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Physica D 157 (2001) 16–39
The global flow of the quasihomogeneous potentials ofManev–Schwarzschild type
Claudia VallsDepartament de Matemática Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
Received 17 April 2000; received in revised form 5 February 2001; accepted 16 February 2001Communicated by C.K.R.T. Jones
Abstract
In this paper, we study two-body problems defined by a potential of the form V (r) = a/r + b/r2 + c/r3, where r is thedistance between the two particles and a, b, c are arbitrarily chosen constants. The Hamiltonian
H(r, pr , θ, pθ ) = 1
2
(p2
r + p2θ
r2
)+ a
r+ b
r2+ c
r3
and the angular momentum pθ = r2θ are two first integrals, independent and in involution. Let Ih (respectively Im) be theset of points on the phase space on which H (respectively pθ ) takes the value h (respectively m). Since H and pθ are firstintegrals, the sets Ih, Im and Ihm = Ih ∩ Im are invariant under the systems associated to the Hamiltonian H . We characterizethe global flow of the systems when a, b, and c vary, describing the foliation of the phase space by the invariant sets Ih,the foliation of Ih by the invariant sets Ihm and the movement of the flow over Ihm. © 2001 Elsevier Science B.V. All rightsreserved.
Keywords: Global flow; Quasihomogeneous; Manev–Schwarzschild type
1. Introduction
In this paper, we describe the global phase portraits of a two-body problem defined by a potential of the forma/r + b/r2 + c/r3, where r is the distance between the two particles and a, b, c are constants.
The study of the motion of these two-body problems has a long history. Newton himself, guided by the discrep-ancies between the observed and theoretical apsidal motion of the moon within the framework of the inverse-squareforce model, was the first to consider it (with c = 0) in his Principia. These discrepancies motivated the researchof alternative gravitational models and corrections to reconcile these differences and obtain a satisfactory degreeof agreement between the observational evidence and theoretical predictions of the motion of celestial bodies inthe Solar system, mainly the Moon, but also the planets. It was not easy to find a suitable model that maintainsthe advantages of the Newtonian one and also makes the necessary corrections such that orbits coming close tocollisions match theory with observation. In this scenario, assigning the mass M to the particle at the origin, the unit
E-mail address: [email protected] (C. Valls).
0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 1 6 7 -2 7 89 (01 )00240 -8
C. Valls / Physica D 157 (2001) 16–39 17
mass to the rotating particle and accepting the theory of relativity, the particle behaves as if the law of attraction isgiven by
f (r) = µ
r2+ 3µv−2p2
θ
r4,
where v is the velocity of light and µ = MG with G the constant of gravitation. Hence, our model correspond tothe values a = µ, b = 0 and c = µv2p2
θ . This model allows to obtain a very good theoretical approximation of theperihelion advance of Mercury and of the other inner planets as well as an accurate description of the moon’s motion.That is why this potential was used by Einstein himself as an approximation of relativity in order to compute thecorrect perihelion advance of Mercury. For special values of a, b and c, various physical and astronomical problemscan be modelized, see [14] or [4]. The motion in certain post-Newtonian fields, non-relativistic (including Manev’sone) or relativistic (as Fock’s one [10]) are in such situation. The motion around an oblate planet [15], or the motionin a photo-gravitational field [11,16] generated by a luminous also correspond to this model. Connections withatomic physics [17], or astrophysics [5] are also possible.
Taking into account that as pointed out by Hagihara [7], these potentials provide the same good theoreticalapproximations as the relativity (at least in the Solar system), nowadays there has been a series of studies concerningwith this systems, specially in the Celestial mechanics scenario. As a departure point we cite Diacu’s researches[5], then Mioc and Stoica [12,13] obtained the general solution for its regularized equations, while Diacu et al.[6] found the analytic solution and the local flow near collision showing that a black hole effect is present inthis model. The two-body problem for c = 0 (which is the well-known Manev potential and represents the firstapproximation given by the relativity), has been extensively studied by several authors and from different point ofviews. Lacomba et al. [8] studied it in Hamiltonian formalism for negative energy and a, b > 0; Delgado et al. [4]provided its analytic geometrical and physical description; Craig et al. [3] studied it for the anisotropic case; Diacu[5] studied it, pointing out the role of this potential among all quasihomogeneous potentials within the frameworkof the three-body problem; Mioc and Stoica [14] obtained the analytic solution; Aparicio and Florıa [2] studied itusing canonical coordinates and, finally, Llibre et al., give, in a recent paper [9], the general description of the phaseportraits. However, nothing is done in this direction when c �= 0. The aim of this paper is to fill in this gap, i.e.,provide the complete description of the phase portrait of the two degrees of freedom family Hamiltonian systemsof the form
H(r, pr , θ, pθ ) = 1
2
(p2
r + p2θ
r2
)+ a
r+ b
r2+ c
r3, (1.1)
where a, b, c ∈ R, (r, θ) are polar coordinates for the position and (pr , pθ ) for the momenta, respectively. Moreover,if we denote by R+ the open interval (0, ∞) then, (r, θ, pr , pθ ) ∈ E = R+ × S1 × R2.
