14)
The averaged perturbing function R̄ and the coefficients R̄0 and R̄1
of its Taylor expansion have been calculated (via symbolic algebra
manipulation software) by approximating the integral in equation
(9) by a trapezoidal rule
N
1 w j R(τ, x, x2 , E j )(1 − e cos E j ),
R̄(τ, x, x2 ) =
N
2.2 Correction of the rotation number
The implementation of Hadjidemetriou’s method as above, i.e. by
setting the averaged Hamiltonian equal to the perturbing term of
a near-identity generating function, yields an O(T 3short ) error in the
rotation number of the central elliptic point with respect to that of
the Hamiltonian Poincaré surface of section. This can be readily
shown by the following elementary example: Consider the stroboscopic Poincaré map, with period T short , of the harmonic oscillator
Hamiltonian
1
H = ( p 2 + ω2 q 2 ).
(19)
2
The rotation number of the elliptic point q = p = 0 of the stroboscopic map is, simply, rot = T short /T , where T = 2π/ω. On the
j=1
(15)
R̄0 (τ, x2 ) = R̄(τ, x=0, x2 ),
R̄1 (τ, x2 ) =
∂ R̄(τ, x, x2 )
∂x
(17)
where ω is the rotation angle of the central fixed point. The complex
conjugate variables z and z ∗ are defined by
which according to equation (7) leads to the explicit mapping
xn+1 = xn + 2πµ
(16)
i, j=1
Finally, the mapping (16) is expressed in complex form by the
complex conjugate variables z and z ∗ in which the linearized mapping can be written as
W = x n+1 τn − 2πH0 (xn+1 ) − 2πµ R̄0 (τn , x2 )
βi j u in vnj .
(1) calculation of (τ ∗ , x ∗ ) for a given value of e p ,
(2) two-variable Taylor expansion of the generating function (9)
around τ ∗ and x∗ up to power 80 of the new variables u and v,
(3) derivation of the equations of the mapping by using the expanded generating function,
(4) two-variable Taylor expansion of the equations of the mapping around u = 0 and v = 0 up to order 80.
(12)
Denoting the constant term of the above Taylor expansion by
R̄0 (τ, x2 ) and the coefficient of the first-order term by R̄1 (τ, x2 ),
we obtain the generating function
vn+1 =
The above expansion has been obtained in the following steps.
− xn+1 2µ R̄1 (τn , x2 ),
i+ j=80
αi j u in vnj ,
i, j=1
The above equations result in an implicit mapping in the variable
x n+1 . In order to derive an explicit mapping, we developed the averaged perturbing function around x = 0 up to first order:
R̄(x, τ, x2 ) = R̄(x=0, τ, x2 ) + x
i+ j=80
u n+1 =
,
x=0
where E j = 2π j/N , w 1 = w N = 1/2 and wj = 1 for 1 < j < N .
−14
In our calculations we used N = 15, which value
provided a 10
accuracy in evaluating R̄. (We compared R̄ with [0,2π] R dE, where
this integral was evaluated by the internal numerical integration
routine of the symbolic algebra manipulation program.)
The mapping (14) depends on two parameters: the mass parameter µ and x 2 . For a fixed value of x 2 , the equation x2 =
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257
other hand, if one considers the Hadjidemetriou mapping via the
generating function
oscillator example, the linear mapping (21) admits the family of
invariant ellipses
2
1
Tshort pn+1
+ ω2 qn2 ,
2
one readily finds the linear symplectic mapping
2C = p 2 − ω2 Tshort qp + ω2 q 2 ,
W (qn , pn+1 ) = qn pn+1 +
(20)
2
qn+1 = qn + Tshort pn − Tshort
ω2 qn , pn+1 = pn − Tshort ω2 qn ,
(21)
so that the rotation number of the elliptic point at the origin is
r ot =
1
cos−1
2π
π2
Tshort
=
+
T
6
1−
2
ω2
Tshort
2
Tshort
T
3
+O
Tshort
T
5 .
(22)
Now, in the case of the Trojan mapping, we have T short /T =
0.0807. . . at L 4 , and therefore
π2
6
Tshort
T
3
= 8.6 × 10
−4
−3
∼ 10 .
(23)
which differ from the invariant ellipses of the Hamiltonian (19) (with
H = 2C) by the presence of the non-diagonal term qp, which causes
the angular deformation of the ellipses. From a dynamical point of
view, this deformation can be expressed as a phase difference in the
Érdi (1988) model for the long periodic variations of the Trojans’
elements. Namely, if q is identified with q = λ − λ − π/3 (for L 4 ),
then the first equation in Érdi’s model,
1
a = a + dp sin θ + O dp2 , λ = λ + π + Dp cos θ + O Dp2 ,
3
(24)
should be replaced in the mapping case by
a = a + dp sin(θ − θ0 ),
where θ 0 is given by
1
Tshort ω2
,
θ0 = tan−1 −
2
1 − ω2
This difference in the third digit may be thought to be not very important for the dynamics. However, Nekhoroshev estimates depend
crucially on the small divisor structure of formal perturbative series,
and the value of some small divisors changes appreciably due to this
difference (Section 4.2). Therefore, the mappings (14) or (16) have
to be corrected with respect to the rotation number. In order to do
so, we simply consider as new mapping model the composition of
equation (14) with the 2D rotation map U (φ) with angle φ =
−2π × 10−3 .
i.e. for T short T , one has θ 0 = O(T 3short ). However, this change
does not appreciably affect the value of the proper elements d p
and d p , in terms of which the Nekhoroshev estimates below are
expressed. The difference in the proper element values between the
Hamiltonian and mapping cases is given by
q0,mapping − q0,hamiltonian
≡ Dp
q0,hamiltonian
2.3 Comparison of the Hamiltonian and mapping models
For T short /T = 0.0807. . . , one finds D p 0.8 per cent. Therefore,
the proper element D p calculated by the mapping differs by less
than one per cent from that calculated for the same level curve in
the Poincaré section of the precise Hamiltonian.
(b) The mapping sections accurately reproduce the position of
the central periodic orbit as well as of higher-order periodic orbits
of the Poincaré plane of the Hamiltonian model. The period-one elliptic fixed points of the mapping and of the Hamiltonian Poincaré
sections correspond to the periodic orbits (having simple oval shape)
of the short-period family emanating from L 4 . Fig. 2(a) shows the
comparison between its calculated position in the Hamiltonian and
mapping models as a function of the proper eccentricity e p . Increasing the eccentricity, the centre of libration departs from τ 0 = π/3.
The mapping model follows the position of the fixed point with
great precision. This agrees with the results of Namouni & Murray
(2000). Furthermore, as shown in Fig. 2(b), the rotation number of
the fixed point, calculated by the mapping model, shows an excellent
agreement to the one derived by the Hamiltonian model (rotation
numbers in both models are considered as positive).
The mapping model agrees also with the Hamiltonian model as
regards the positions and stability of the chains of fixed points corresponding to the 1/13, 1/14 and 1/15 resonances between the shortand long-period components.
(c) Finally, the extension (maximum variations of τ and x) of the
islands of stability in the Hamiltonian and mapping cases have an
overall agreement, except for very small values of the eccentricity
(e.g. top panels of Fig. 1), where the border of stability in the mapping model extends to a smaller distance from the centre (τ max =
2.1 in the mapping case compared to 2.3 in the Hamiltonian case),
i.e. there is some more chaos at the border of the island of stability
of the mapping model.
In what follows we intend to show that the mapping (14) well reproduces the phase dynamics of the precise Hamiltonian model in
the neighbourhood of the triangular points.
To this end, it can be proven (Sándor et al. 2002) that, in the
co-orbital region, mapping (14) has three period-one fixed points,
which, when e p = 0, correspond precisely to the Lagrangian solutions L 3 , L 4 and L 5 . When e p = 0, these points correspond to
the intersections of period-one periodic orbits that pass from the
neighbourhood of the Lagrangian points by an appropriate Poincaré
surface of section. Thus it is convenient to refer to them as the
fixed points L3 , L4 and L5 of the mapping (14), from which L4
and L5 are elliptic fixed points, while L3 is a hyperbolic fixed
point.
The comparison between the phase portraits of the precise Hamiltonian and of the mapping is shown in Fig. 1. The three left-hand
panels show the Poincaré surface of section (τ , x) when = 0
and ˙ < 0 in the Hamiltonian (6) of the planar and circular RTBP,
for three different values of the Jacobi constant E J corresponding
to three characteristic values of the proper eccentricity. The variation of the eccentricity as we move from the central fixed point L4
outwards is written above each panel. The three right-hand panels
show the corresponding mapping sections derived by the explicit
mapping (14) for the values of the eccentricity e p = 0.001 (top),
e p = 0.06 (middle) and e p = 0.15 (bottom). The following can be
observed:
(a) The invariant curves in the mapping sections are angularly
deformed with respect to those of the Poincaré section of the
Hamiltonian case. This deformation is due to the method of production of the mapping. Namely, referring again to the harmonic
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2005 The Authors. Journal compilation π2 Tshort 2
1
= 1 − .