The flow associated to (1.1) satisfies the ordinary differential equation
r = ∂H
∂pr
, θ = ∂H
∂pθ
, pr = −∂H
∂r, pθ = ∂H
∂θ. (1.2)
The above equations of motion can be rewritten in the following way. Let us define the matrix J as
J =(
0 I
−I 0
),
where I is the identity matrix 2 × 2. With this matrix J , the equations of motion (1.2) takes the form
z = J ∇H(z), z = (r, θ, pr , pθ ).
18 C. Valls / Physica D 157 (2001) 16–39
The matrix J is antisymmetric (J T = −J ) and is non-singular (det J = 1). Thus, it defines a symplectic form, ω0,over the vectorial space R4 : ω0(u, v) = uTJv with u, v ∈ R4.
At this point, let us introduce some definitions and results that will be used all through the paper.
Definition 1.1. A non-constant function F = U ⊂ R2n → R is called a first integral of a Hamilton system givenby a Hamiltonian H , if it is constant over the solution of the differential equation associated to H . It is easy to seethat this is equivalent to have {F, H } = 0.
Definition 1.2. The functions F1, F2, Fi = U ⊂ R4 → R, i = 1, 2 are in involution if {Fi, Fj } = 0 for1 ≤ i �= j < 2. Moreover, they are said to be independents if the one-forms dF1, dF2 are linearly independentsover a full Lebesgue measure subset of U .
Definition 1.3. (x, y) ∈ U ⊂ R4 is a critical point of (F1, F2) : U → R2 if the one-forms dF1(x, y) and dF2(x, y)
are linearly dependents. The points which are not critical points, are called regulars. Moreover, if (x, y) ∈ U is acritical point, then F1(x, y), F2(x, y) ∈ R2 is a critical value of the function (F1, F2) : U → R
2.
Theorem 1.4 (Sard’s theorem). The subset of critical points of a differentiable function has zero Lebesgue measure.
Theorem 1.5. If f ∈ R is a regular value for F : U ⊂ R4 → R, then If = F−1(f ) is a three-differentiablesub-manifold of U. Moreover, if (f1, f2) ∈ R2 is a regular value for (F1, F2) : U ⊂ R4 → R
2, then If1,f2 =(F1, F2)−1(f1, f2) is a 2-differentiable sub-manifold of U.
Let us observe that H and the angular momentum pθ are two first integrals due to the fact that they are constantover the solutions of (1.2), and hence, the sets
Ih = {(r, θ, pr , pθ ) ∈ E : H(r, θ, pr , pθ ) = h}, Im = {(r, θ, pr , pθ ) ∈ E : pθ(r, θ, pr , pθ ) = m}and
Ihm = Ih ∩ Im
are invariant under the flow (1.2).Furthermore, we claim that H and pθ are independent on E. Indeed, the one-forms ∇H and ∇pθ ,
∇H =(
p2θ
r3− V ′(r), 0, pr ,
pθ
r2
), ∇pθ = (0, 0, 0, 1)
are linearly dependents if and only if, there exists λ �= 0 such that ∇H = λVpθ . In that case, this gives the followingset:
C ={(r, θ, pr , pθ ) ∈ E : V ′(r) = pθ
r3, θ ∈ S1, pθ = λr2
}. (1.3)
By the definition of critical points, C is the set of critical points of the function
(H, pθ ) : E → R2
and by Sard’s theorem, it has zero Lebesgue measure. Thus, H and pθ are linearly independents on E.Moreover, using J , the Poisson bracket of H and pθ is
{H, pθ } = ∇H TJ ∇pθ = ∂H
∂r
∂pθ
∂pr
+ ∂H
∂θ
∂pθ
∂pθ
− ∂H
∂pr
∂pθ
∂r− ∂H
∂pθ
∂pθ
∂θ= 0.
Hence, H and pθ are in involution.