= 8
2
T
1 − ω2 Tshort
/4
(25)
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C. Efthymiopoulos and Z. Sándor
Figure 1. Comparison of the Poincaré surface of section of the Hamiltonian of co-orbital motion in the circular RTBP (equation 6, left-hand panels) with the
mapping sections (right-hand panels) for e = e p = 0.001 (top), e = 0.06 (middle) and e = 0.15 (bottom).
in which the point z = 0 is an elliptic fixed point, i.e. F non−linear
(0, 0) = 0 and ω0 ∈ R. Assuming that the quantity ω 0 /2π, called
the rotation number, is an irrational number, the aim of the nonresonant normal-form construction is to find a normalizing canonical
transformation z = (ζ , ζ ∗ ), such that in the new variables (ζ , ζ ∗ )
the mapping (26) is transformed to a twist mapping
In conclusion, the mapping model (14) reproduces the dynamics
of the Poincaré mapping of the Hamiltonian model of co-orbital
motion (equation 6) with good precision. We shall now use this
mapping to obtain analytical Nekhoroshev estimates for the stability
around the family of fixed points L4 (or L5 ).
ζ = U (ζ, ζ∗ ) ≡ eiω(ζ ζ∗ ) ζ,
3 BIRKHOFF NORMAL FORM AND
O P T I M I Z AT I O N O F N E K H O RO S H E V
E S T I M AT E S
(27)
Consider a symplectic mapping of the plane given in complex conjugate canonical variables (z, z ∗ ),
where the function ω(ζ ζ ∗ ) is equal to ω 0 plus a correction,
i.e. ω(ζ ζ ∗ ) = ω 0 + ω 1 (ζ ζ ∗ ). If a transformation with the
above properties exists, and the inverse function ζ = (z, z ∗ ) =
−1 (z, z ∗ ) can be defined in an open domain including the origin,
then the mapping (26) is called integrable. The exact integral of the
mapping (27) is I = ζ ζ ∗ , or, in the old coordinates,
z = F(z, z ∗ ) = eiω0 z + Fnon-linear (z, z ∗ ),
I = (z, z ∗ )∗ (z, z ∗ ).
3.1 Birkhoff normal form
(26)
C
(28)
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259
Figure 2. Comparison of the Hamiltonian (solid curves) and mapping (dashed curves with squares) models: (a) position of the elliptic fixed point corresponding
to a tadpole periodic orbit on the Poincaré surface of section, and (b) rotation number of the elliptic fixed point.
The level curves I = constant are invariant circles on the plane
defined by the coordinates (ζ , ζ ∗ ), which correspond to closed invariant curves on the plane of the old coordinates (z, z ∗ ).
The definition of the normalizing transformation and of the normal form follows by the solution of the so-called conjugation equation (Servizi et al. 1983; Bazzani et al. 1990). Namely, one may
write z as z = (ζ , ζ ∗ ) = (U (ζ , ζ ∗ ), U ∗ (ζ , ζ ∗ )), or as z =
F(z, z ∗ ) = F((ζ , ζ ∗ ), ∗ (ζ , ζ ∗ )). Equating the right-hand
side (RHS) of these equations yields (U (ζ , ζ ∗ ), U ∗ (ζ , ζ ∗ )) =
F((ζ , ζ ∗ ), ∗ (ζ , ζ ∗ )), or
( ◦ U ) (ζ, ζ∗ ) = (F ◦ ) (ζ, ζ∗ ).
(29)
The solution to equation (29) is obtained, step by step, by an
iterative procedure. Let the mapping, normalized transformation
and normal form be expanded in polynomial series of the canonical
variables around the origin,
F(z, z ∗ ) = e
iω0
z + F2 (z, z ∗ ) + F3 (z, z ∗ ) + · · · ,
U (z, z ∗ ) = e
iω0
ζ + U2 (ζ, ζ∗ ) + U3 (ζ, ζ∗ ) + · · · ,
(30)
(ζ, ζ∗ ) = z + 2 (ζ, ζ∗ ) + 3 (ζ, ζ∗ ) + · · · .
Then it can be readily shown that the solution of the conjugation
equation (29) for the terms of order N is equivalent to the solution
of the homological equation
N eiω0 ζ, e−iω0 ζ∗ − N (ζ, ζ∗ ) + U N (ζ, ζ∗ ) = Q N (ζ, ζ∗ ),
where
QN =
N
N −1
ω0 ζ κ ζ∗λ = ei(κ−λ)ω0 − eiω0 ζ κ ζ∗λ ,
i.e. any monomial of the form ζ κ ζ λ∗ is an eigenfunction of the operator ω0 with eigenvalue equal to (ei(κ−λ) ω0 − eiω0 ). This means
that the operator ω0 is invertible in the subspace of monomials,
R(ω0 ) ≡ ζ κ ζ∗λ , (κ, λ) ∈ Z 2 , κ − λ = 1 ,
(35)
called the range of ω0 . The inverse operator has the same eigenfunctions as ω0 with inverse eigenvalues, i.e. (ei(κ−λ)ω0 − eiω0 )−1 .
On the other hand, the image of the subspace of monomials,
N (ω0 ) ≡ ζ κ ζ∗λ , (κ, λ) ∈ Z 2 , κ − λ = 1 ,
(36)
under the action of ω0 is just the null space, and thus N (ω0 ) is
identified with to the kernel of ω0 . Following these definitions, the
general solution of equation (34) is given by the relations
U N = kernel terms ofQ N with respect toω0
N = −1
ω0 (Q N − U N ) + arbitrary kernel terms wrtω0 .
(37)
For N even, we simply have UN = 0 and there can be no arbitrary
kernel terms in N . For N odd, we have in general UN = 0, while
there can be one arbitrary term in N , namely
[N /2]+1 [N /2]
ζ∗ ,
N = −1
ω0 (Q N − U N ) + b N ζ
N
[ f ]N ≡ f N ,
(32)
with f i meaning polynomial terms of degree i in the canonical variables for any complex function f .
By introducing the linear difference operator ω0 defined as
ω0 f (ζ, ζ∗ ) = f (eiω0 ζ, e−iω0 ζ∗ ) − f (ζ, ζ∗ ),
C
The definition of ω0 yields that
∂[]N ∂[∗ ]N
∂[]N ∂[∗ ]N
−
∂ζ
∂ζ∗
∂ζ∗
∂ζ
(38)
= 1 + O(N ).
and the following notation is used:
[ f ]N ≡ f 1 + f 2 + · · · + f N ,
(34)
{[]N , [∗ ]N } =
k ([U ]N −1 , [U∗ ]N −1 )
k=2
ω0 N + U N = Q N .
where bN is arbitrary. However, there is one more restriction imposed
by the request that be a symplectic transformation. Namely, the
real part of bN is determined by the request that [] N be symplectic
up to order N − 1, that is
Fk ([]N −1 , [∗ ]N −1 )
k=2
−
(31)
equation (31) takes the form
(33)
C 2005 RAS, MNRAS 364, 253–271
2005 The Authors. Journal compilation The imaginary part of bN is left undetermined even after the symplecticity condition and it can be assigned arbitrary values. The most
common choice is to set Im(bN ) = 0.
3.2 Remainder
We consider now the iterative procedure after r − 1 steps, i.e. U and
are defined up to r, called the order of truncation of the normal
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C. Efthymiopoulos and Z. Sándor
the same parity as k. In particular, the properties of the normal form
imply that
form. According to (38), the normalizing transformation
z = [(ζ, ζ∗ )]r
(39)
gk(r ) (φ) = 0, if k r , k even,
is symplectic only up to terms of order r. The mapping F(z, z ∗ ),
which in general contains terms of any order beyond r, is transformed, via the normalizing transformation (39), to a new mapping
in the variables (ζ , ζ ∗ ), which can be written as
ζ = f (r ) (ζ, ζ∗ ) = [U ]r (ζ, ζ∗ ) + R (r ) (ζ, ζ∗ ).
gk(r ) (φ) = 1, if k r , k odd.
Taking the imaginary part of the principal logarithm of (44) yields
the mapping of the angles
(40)
φ = φ + ω0 + Im
(r )
The mapping (40) is symplectic also up to order r. The quantity R ,
called the remainder at the rth order of truncation, is an infinite series that contains all the terms beyond r produced by the terms of
the original mapping F(z, z ∗ ). The fact that R (r ) is a non-symplectic
perturbation to the integrable mapping [U ] r (ζ , ζ ∗ ) has no influence on the study of dynamics in the normalized variables (ζ , ζ ∗ ),
because the mapping (40) is a precise transformation of the original
symplectic mapping [see, for example, the use of (40) in the proof
of Nekhoroshev stability by Bazzani et al. (1990)].
The mapping (40) is determined as follows: The inverse transformation ζ = (r ) (z, z ∗ ) of (39) is first determined by inverting the
polynomial (39). The relevant homological equation to be solved is
obtained by the identity condition
z = ( (r ) (z, z ∗ ), ∗(r ) (z, z ∗ ))
r
,
N(r )
=−
r
k ( (r )
,
N −1
2
ρ =ρ
∗(r ) N −1 )
,
f (r ) = (r ) ◦ F ◦ []r .
R (rJ ) =
and
Rφ(r ) =
,
(43)
.
(47)
)
R (rJ ,sk
J s/2 eikφ
(48)
(r )
Rφ,sk
J s/2 eikφ
(49)
3.4 Nekhoroshev estimates with the one-turn remainder
In order to obtain an estimate of the size of the domain of Nekhoroshev stability, for a time equal to T = 109 periods of Jupiter, we
first implement the discrete analogue of the analytical treatment of
the normal form of the Hamiltonian case as in Giorgilli & Skokos
(1997). Essentially, one tries to prove that the remainder term R (r )
in the mapping (40) produces a quite small perturbation to the integrable dynamics induced by the normal-form term U r in the same
mapping. This means that the cumulative effect of R (r ) up to the
time T should be small enough that the orbits cannot escape from a
properly defined area of stability.
The above concepts are quantified as follows. Consider the annulus D(ρ 0 , ρ) = {ζ : 2ρ 0√
−ρ |ζ | ρ}√
determined by two nearby
circles with radii ρ0 = 2J0 and ρ = 2J > ρ0 on the plane of
the mapping (40). If the remainder term in the mapping (40), or its
equivalent terms in the mapping (47), were absent, an orbit started
anywhere on the circle |ζ | = ρ 0 would remain on the same circle for
all times. However, the presence of the remainder term causes a drift
of an orbit, at every step, from its initial circle to a nearby circle. If
ρ is set equal to the radius of the most distant circle visited by the
orbit within a time T, then the cumulative effect of the remainder
ρ k−1 gk(r ) (φ)
(46)
k=2
)
(r )
, Rφ,sk
∈ C.
with s, k ∈ Z, and R (rJ ,sk
The mapping (40) is next converted to a mapping in polar coordi
nates. Setting ζ = ρ eiφ and ζ = ρ eiφ , and taking into account that
the normal form U starts with the term U1 = eiω0 ζ , equation (40)
takes the form
1+
(r )
ρ k−1 g∗,k
(φ)
sr +2, |k|s
3.3 Remainder in action–angle variables
ρe
1+
∞
sr +2, |k|s
The polynomial truncation [ f (r ) (ζ , ζ ∗ )] r of the series f (r ) (ζ , ζ ∗ )
should coincide with the polynomial [U ] r up to terms of order r. In
the computer implementation, the two series turn out to be different
as a result of cumulative error due to round-offs. Thus, by comparing
the polynomials [ f (r ) ] r with [U ] r , we determine the size of the
round-off error at the order of truncation r. We then check that the
error, normalized to the size of [U ] r , is smaller than a prescribed
tolerance (10−12 ).
=e
ρ k−1 gk(r ) (φ)
where the remainders RJ and R φ are both derived from the remainder
of the mapping (27). In particular, since the remainder R (r ) is a sum
of homogeneous polynomials in the variables (ζ , ζ ∗ ), the functions
RJ and R φ are given by series of the form
(42)
or, simply,
ρ e
1+
∞
φ = φ + ω(r ) (J ) + Rφ(r ) (φ, J ),
= (r ) (F([]r (ζ, ζ∗ ), [∗ ]r (ζ, ζ∗ )),
∞
(45)
J = J + R (rJ ) (φ, J ),
ζ = (r ) (z , z ∗ ) = (r ) (F(z, z ∗ ), F∗ (z, z ∗ ))
,
which is again expanded as a trigonometric series.
Finally, defining the action variable J = ρ 2 /2, the mappings (46)
and (45) take the form of a perturbed twist mapping in action–angle
variables,
N
iφ
2
k=2
for N = 2, . . . , ∞. In computer implementations, the iterative procedure is stopped at some high order M with N M and M r .
A criterion for selecting the value of M is discussed below.
Following the determination of (r ) , the mapping (40) is obtained
by the old mapping (26) via the composition
iω0
ρ k−1 gk(r ) (φ)
which can be expanded in a trigonometric series by Taylor expansion
of log (1 + x) around zero. On the other hand, multiplying equation (44) by its complex conjugate equation yields the mapping of
polar radii
(41)
k=2
iφ log 1 +
∞
k=2
which leads to the recursive formulae
1(r ) = z,
(44)
k=2
For any fixed value of k, the functions g (rk ) (φ) are sums of trigonometric terms of the form e iK φ with K ∈ Z, |K | k and K having
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after T steps causes an uncertainty of the position of the orbit within
the domain D(ρ 0 , ρ).
Defining now the norm
(r ) RJ D(ρ0 ,ρ)
= sup R (rJ ) (φ, J )
(50)
D(ρ0 ,ρ)
and assuming the expansion (48), yields
(r ) RJ D(ρ0 ,ρ)
=
sr +2, |k|s
(r ) s (r ) R J ,sk ρ = R J .
ρ
(51)
The quantity ||R (rJ ) || ρ is called the norm of the remainder at distance
ρ. After N iterations of the mapping (47), the maximum distance
travelled in action space is
|J − J0 | N R (rJ ) D(ρ
(52)
0 ,ρ)
or
ρ 2 − ρ02 2N R (rJ ) D(ρ
0 ,ρ)
.
(53)
If we now fix the radii ρ 0 and ρ, the time N (number of iterations)
needed to cross the annulus is bounded from below by a minimum
time N min :
ρ2 − ρ2
N Nmin = (r ) 0 .
2 R J D(ρ ,ρ)
(54)
0
According to equation (51), the value of |R (rJ ) | D(ρ0 ,ρ) depends only
on the outer radius ρ of the annulus, and it is an increasing function
of ρ. Now, at the limit ρ = ρ 0 , the RHS of equation (54) yields
N min = 0. This reflects the trivial fact that an orbit will certainly
depart from its initial condition at the next iteration. On the other
hand, at the limit ρ → ∞ one has limρ→∞ |R (rJ ) | D(ρ0 ,ρ) = ∞, yielding again N min = 0. In other words, if one looks at how long it takes
an orbit to go from ρ 0 to ρ ρ 0 , one has to overestimate the speed
of diffusion by its upper value at ρ, thus underestimating N min . This
implies that an optimal estimate for the minimum Nekhoroshev time
N min is given by the maximum of the RHS of equation (54) with
respect to ρ. We set
ρ2 − ρ2
Nnek = max (r ) 0
ρ 2 R J D(ρ ,ρ)
(55)
0
Now, standard arguments of the normal-form theory (e.g. Bazzani
et al. 1990) assert that the series R (rJ ) is convergent in a disc 0 < ρ ρ c . This allows one to express the norm of the remainder |R (rJ ) | D(ρ0 ,ρ) ,
given by equation (51), in terms of the size of the first-order term of
the remainder only. Precisely, for any ρ ∗ ρ c , there is a positive
number CJ (r , ρ F , ρ ∗ ), depending on the order of the optimization r
and the radius of convergence ρ F of the original mapping, such that
for any ρ with 0 < ρ < ρ ∗ one has
(r ) RJ D(ρ0 ,ρ)
C J (r , ρ F , ρ∗ )ρ r +2 .
(56)
Essentially, CJ (r , ρ F , ρ ∗ ) is equal to the first order of the remainder
)
|R r(r+2
| times a coefficient that depends on the value of the radius
of convergence of |R (r ) |. In particular, replacing equation (56) into
equation (55) yields the optimal Nekhoroshev time
Nnek = max
ρ
ρ 2 − ρ02
.
2C J (r , ρ F , ρ∗ )ρ r +2
(57)
However, the maximum in the above equation can be calculated by
simply differentiating the RHS of equation (57) with respect to ρ.