C. Valls / Physica D 157 (2001) 16–39 19
Therefore, the well-known Lioville–Arnold theorem, asserts that, since H is a 2-degree of freedom Hamiltoniansystem with two first independent integrals in involution, then it is integrable. Let us give a more precisely statementof this result, in the following theorem, based only towards the situations needed in this paper.
Theorem 1.6 (Liouville–Arnold theorem). The Hamiltonian system (1.1) with two degrees of freedom defined onthe phase space E has the Hamiltonian H and the angular momentum pθ as two independent first integrals ininvolution. If Ihm �= ∅ and (h, m) is a regular value of the map (H, pθ ), then the following statements hold.
1. Ihm is a two-dimensional sub-manifold of E invariant under the flow (1.2).2. The flow on a connected component I ∗
hm of Ihm is diffeomorphic either to the torus S1 × S1, or the cylinderS
1 ×R. We note that if I ∗hm is compact (i.e. I ∗
hm ≈ S1 ×S1), then the flow on it is always complete, i.e., is definedfor all t ∈ R.
3. Under the hypothesis 2, the flow on I ∗hm is conjugated to a linear flow either on S1 × S1, or on S1 × R.
For more details about Hamiltonian systems and the proof of the previous theorem, see [1]. We remark that, sincethe Manev systems are symmetric with respect to the variable θ , the manifolds I ∗
hm must have a factor S1.Roughly speaking, from the Liouville–Arnold theorem, we conclude that, if (h, m) is a regular value of the map
(H, pθ ) and Ihm �= 0, then they are two-dimensional sub-manifolds of E invariant under the flow. Moreover, if theflow on such Ihm is complete, i.e., its solutions are defined for all time, then Ihm are diffeomorphic to the union ofgeneralized cylinders Si × Rj , i + j = 2, i �= 0, where the flow on them is conjugated to a linear flow.
However, the Liouville–Arnold theorem does not give the complete description of the phase portraits for theHamiltonian given in (1.1) when a, b, c varies. More concretely, it does not provide any information of which is thetopology of the invariant sets Ihm when (h, m) is a critical value of the map (H, pθ ) and how is the flow on theseinvariant sets. Furthermore, it does not mention anything about how is the foliation of the phase space E by meansof the energy levels Ih and Ih by the invariant sets Ihm.
The main result of this paper is to give a good description of the phase portraits of the Hamiltonian family givenin (1.1) for any value of a, b, c ∈ R. More precisely, we give the description of the phase space E by the invariantsets Ih; the invariant sets Ih by Ihm and Ihm by the flow of the Hamiltonian system (1.2).
The paper is organized as follows. In Section 2, the set of critical points and critical values for the map H : E → R
is defined. In Section 3, we give the classification of the Hill regions, Rh. These regions will be one of the main toolsin order to describe the invariant sets Ih. In Section 4, guided by the classification of the Hill regions in Section 3,we derive the topology of Ih according with the different values of a, b and c. In Section 5, we describe the topologyof the sets Ihm and how they foliate Ih. Finally, in Section A we provide an appendix which summarizes the rulesused to derive the tables given in this paper.
We want to point out that, since this paper can be seen as a generalization of the one given by Llibre et al. [9],we have followed the same scheme as well as the notation used therein.
2. Critical values of H
As it has been pointed out in Section 1, from (1.2) and r > 0, system (1.3) reduces to
ar2 + 2br + 3c = 0, pr = pθ = 0.
Hence, the set of critical points of H , denoted by C, is given by
C{(r, θ, 0, 0) ∈ E : ar2 + 2br + 3c = 0, θ ∈ S1}.
20 C. Valls / Physica D 157 (2001) 16–39
Therefore, for any a, b, c ∈ R, C is equal to
1. {(r, θ, 0, 0) ∈ E : (r, θ) ∈ R+ × S1}, if a2 + b2 + c2 = 0.2. ∅, if either of the following six possibilities take place:
• a = 0 and bc ≥ 0 with b2 + c2 �= 0.• a �= 0, b = 0 and ca > 0 or a �= 0, c = 0 and ab > 0.• b2 < 3ac.• b2 = 3ac and ab > 0.• b2 > 3ac and ab > 0 with ac > 0.
3. {−3c/2b, θ, 0, 0) ∈ E : θ ∈ S1}, if a = 0 and bc < 0.4. {√−3c/a, θ, 0, 0) ∈ E : θ ∈ S1}, if b = 0 and ca < 0.5. {(−2b/a, θ, 0, 0) ∈ E : θ ∈ S1}, if c = 0 and ab < 0.6. {((−b + √
b2 − 3ac)/a, θ, 0, 0) ∈ E : θ ∈ S1}, if b2 = 3ac, ba < 0 or b2 > 3ac, ac < 0, a > 0 or b2 > 3ac,ac > 0, ab < 0.