The radius of the maximum is
ρm =
C
r +2
ρ0 .
r
(58)
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2005 The Authors. Journal compilation 261
Substituting equation (58) into equation (57) yields the optimal
Nekhoroshev time, as a function of the initial radius,
1
Nnek =
.
(59)
r (1 + 2/r )(r +2)/2 C J (r , ρ F , ρ∗ )ρ0r
Fixing the value of N nek (=109 periods), we have the following
lemma.
Lemma 1. Assume ρ ∗ ρ c , where ρc is the radius of convergence
of |R (rJ ) |. Then, for any time t N nek , an orbit of the mapping (47)
with initial conditions on the circle J = ρ 20 /2, where
ρ0 =
1
r (1 + 2/r )(r +2)/2 Nnek C J (r , ρ F , ρ∗ )
1/r
,
(60)
cannot travel a distance greater than ρ m − ρ 0 , where ρ m is given
by equation (58), provided that ρ m ρ ∗ .
In the computer implementation, in order to obtain an optimal
estimate, one fixes the order of normalization r and the time N nek
to some value, and then proceeds in the following way:
(1) A first value of ρ ∗ is obtained by implementing equation (60),
where CJ (r , ρ F , ρ ∗ ) is replaced by the first order of the remainder
|R (rJ,r) +2 |, then equation (58), and then by setting ρ ∗ = ρ m . This is
an overestimate, because |R (rJ,r) +2 | CJ (r , ρ F , ρ ∗ ).
(2) The value of CJ (r , ρ F , ρ ∗ ) is calculated as follows. Provided
that ρ ∗ is smaller than the radius of absolute convergence of the
remainder series ρ c , the norm of the remainder series at ρ ∗ is given
by
(r ) R J = R (rJ ,r) +2 ρ∗r +2 + R (rJ ,r) +3 ρ∗r +3 + · · ·
ρ∗
∞
(r ) s
R J ,M ρ∗ ,
) M
ρ∗ +
+ R (rJ ,M
(61)
s=M+1
where M is the maximum order of computer expansion. The terms
in equation (61) are given by the computer expansion, but the terms
of order beyond M are not known explicitly. The size of these
terms must be bounded from above analytically. It turns that this
size is quite small, compared to the whole size of the remainder,
provided that r is not very close to M. In our implementation, a
coefficient B ∗ (ρ F ) is determined, where ρ F is the radius of convergence of the initial mapping (26), so that the size of the terms
beyond N is incorporated into the last term of the expansion known
explicitly. Namely, the following expression is precise:
(r ) R J R (rJ ,r) +2 ρ∗r +2 + R (rJ ,r) +3 ρ∗r +3 + · · ·
ρ∗
) M
ρ∗ .
+ B∗ (ρ F ) R (rJ ,M
(62)
The determination of B ∗ (ρ F ) can be done analytically by the
rigorous bounds provided in Bazzani et al. (1990). In all cases we
find B ∗ (ρ F ) 2. Thus, in the computer implementation we simply
substitute B ∗ (ρ F ) with a safety factor equal to 2.
If, now, we determine the coefficient
(r ) r +2
1
R J ,r +2 ρ∗
C J (r , ρ F , ρ∗ ) = (r ) R J ,r +2 ρ∗r +2
) M
+ R (rJ ,r) +3 ρ∗r +3 + · · · + B∗ (ρ F ) R (rJ ,M
ρ∗ ,
(63)
then, for any 0 < ρ < ρ ∗ , the statement of equation (56) holds true.
(3) Having fixed the value of CJ (r , ρ F , ρ ∗ ), the algorithm returns
to equation (60) and determines new values for ρ 0 and ρ m . Then
the algorithm goes to step 2, determining a new ρ ∗ = ρ m and a new
CJ (r , ρ F , ρ ∗ ). Since the old ρ ∗ and CJ (r , ρ F , ρ ∗ ) were overestimated, the new ρ 0 , ρ ∗ = ρ m and CJ (r , ρ F , ρ ∗ ) are underestimated.
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C. Efthymiopoulos and Z. Sándor
Let us now temporarily ignore the term S (r,N ) (J , φ) and focus on
the first term. This function
Thus the algorithm repeats the previous steps iteratively until the
differences in successive values of CJ (r , ρ F , ρ ∗ ) become smaller
than a prescribed tolerance.
R (rJ ,N ) (J , φ) =
The value of ρ 0 determined in the last iteration determines an area
of Nekhoroshev stability for which Lemma 1 holds. This is a value
of ρ 0 corresponding to a particular Nekhoroshev time N nek , but also
to a particular choice of normalization order r. The calculation is
repeated for different normalization orders in the range 3 r M,
and the optimal radius ρ nek
0 is selected as the maximum of the values
ρ 0 found for different orders r. In practice, in our calculation we
set M = 80, but we do not consider normalization orders beyond
r = 60. Thus, the terms of at least the first 20 orders (= 80–60) of
the remainder series R (r ) are known explicitly, the remaining terms
representing a negligible contribution. This is an improvement as regards the precision of our estimates compared to previous estimates
in the literature. For example, in Giorgilli & Skokos (1997), the total
size of the remainder is estimated as twice the size of its first-order
)
term R r(r+1
. Our calculations show that this is an underestimate by a
factor of 1000 at the order of truncation r = 38, which results in an
overestimate of the radius ρ 0 by a factor of 10001/38 1.2.
)
R (rJ ,sk
J s/2
sr +2, |k|s
×
N −1
exp ik φ + mω (J )
(r )
.
(67)
m=0
The sum in large parentheses is simply
1 − exp iN kω(r ) (J )
× eikφ ,
1 − exp ikω(r ) (J )
and thus
R (rJ ,N ) (J , φ)
=
sr +2, |k|s
1 − exp iN kω(r ) (J )
1 − exp ikω(r ) (J )
)
× R (rJ ,sk
J s/2 exp(ikφ).
(68)
Equation (68) implies that the Fourier coefficients of the iterated
remainder R (r,N ) differ from the respective coefficients in the oneturn remainder by the quantity
1 − exp iN kω(r ) (J )
,
1 − exp ikω(r ) (J )
R (rJ ) J (m−1) , φ + (m − 1)ω(r ) (J ) .
which for most frequencies ω(r ) (J ) is O(1). However, the most
important Fourier terms in the remainder are those in which the
wavenumber k is a multiple of q ∈ N ∗ , where q is the denominator of a rational approximation to ω(r ) (J ), namely ω(r ) (J ) =
2π p/q + . For these terms, it can be easily shown that, if N is
equal to the denominator of a higher order rational approximation
to w(r ) (J ), then
(64)
m=1
Since the variation of the action is small, the actions J (m) for m =
0, . . . , N − 1 are given by J (m) = J + mJ (m) , where J (m) is
a quantity of the order of O(|R (rJ ) |). Thus, equation (64) takes the
form
RJ
R (rJ ,N ) (J , φ) =
while the first of equations (47) takes the form
J (N ) − J =
)
R (rJ ,sk
J s/2 eikφ .
sr +2, |k|s
φ (N ) = φ + N ω(r ) (J ),
(r )
(66)
Equation (66) then gives
If [J (N ) , φ (N ) ] denotes the Nth image of some initial point (J , φ) of
the mapping, we readily obtain
N
R (rJ ) =
φ = φ + ω(r ) (J ).
will be called the N-iterated remainder of order r of the mapping
(47). The key point in this definition is that N − 1 successive values
of the one-turn remainder are added algebraically rather than by
their absolute values. This is motivated by the physically relevant
picture that the drift in action space is not realized by steps that are
all in the same direction, i.e. of increasing or decreasing J (m). On
the contrary, the drift in action space is the overall sum of a number
of steps, each of which may cause a positive or negative variation
of the action.
The long-term behaviour of the N-iterated remainder can be determined by recalling that the one-turn remainder R (rJ ) is a sum of
Fourier terms (equation 48)
The above described estimates by the remainder of the ‘one-turn’
map can be improved further by considering an ‘iterated remainder’,
which is an improvement of the iterated remainder method used by
Bazzani et al. (1990). This section contains the basic formulae used
in the implementation of the new method, while the technical details
are deferred to the Appendix.
Let us assume that for a number of iterations N N nek the
variation of the action J = J fin − J from an initial value J is small
enough that the frequency ω in equation (47) can be considered as
practically constant, equal to its initial value ω(r ) (J ). Under this
simplifying assumption, the angular part of the mapping (47) is
obtained by neglecting the term R φ in equation (47) and also the
variation of the action. One then has
J (N ) − J =
R (rJ ) J , φ + (m − 1)ω(r ) (J )
m=1
3.5 Nekhoroshev estimates with an iterated remainder
N
N
1 − exp iN kω(r ) (J )
= O(k 2 /N ).
1 − exp ikω(r ) (J )
J , φ + (m − 1)ω(r ) (J ) + S (r ,N ) (J , φ). (65)
This means that the coefficient of a nearly resonant term in the
iterated remainder can be even smaller than the coefficient of the
same term in the one-turn remainder, at values of N that are multiples
of the multiplicity of the respective near-resonance.
In conclusion, the size of the N-iterated remainder is expected
to be very similar to, or even smaller than, the size of the one-turn
remainder, for N not very small. This, in turn, leads to improved
estimates of the size of the area of stability. In fact, any estimate
of this size based on equation (60) is improved by essentially the
m=1
Equation (65) is precise if the quantity S (r,N ) (J , φ) is calculated
by the full mapping (47), without the above approximations. In the
Appendix, it is shown that the size of S (r,N ) (J , φ) is of the order of
O(N 3 |R (rJ ) (J , φ)|2 ). Thus, if the quantity |R (rJ ) (J , φ)| is small, the
second term on the RHS of equation (65) is much smaller than the
first term, which is O(|R (rJ ) (J , φ)|), even for N not very small (in
our calculations we find that N can be as high as N = 2000).
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factor of N 1/r , since the Nekhoroshev time for the iterated mapping
is N nek = N nek /N .
The precise formulae used in the computer implementation of the
iterated remainder are given in the Appendix.
Table 1. Optimal Nekhoroshev stability estimates.
4 R E S U LT S
0.001
0.03
0.06
0.09
0.12
0.15
0.18
0.21
e
This section presents the results of the determination of the area of
Nekhoroshev stability on the plane of the canonical variables (τ ,
x), for the mapping models determined in Section 2. An explicit
symplectic mapping, as well as the normal form, remainder and
iterated remainder, were calculated for each of the values of the
proper eccentricity e p = 0.001, 0.03, 0.06, . . . , 0.21.
Optimal
order
39
41
45
53
>60
>60
>60
>60
D p (degree)
one-turn
iterated
11.61
11.76
11.47
11.32
12.28
10.52
7.01
5.44
12.84
12.89
12.64
12.23
12.87
11.30
7.49
5.83
263
Number of
remainder
iterations N
700
800
600
700
600
1300
1500
1500
4.1 The domain of Nekhoroshev stability in the space
of proper elements
The domain of Nekhoroshev stability is first determined as a disc
on the plane of the normalized canonical variables (ζ , ζ ∗ ) centred
at the origin (Section 3.1). All asteroids located within this disc are
shown to be Nekhoroshev-stable for a time equal to at least N nek =
109 periods of Jupiter. The radius ρ 0 of the disc is determined by
equation (60), in which the values of the optimal order of truncation
r and of the reference radius ρ ∗ are determined by steps 1 to 3
of the algorithm of Section 3.4. Two different radii are calculated,
corresponding to the optimal values found by the use of the one-turn
remainder and of the N-iterated remainder, respectively. In the case
of the one-turn remainder, the quantity CJ appearing in equation (58)
is given by equation (63). In the case of the N-iterated remainder, for
a fixed value of N, the radius of Nekhoroshev stability is corrected
for the effect of the term S (r ,N ) ρ∗ as discussed in the Appendix.
The results are transformed to proper element values by the following procedure: The disc of radius ρ 0 corresponds to a deformed
disc in the original complex canonical variables (z, z ∗ ) with an outermost boundary given by
z = (r ) (ρ0 eiφ , ρ0 e−iφ ),
0 φ < 2π.
(69)
It follows that this boundary is given in the canonical variables (u,
v) (Section 2.1) by
u(ρ0 , φ) = U (r ) ρ0 eiφ , ρ0 e−iφ , (r∗ ) ρ0 eiφ , ρ0 e−iφ
v(ρ0 , φ) = V (r ) ρ0 eiφ , ρ0 e−iφ , (r∗ ) ρ0 eiφ , ρ0 e−iφ
,
(70)
for 0 φ < 2π. Finally, the maximum libration amplitude for which
an orbit is Nekhoroshev-stable is given by
Dp =
1
(u max − u min ),
2
where
(71)
u max = max u(ρ0 , φ), 0 φ < 2π ,
φ
u min = min u(ρ0 , φ), 0 φ < 2π .
φ
One value of D p is found for each different mapping derived for
one value of e p (Table 1).
Fig. 3 shows the final result. The left-hand solid line joins the
points (e p , D p ) corresponding to the upper limit of D p for which
we have Nekhoroshev stability with the one-turn remainder criterion. The right-hand solid line joins the points of Nekhoroshev
stability determined by the N-iterated remainder. The dashed line
corresponds to a criterion with the resonant Birkhoff normal form
discussed below. It should be noted, however, that the precise calculation was done only on the points of Table 1 along the above lines.
C
C 2005 RAS, MNRAS 364, 253–271
2005 The Authors. Journal compilation Figure 3. Calculated area of Nekhoroshev stability in the space (D p , e p )
with the one-turn remainder criterion (left-hand solid line), the N-turn remainder criterion (right-hand solid line) and a non-rigorous estimate with the
resonant Birkhoff normal form (dashed line). Open circles correspond to real
asteroids with 0 I p 5◦ which are Nekhoroshev-stable for 1010 yr with
the N-turn remainder criterion, while the squares correspond to asteroids
outside the area of Nekhoroshev stability.
The domain shown to be Nekhoroshev-stable for 109 periods of
Jupiter has an area that is about 50 per cent of the total area occupied
by real asteroids on the (D p , e p ) plane. However, this area is not
uniformly populated by asteroids. We have used the data of 667
numbered objects from the AstDys data base.2 It turns out that the
percentage of asteroids in this catalogue shown to be Nekhoroshevstable is smaller. Precisely, 28 per cent of the asteroids (29 out of
102) with proper inclinations I p 5 are shown to be Nekhoroshevstable with the one-turn remainder criterion, and 35 per cent (36 out
of 102) with the iterated remainder criterion. Our previous result
was Dp = 10.◦ 6 with a simplified mapping model for e p = 0. In the
AstDys catalogue, this corresponds to a percentage of 11 per cent
of asteroids in the range I p 5◦ , e p 0.03.
We would like to point out an apparent similarity of our analytical
result to recently obtained results on the stability of Trojans by semianalytical or numerical methods (Sándor & Érdi 2003; Tsiganis
et al. 2005; Robutel et al. 2005), in the framework of a much more
realistic model, i.e. the elliptic RTBP including the effects of Saturn. Fig. 3 shows a rough agreement with the separation of asteroids
into stable or chaotic according to a Lyapunov time criterion (T L >
400 000 yr for the stable Trojans; Tsiganis et al. 2005). This agreement may not be totally coincidental. In particular, as demonstrated
by Sándor & Érdi (2003) and Robutel et al. (2005), the interaction
of the indirect effects of Saturn broaden the already existing chaotic
2
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264
C. Efthymiopoulos and Z. Sándor
of the size of the remainder, as a function of r, can be found by a
detailed analysis of the propagation of small divisors in the various terms of the series [see Servizi et al. (1983) and Efthymiopoulos, Contopoulos & Giorgilli (2004) for detailed discussions of the
mapping and Hamiltonian cases, respectively]. The outcome of this
analysis is that the series ||R (r ) || appears as ‘piecewise geometrical’,
namely, the ratio λ r =||R (r +2) /R (r ) || is almost constant within fixed
intervals of values of r, but it increases by abrupt steps at particular values of r. These values are equal to r = qn + dn , with qn
and dn integers, qn being the denominator of the nth member of the
continued fraction sequence of ω 0 . The integer dn qn is called
‘retardation’ (Turchetti 1989) or ‘delay’ [see Efthymiopoulos et al.
(2004) for an analysis and quantitative estimate of dn ]. In the case
of mappings, the piecewise constant geometrical ratio λr can be
defined by a generalization of the formula for the Hamiltonian case
(Efthymiopoulos et al. 2004)
layers in the elliptic RTBP, thus creating large overlapping domains
where chaotic diffusion may take place. However, precisely because
of this extended overlapping, the relevant region of phase space does
not have a ‘Nekhoroshev-type’ structure (Froeschlé, Guzzo & Lega
2000). On the other hand, it is likely that the small divisors generated
by the circular RTBP are the ones that produce the dominant effect of
Nekhoroshev stability in most of the open circles domain of Fig. 3,
even after the addition of terms induced by the elliptic problem or
the effects of Saturn. This is because the relevant frequencies are
well separated (Beaugé & Roig 2001), so that the divisors induced
by the extra terms are expected to affect the Birkhoff series at orders
of normalization well beyond the optimal orders of truncation cited
in Table 1. We are currently investigating this subject in detail.