7. {((−b − √b2 − 3ac)/a, θ, 0, 0) ∈ E : θ ∈ S1}, if b2 = 3ac, ba < 0 or b2 > 3ac, ac < 0, a < 0 or b2 > 3ac,
ac > 0, ab < 0.
Furthermore, reminding the definition of the critical values for the map H : E → R, we have that as a functionof a, b and c they are
1. 0, if a2 + b2 + c2 = 0.2. 4b3/27c2, if a = 0 and bc < 0.3. 2a
√a/3
√−3c, if b = 0 and ca < 0.4. −a2/4b, if c = 0 and ab < 0.5. −a2[2ca + b(
√b2 − 3ac − b)]/(−b + √
b2 − 3ac)3, if b2 > 3ac, ac < 0, a > 0 or b2 > 3ac, ac > 0, ab < 0.6. a2[2ca − b(
√b2 − 3ac + b)]/(b + √
b2 − 3ac)3, if b2 > 3ac, ac < 0, a < 0 or b2 > 3ac, ac > 0, ab < 0.
On the other hand, by means of Theorem 1.5, if h is a regular value for the map H : E → R, then Ih is athree-dimensional manifold.
3. Hill regions
Let us begin this section by defining in a concrete way the notion of Hill region of Ih, denoted by Rh and givenin Section1.
Definition 3.1. Let Π : E → R+ × S1 be the natural projection from the phase space E to the configuration space
R+ × S1. Then, for each h ∈ R, the Hill region Rh of Ih is defined by Rh = π(Ih).
Note that by definition, Rh is the region of the configuration space where the motion of all orbits having energyh takes place.
From our Hamiltonian family (1.1), Rh takes the form:
Rh ={
(r, θ) ∈ R+ × S1 :a
r+ b
r2+ c
r3≤ h
}≈ {r ∈ R+ : hr3 − ar2 − br − c ≥ 0} × S1,
where we have used the notation a ≈ b to express that a is diffeomorphic to b.Depending on the values of a, b, c and the critical values of H described in Section 2, Rh is diffeomorphic to
different topological objects. These objects are given in Tables 1–4. We point out that case c = 0 was previouslyobtained by Llibre et al. [9].
C. Valls / Physica D 157 (2001) 16–39 21
Table 1Rh for c = 0 and a, b, h ∈ Ra b h Rh
a < 0 b > 0 h < −a2/4b ∅h = a2/4b {a/2h} × S1
h ∈ (a2/4b, 0) [r1, r2] × S1
h = 0 [−b/a, ∞) × S1
h > 0 [r1, ∞) × S1
b = 0 h < 0 (0, a/h] × S1
h ≥ 0 R+ × S1
b < 0 h < 0 (0, r2] × S1
h ≥ 0 R+ × S1
a = 0 b > 0 h ≤ 0 ∅h > 0 [
√b/h, ∞) × S1
b = 0 h < 0 ∅h ≥ 0 R
+ × S1
b < 0 h < 0 (0,√
b/h] × S1
h ≥ 0 R+ × S1
a > 0 b > 0 h ≤ 0 ∅h > 0 [r1, ∞) × S1
b = 0 h ≤ 0 ∅h > 0 [a/h, ∞) × S1
b < 0 h < 0 (0, r2] × S1
h = 0 (0, −b/a] × S1
h ∈ (0, −a2/4b) {(0, r2] ∪ [r1, ∞)} × S1
h ≥ −a2/4b R+ × S1
Table 2Rh for a = 0, c �= 0 and b, h ∈ Rc b h Rh
c > 0 b > 0 h ≤ 0 ∅h > 0 [x1, ∞) × S1
b = 0 h ≤ 0 ∅h > 0 [(c/h)1/3, ∞) × S1