4.2 Small divisors and the divergence of the Birkhoff series
It is well known that the procedure for the construction of
the Birkhoff normal form (Section 3) is generically divergent
(Rüssmann 1959). The precise meaning of this statement is the following: Except for a residual set of integrable mappings of the form
of equation (26), one has that, while for any fixed order of truncation
r, the remainder R (r ) is an absolutely convergent series, the radius of
its convergence ρ (rc ) tends to zero as r tends to infinity. Therefore, it
is impossible to transform a non-integrable mapping to an integrable
one by this formal construction.
The divergence of the Birkhoff series is attributed to the mechanism of accumulation of the so-called small divisors. The latter are
quantities ar defined by
1
1
≡ iK ω
ar
|e r 0 − 1|
λr = max
ρ
r
(72)
r
k
k=2
ak
,
(74)
0 1
2
3
5
,
,
,
,
, ....
1 12 25 37 62
Thus, new small divisors ar appear at the orders r = 1, 12, 25, 37,
62, . . . . The values of these small divisors correspond to the local
minima of the curve ar (equation 72) as a function of r (Fig. 4a). The
terms containing these divisors determine a ‘piecewise geometrical’
growth of the remainder in the Birkhoff series. The corresponding
ratio λ r , calculated by equation (74), is shown in Fig. 4(b). One sees
that each jump to a new minimum divisor, as r increases (Fig. 4a),
causes a respective jump to a higher value of the geometrical ratio
λ r in Fig. 4(b).
The behaviour of λ r shown in Fig. 4(b) determines the growth of
the size of the remainder terms in the Birkhoff series. Fig. 4(c) shows
+2)
)
the ratio of the first-order terms of the remainders ||R r(r+3
/R r(r+1
|| for
the Birkhoff normal form of the mapping with e p = 0.001. This ratio
forms ‘plateaux’ corresponding to an almost constant value for an
interval of values of r. The transition from one plateau to the next is
by quite abrupt steps. However, consecutive transitions appear with
a delay with respect to the orders of appearance of the corresponding
new small divisors. Thus, the transition to the first plateau is around
the order r = 18, while the responsible divisor is a 12 , appearing
at r = 12. Similarly, the transition to the second plateau is around
the order r = 41, while the responsible divisor is a 25 , appearing at
r = 25. Finally, the transition to the third plateau is around the order
r = 70, while the responsible divisor is a 37 , appearing at r = 37.
Thus the delay increases at consecutive transitions, in accordance
with the theoretical predictions of Efthymiopoulos et al. (2004).
+2)
)
If one calculates the ratio of the value ||R r(r+3
/R r(r+1
|| (Fig. 4c)
over the theoretically calculated value λ r (Fig. 4b), one finds a ratio
A = 7.02 for the first plateau, A = 7.93 for the second plateau,
= O(r !a ρ r ),
where A is a real positive constant depending on the size of the
non-linear part of the mapping. The ratio λr changes value abruptly
whenever r = qn , the denominator of the next fraction in the sequence of continued fraction approximations of ω 0 , or at some linear combination of the form r = mn qn + m n q n , with mn and m n
integers.
The above analysis reproduces quite accurately the asymptotic
behaviour of the Birkhoff series in the case of the mapping (14) for
the Trojan asteroids. Let us consider, for example, the small divisor
structure in the case of the mapping for e p = 0.001. The rotation
angle is ω 0 = 2π × 0.080 729 943 153 368 41. The first members of
the continued fraction sequence of ω 0 /2π are
that appear in the formal construction at every iterative step. Precisely, the form of the second recursive equation in (37) implies that
the term N of the generating transformation will contain divisors
of the form eiω0 ar with K r = κ − λ − 1, with κ + λ − 1 = r , which
are the eigenvalues of the operator ω0 .
According to well-known results of number theory (see e.g. Berry
1978), the real number ω 0 /2π is most rapidly approximated by the
sequence of rational numbers pn /qn belonging to its continued fraction representation. In particular, if ω 0 /2π is irrational, the quantity
ar in equation (72), being in general of order O(1), becomes though
quite small [of the order of O(1/ pn )] at the precise values K r = qn . A
divisor ar first appears in the series at order r = qn , and it repeatedly
appears at all successive orders beyond qn . As r increases, new divisors ar , smaller than the previous ones, appear in the series whenever
r becomes equal to the denominator qn of the succeeding fractions
in the continued fraction sequence pn /qn . This property generates
the effect of accumulation of small divisors. One can show that the
size of the generating transformation, at order r, is bounded by a
quantity
(r ) O
ρ
A|κ − λ|
, κ, λ ∈ N , κ + λ r
|e(κ−λ)ω0 − 1|
(73)
where the exponent a has the value a = 2 in the case of twodimensional mappings. A similar formula holds for the size of the
remainder R (r ) . The asymptotic character of the Birkhoff series follows from equation (73) [see, for example, the elementary example
given in Méthodes Nouvelles (Poincaré 1892), with a = 1, ρ =
1/1000].
It should be noted that, while the bound (73) can be rigorously
proven, it is by no means optimal. Optimal estimates of the growth
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2005 The Authors. Journal compilation Nekhoroshev stability estimates for Trojan asteroids
265
Figure 4. Growth of the remainder in the mapping for e p = 0.001. (a) The minimum divisor of the form (72) as a function of the order r. (b) The ratio
(r +2)
(r )
λ r (equation 74) as a function of r. (c) Precise calculation of the ratio of the size of the first terms of the remainder ||R r +3 /R r +1 || as a function of r. (d)
Comparison of the growth of the size of the remainder (solid line with small asterisks) to the growth of the size of the mapping terms (line of open circles) in a
base-10 logarithmic scale. In the latter case the growth is geometrical, while in the former case the growth ratio changes near the orders 18, 41 and 70.
and A = 5.40 for the third plateau. Thus, A is not a constant. This
is due to the fact that the non-linear part of the mapping contains
terms beyond the order r = 3. However, the variation of A has only
a small effect in the growth of the series, which is determined by
the large variation of the value of the λ r at each transition to a new
minimum divisor.
At orders r < 12 there seems to be a small plateau around the
+2)
)
value ||R r(r+3
/R r(r+1
|| = 100 (Fig. 4c). This plateau is not due to a
small divisor, but it reflects the geometrical growth of the remainder,
which follows the geometrical growth of the non-linear terms of the
mapping up to the order of appearance of the first small divisor. That
is, in the interval of values of r (1 r < 12), the growth of the size
of the remainder is dominated by the appearance, in the Birkhoff
series, of new terms generated by respective terms of the mapping
(26). These terms are produced by the second term on the RHS of
the solution (31) to the homological equation, while repetitions of
small divisors are produced by the first term on the RHS of the same
equation. For the first few orders r, the two terms have a comparable
size, because the divisor (a 1 = 0.6) is of the order of unity. This effect
is demonstrated in Fig. 4(d). The straight line of open circles gives
the norm ||Fr || of the terms of the mapping expansion at order r. This
is a convergent series, which is well represented by a geometrical
progression. A power-law fit yields
Fr = 0.955 × 9.226r
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(75)
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2005 The Authors. Journal compilation for r > 1, i.e. the radius of convergence of the polynomial mapping
expansion is rF = 1/9.226 = 0.1083.
Now, as shown in Fig. 4(d), the growth of the size of the remainder
(upper solid curve with small asterisks) up to order r = 12 is very
similar to the law (75), i.e. the two curves coincide. However, for
r beyond r = 12, the series ||R (r ) || grows by a larger geometrical
ratio than ||Fr ||. Furthermore, this ratio increases by an abrupt step
at order 41, producing a straight segment with larger slope than
the previous segment corresponding to 12 r 41. A less clear
second change of the slope is at r = 70. New segments of higher
and higher slope will appear at still higher orders, so that the radius
of convergence of the series ||R (r ) || tends to zero as r → ∞, i.e. the
Birkhoff series is asymptotically divergent.
Similar phenomena are observed in all the different mappings
for the values of e p = 0.3, 0.6, . . . , 0.21. The most important phenomenon is the increasing importance of the resonance 1/12 as e p
increases. This is shown in Figs 5 and 6.
Fig. 5 refers to the mapping for e p = 0.06, ω 0 = 2π ×
0.081 331 286 276 195 97). The continued fraction sequence of
ω 0 /2π is now 1/12, 3/37, . . . . Therefore, as shown in Fig. 5(c),
there are now two plateaux in the remainder ratio up to order 80.
The appearance of the second plateau is at order 60, meaning that the
growth of the remainder is dominated by the ratio λ 12 corresponding
to the divisor a 12 . As e p increases, ω 0 /2π approaches closer to the
value 1/12 = 0.0833. . . (Fig. 2b). Thus, when e p = 0.15, we have
266
C. Efthymiopoulos and Z. Sándor
Figure 5. Same as Fig. 4, for the mapping with e p = 0.06. The growth ratio of the remainder changes near the orders 14 and 52.
ω 0 /2π = 0.082 314 483 924 536 93. The dominance of the divisor
a 12 at this value of e p is clearly seen in Fig. 6. We found that the
plateau due to a 12 extends up to r = 60 already at e p = 0.12. This
explains why the optimal order of truncation of the series is beyond
r = 60 when e p 0.12 (Table 1). Furthermore, the same phenomenon causes an abrupt turn of the boundary of the Nekhoroshev
stability region to the left (Fig. 3) for e p 0.12.
normalized canonical variables. This is because the level curves
of formal integrals of the resonant normal form are not circles.
Precisely, the new formal integrals are the level curves of the Lie
generating function H (or ‘interpolating Hamiltonian’) for time t =
1, of a mapping Û which is equal to the normal-form mapping U
rotated by an angle −2π p/q. To find H, one solves, order by order,
the equation
e D H ζ = Û (ζ, ζ∗ ),
4.3 A non-rigorous estimate with the resonant Birkhoff
normal form
(76)
where DH stands for the Poisson bracket operator DH ≡ { · , H } and
e D H is the exponential operator determined by DH . The level curves
of H give ‘resonant’ formal integrals that account for both the nonresonant dynamics and the dynamics of one particular resonance.
In particular, the level curves of H yield deformed circles away
from the resonant domain, and island chains within the resonant
domain.
Even if Lemma 1 does not apply in the resonant construction, the
radius given by equation (60) is expected to be a good approximation
of the real size of the domain of stability if CJ is calculated by the
remainder of the resonant normal form. This is because, as shown in
the phase portraits of Fig. 1, the produced island chains are elongated
and their shape does not deviate much from that of arcs of circles,
i.e. the shape of the domain of Nekhoroshev stability is not expected
to deviate much from a disc. We thus naively proceed by considering
a disc of stability obtained in the following way: For each mapping,
we calculate the size of the remainder ||R (rres) || of the resonant normal
form and find the ratio of this quantity to the size of the remainder
of the non-resonant normal form ||R (r ) ||, as a function of the order
The results of the previous section strongly suggest that the rigorous estimate of the domain of Nekhoroshev stability given in Fig. 3
could be further improved by considering a resonant formal construction of the Birkhoff series (Bazzani et al. 1993). The essence
of this construction is to eliminate the effect of just one divisor, by
constructing a formally integrable mapping that contains resonant
terms corresponding to that particular divisor. In order to do so,
one solves equation (31) again by replacing the operator ω0 by a
new operator µ with µ = 2π p/q, where p and q are integers and
p/q is the resonance eliminated from the Birkhoff series. This new
definition changes also the splitting of the space of monomials to a
direct sum of kernel and range, by including new terms in the kernel.
These are the terms of the form ζ k+q+1 and ζ k∗ , where k is integer
and q is the denominator of the resonance (see Bazzani et al. 1993,
for details).
In the case of the resonant construction, Lemma 1 cannot be
applied to obtain a disc of Nekhoroshev stability on the plane of
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267
Figure 6. Same as Fig. 4, for the mapping with e p = 0.15. The growth ratio of the remainder changes near the order 12.
of truncation r. We then calculate a new radius of Nekhoroshev
stability
ρ =ρ
(r ) 1/ropt
Rresopt R (ropt ) ,
(77)
where ρ is the radius of stability found with the non-resonant normal
form at the optimal order of truncation = r opt given in Table 1.
Fig. 7 shows this procedure in two examples referring to the
mappings for e p = 0.001 (Figs 7a and b) and e p = 0.12 (Figs 7c and
d). The optimal order of truncation of the mapping for e p = 0.001
is r opt = 39 (Table 1). From Fig. 4(c) we find that just before r =
41 there is a rise of the remainder ratio due to the new small divisor
a 25 . We thus construct the resonant Birkhoff normal form for the
resonance p/q = 2/25. The remainder ratio for this construction
is shown as the dashed curve in Fig. 7(b). Clearly, in the resonant
construction, the plateau due to the previous divisor (a 12 ) survives
up to r = 50, where a new jump occurs due to the divisor a 37 . On the
contrary, in the non-resonant construction (solid curve with small
asterisks), the plateau due to a 25 dominates the series growth for at
least 10 consecutive orders below r = 50. This difference is reflected
in the growth of the size of the two remainders for r 40 (Fig. 7a).
Namely, the remainder of the resonant normal form at orders r 40 grows more slowly than that of the non-resonant normal form,
resulting in a difference of three orders of magnitude at r = 50.
This phenomenon becomes more pronounced as e p increases.
Fig. 7(d) shows the difference of the size of the remainders of the
non-resonant and resonant normal forms at orders r 40, which
C
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2005 The Authors. Journal compilation is by about six orders of magnitude. In this case, the resonant construction corresponds to the resonance p/q = 1/12, since the plateau
due to a 12 dominates in the non-resonant construction up to r = 40
(Fig. 7d). The elimination of the divisor a 12 in the resonant construction causes a disappearance of this plateau and a much slower
growth of the remainder of the formal series.
By implementing equation (77) in all the mappings, we obtain a
non-rigorous estimate of the domain of Nekhoroshev stability in the
space (D p , e p ) for all the mappings with 0 e p 0.21. This is shown
as a dashed line in Fig. 3. Of the asteroids with I p 5◦ from AstDys,
48 per cent (49 out of 102) are within this so-determined domain
of stability. We are presently working on a rigorous treatment of
Nekhoroshev estimates with the resonant Birkhoff normal form.
5 CONCLUSIONS
This paper reports analytic estimates of the Nekhoroshev stability
of the orbits of Jupiter’s Trojan asteroids, by calculating a region of
stability in the space of proper elements (D p , e p ) for a Nekhoroshev
time t = 1010 yr. The model used is a family of Hadjidemetriou
mappings, for different values of the proper eccentricity e p , corresponding to the Hamiltonian of co-orbital motion in the planar and
circular restricted three-body problem. Our main conclusions are as
follows:
(i) Except for a phase difference, the derived mappings reproduce with good precision the Poincaré surfaces of section of the
Hamiltonian model. In particular, the position and rotation numbers
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C. Efthymiopoulos and Z. Sándor
Figure 7. Comparison of the growth of the size of the remainder with the resonant and non-resonant normal forms. (a) The size of the remainder of the resonant
normal form (dashed line, for the resonance p/q = 2/25) and of the non-resonant normal form (solid line), as a function of r, for r 40, in the mapping with
e p = 0.001. (b) Comparison of the growth ratios for the resonant (dashed) and non-resonant (solid) normal forms in the same mapping. Panels (c) and (d) are
the same as (a) and (b), for the mapping with e p = 0.15.
regular orbits found in other studies based on numerical integrations
in the framework of the much more realistic model (elliptic RTBP
including the effects of Saturn). This agreement may not be totally
coincidental, because the large chaotic domains generated by the
overlapping of secular resonances lie for the most part outside the
domain of Nekhoroshev stability determined by the small divisors
structure of the circular RTBP model. A detailed comparison of
the domains of stability determined by various methods in various
models is in progress.
of the family of fixed points corresponding to the short-period orbit
are precisely reproduced by the mappings. Furthermore, the mappings generate the same families of higher-order (long-period) island chains as in the Hamiltonian model, and at the same position.
Finally, the difference in the calculation of proper element values in
the mapping and Hamiltonian models is smaller than 1 per cent.
(ii) An analytical theory for a rigorous calculation of a domain of
Nekhoroshev stability is developed in the framework of the Birkhoff
normal form for symplectic mappings. Optimized estimates are obtained by use of an iterated remainder criterion. The Birkhoff normal
form is constructed for the mapping models with a computer algebra
program, up to the order of expansion r = 80.
AC K N OW L E D G M E N T S
(iii) The calculated domain of Nekhoroshev stability (Fig. 3)
comprises 35 per cent of the asteroids from the AstDys catalogue
with I p 5◦ . A non-rigorous extension of the boundary of stability,
by use of the resonant Birkhoff normal form, results in increasing
this percentage to 48 per cent.
we would like to thank Professor G. Contopoulos and Drs F. Roig
and K. Tsiganis for useful discussions on the Trojan stability problem. CE received support from an Archimidis (EPEAEK) project
of the Greek Ministry of Education. Part of the normal-form calculations were done on a virtual parallel machine cluster funded by
Empirikion Foundation. ZS gratefully acknowledges the support of
the Hungarian Scientific Research Fund (OTKA) under the grants
D048424 and T043739.
(iv) The accumulation of small divisors, responsible for the divergence of the Birkhoff series, is analysed. This analysis is in agreement with the theoretical predictions of Servizi et al. (1983) and with
the analytical formulae for the growth of the size of the remainder
given in Efthymiopoulos et al. (2004). The ‘delay’ mechanism in the
accumulation of small divisors plays an important role in the choice
of quasi-resonance for the construction of the resonant Birkhoff
normal form.
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(v) The domain of Nekhoroshev stability determined analytically
by our model shows an apparent agreement with the domain of
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APPENDIX
Transformation to normal coordinates
Let the primed variables τ and x denote one iterate of the mapping (10) with initial values τ and x. Mapping (14) can be written formally as
τ = τ (τ, x),
x = x (τ, x).
(78)
Linearization around τ = τ 0 and x = x 0 , or u = v = 0, yields
u =u
∂τ ∂τ
+v
τ0 ,x0
∂τ ∂x
,
v =u
τ0 ,x0
∂x ∂τ
+v
τ0 ,x0
Denoting the elements of the Jacobian matrix by
a11 =
∂τ ∂τ
,
a12 =
τ0 ,x0
∂τ ∂x
,
a21 =
τ0 ,x0
∂x ∂τ
∂x ∂x
.
(79)
τ0 ,x0
,
τ0 ,x0
a22 =
∂x ∂x
,
(80)
τ0 ,x0
the following steps lead to the normal coordinates X and Y:
1
ϕ0 = arctan
2
b=
a22 − a11
,
a21
ωx =
X=
C
a11 − a22
a12 + a21
c=−
,
(81)
a12
,
a21
(82)
1
cos2 ϕ0 + c sin ϕ0 − b sin 2ϕ0 ,
2
ωy =
2
ωx
(u cos ϕ0 − v sin ϕ0 ),
ωy
Y =
1
sin2 ϕ0 + c cos2 ϕ0 + b sin 2ϕ0 ,
2
ωy
(u sin ϕ0 + v cos ϕ0 ).
ωx
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(84)
270
C. Efthymiopoulos and Z. Sándor
Formulae for the N-iterated remainder
We first define convenient norms for the remainder functions RJ and R φ of the mapping (47) given by equations (48) and (49), respectively.
The norm of the function R (rJ ) at the value of action J is defined by
(r ) RJ =
J
(r ) s/2
R J ,sk J ,
(85)
sr +2, |k|s
with a similar definition for the norm ||R (rφ ) || J . Both norms are real analytic functions of J in the disc J < J 0 , where J 0 is the minimum of
the radii of absolute convergence of the series R (rJ ) and R (rφ ) . Furthermore, for any J 1 and J 2 in the same disc, we have R (rJ ) J1 R (rJ ) J2 if
and only if J 1 J 2 . Similarly for the norm of R (rφ ) .
Let us consider now a fixed value of the action J ∗ , and a second value J = J∗ − N R (rJ ) J∗ where N is a fixed number of iterations. We
assume that J ∗ is small enough that J∗ − 2N R (rJ ) J∗ 0. Finally, we consider initial conditions (J (0) , φ (0) ) on the circle J (0) = J and call
(J (m) , φ (m) ), m = 1, . . . , N , the iterates of the mapping (47) with these initial conditions.
Defining now the quantity J (m) = J ∗ − J (m) , and taking into account the above inequalities on the norms, yields
|J (m) | 2N R (rJ ) J ,
(86)
∗
meaning that all the iterates (J (m) , φ (m) ), m = 1, . . . , N , belong to an annulus around the central circle J =J (0) with half-width bounded from
above by the value of R (rJ ) J∗ .
Furthermore, the angular part of the mapping (47) can be written as
φ (m+1) = φ (m) + ω(r ) J (m) + Rφ(r ) J (m) , φ (m) .
(87)
However, in view of the mean value theorem for real analytic functions, there is a positive real number ξ (m) in the interval J ∗ − J (m) ξ (m) J ∗ such that
dω(ξ (m) )
J (m) .
dJ
ω(r ) J (m) = ω(r ) (J∗ ) +
Thus we get
dω(ξ (m) )
J (m) + Rφ(r ) J (m) , φ (m) .
dJ
Applying the above equation N times, for m = 0, 1, . . . , N − 1, and adding all the equations yields
φ (m+1) = φ (m) + ω(r ) (J∗ ) +
φ (N ) = φ + N ω(r ) (J∗ ) + φ (N ) ,
where the quantity φ
(N )
dω(ξ (m) )
is defined as
N −1
φ (N ) =
dJ
m=0
(88)
J (m) +
N −1
Rφ(r ) J (m) , φ (m) ,
(89)
m=0
where φ ≡ φ (0) . The following inequality then holds for the norms:
(N ) φ 2N 2 R (rJ ) + N Rφ(r ) .
J∗
J∗
(90)
Writing the mapping of the actions as in (47), i.e. J
m = 0, 1, . . . , N − 1, yields
J (N ) = J +
N −1
(m+1)
=J
(m)
+
R (rJ )
(J
(m)
,φ
(m)
), and adding the first N equations of this mapping for
R (rJ ) (J (m) , φ (m) ).
m=0
(m)
However, implementing again the mean value theorem for real analytic functions, there are positive numbers s (m)
s (m)
J with J ∗ − J
J (m)
(m)
(r )
(r )
(m)
J ∗ and s φ with φ + mω (J ∗ ) s φ φ + mω (J ∗ ) + φ such that
R (rJ ) J (m) , φ (m) = R (rJ ) J∗ , φ + mω(r ) (J∗ ) +
∂R (rJ ) (m) (m) (m) ∂R (rJ ) (m) (m) (m)
s J , sφ J +
s J , sφ φ .
∂J
∂φ
Thus, the action mapping takes the form
J (N ) − J = R (rJ ,N ) (J∗ , φ) + S (r ,N ) (J , φ; J∗ ),
where the function
S (r ,N ) (J , φ; J∗ ) =
)
(J ∗ ,
R (r,N
J
φ) is the N-iterated remainder defined in equation (67), while the function S
N −1
∂R (r ) J
m=0
∂J
(91)
(m)
s (m)
J (m) +
J , sφ
∂R (rJ )
∂φ
(r,N )
(J , φ;J ∗ ) is defined by
(m)
s (m)
φ (m) ,
J , sφ
(92)
with all the quantities appearing in equation (92) being defined in the equations above it. Taking into account the bounds given by equations
(86) and (90), it follows that the size of the variation of the action after N iterations is bounded by the inequality
|J
(N )
(r ) (r ) (r ) (r ,N ) ∂ω N 2 ∂R (rJ ) (r ) 3 ∂R J (r ) 2 ∂R J (r ) − J | RJ
+
N
R
sup
+
R
+
2N
RJ J ,
φ
J
J∗
J∗
J∗
∗
∂φ
∂J
2
∂φ
∂J D
J∗
J∗
J∗
(93)
J∗
where D J∗ is the domain defined by J∗ − 2N R (rJ ) J∗ J J∗ .
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271
The analytic calculation of the domain of Nekhoroshev stability with the iterated remainder criterion is based on the following algorithm:
(1) For given order of truncation r, we repeat the steps 1–3 of the algorithm of Section 3.4, with the one-turn remainder R (rJ ) replaced by
)
the iterated remainder R (r,N
(J ∗ , φ), and J ∗ set equal to J ∗ = ρ 2∗ /2. The radius ρ 0 is then corrected to a new value
J
ρ0 =
ρ02 − 2 S (rJ ,N ) J ,
∗
where
(r ) (r ) (r ) (r ) 2 (r ,N ) S J = N 3 ∂R J R (rJ ) sup ∂ω + N ∂R J Rφ(r ) + 2N 2 ∂R J R (rJ ) ,
J∗
J∗
J∗
J∗
∂φ
∂J
2
∂φ
∂J D
J∗
J∗
J∗
J∗
in view of the inequality (93)
(2) For fixed r, the procedure is repeated for many values of N, until the value yielding the maximum ρ 0 is identified.
(3) Steps 1 and 2 are repeated for different orders of truncation r, until the optimal order, corresponding to the maximum value of ρ 0 , is
found.
This paper has been typeset from a TEX/LATEX file prepared by the author.
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2005 The Authors. Journal compilation