b < 0 h < 4b3/27c2 ∅h = 4b3/27c2 {x3} × S1
h ∈ (4b3/27c2, 0) [x2, x1] × S1
h = 0 [−c/b, ∞) × S1
h > 0 [x1, ∞) × S1
c < 0 b > 0 h < 0 (0, x1] × S1
h = 0 (0, −c/b] × S1
h ∈ (0, 4b3/27c2) {(0, x2] ∪ [x1, ∞)} × S1
h ≥ 4b3/27c2R
+ × S1
b = 0 h < 0 (0, (c/h)1/3] × S1
h ≥ 0 R+ × S1
b < 0 h < 0 (0, x1] × S1
h ≥ 0 R+ × S1
22 C. Valls / Physica D 157 (2001) 16–39
Table 3Rh for b = 0, a, c �= 0, h ∈ R and b �= 0, ac < 0, h ∈ Rc a b h Rh
c > 0 a < 0 b = 0 h < −2a√
a/3√−3c ∅
h = −2a√
a/3√−3c {x3} × S1
h ∈ (−2a√
a/3√−3c, 0) [x2, x1] × S1
h = 0 [√−c/a, ∞) × S1
h > 0 [x1, ∞) × S1
b �= 0 h < h2 ∅h = h2 {x3} × S1
h ∈ (h2, 0) [x2, x1] × S1
h = 0 [r1, ∞) × S1
h > 0 [x1, ∞) × S1
a > 0 b = 0 h ≤ 0 ∅h > 0 [x1, ∞) × S1
c < 0 a < 0 b = 0 h < 0 (0, x1] × S1
h ≥ 0 R+ × S1
a > 0 b = 0 h < 0 (0, x1] × S1
h = 0 (0,√−c/a] × S1
h ∈ (0, 2a√
a/3√−3c) {(0, x2] ∪ [x1, ∞)} × S1
h ≥ 2a√
a/3√−3c R
+ × S1
b �= 0 h < 0 (0, x1] × S1
h = 0 (0, r2] × S1
h ∈ (0, h1) {(0, x2] ∪ [x1, ∞)} × S1
h ≥ h1 R× S1
To give the tables, we have introduced the notation
r1,2 = a ± √a2 + 4bh
2h, r1,2 = b ± √
b2 − 4ac
−2a, h1,2 = −b(9ac − 2b2) ± 2(b2 − 3ac)3/2
27c2
= a2[2ca − b(±√b2 − 3ac + b)]
b ± √b2 − 3ac)3
and
x1 = u + v + a
3h, x2,3 = −
(u + v
2
)+ a
3h± i
√3
(u − v
2
),
where
u = (− 12q +
√D)1/3, v = (− 1
2q −√
D)1/3, D = (− 13p)3 + ( 1
2q)2,
being
p = −3bh + a2
3h2and p = −
(2a3
27h3+ ab
3h2+ c
h
).
Furthermore, when D < 0, we denote by x1, x2:
x1 = max{x1, x2, x3}, x3 = min{x1, x2, x3}
and x2 = x ∈ {x1, x2, x3} with x �= x1, x3.
C. Valls / Physica D 157 (2001) 16–39 23
Table 4Rh for ac > 0, b �= 0 and h ∈ Rc, a sg(b) b h Rh
a, c > 0 b > 0 h ≤ 0 ∅h > 0 [x1, ∞) × S1
a, c > 0 b > 0 b3 ≤ 3ac h ≤ 0 ∅h > 0 [x1, ∞) × S1
b2 ∈ (3ac, 4ac) h ≤ 0 ∅h ∈ (0, h2] [x1, ∞) × S1
h ∈ (h2, h1) {[x3, x2] ∪ [x1, ∞)} × S1
h ≥ h1 [x1, ∞) × S1
b2 = 4ac h < 0 ∅h = 0 {−b/2a} × S1
h ∈ (0, 4√
a3c3/27c2) {[x3, x2] ∪ [x1, ∞)} × S1
h ≤ 4√
a3c3/27c2 [x1, ∞) × S1
b2 > 4ac h < h2 ∅h = h2 {x3} × S1
h ∈ (h2, 0) [x2, x1] × S1
h = 0 [r1, r2] × S1
h ∈ (0, h1) {[x3, x2] ∪ [x1, ∞)} × S1
h ≥ h1 [x1, ∞) × S1
a, c < 0 b < 0 h < 0 (0, x1] × S1
h ≥ 0 R× S1
a, c < 0 b > 0 b2 ≤ 3ac h < 0 (0, x1] × S1
h ≥ 0 R× S1
b2 ∈ (3ac, 4ac) h ≤ h2 (0, x1] × S1
h ∈ (h2, h1) {(0, x3] ∪ [x2, x1]} × S1
h ∈ [h1, 0) (0, x1] × S1
h ≥ 0 R× S1
b2 = 4ac h ≤ −4√
a3h3/27c2 (0, x1] × S1
h ∈ (−4√
a3c3/27c2, 0) {(0, x3] ∪ [x2, x1]} × S1
h ≥ 0 R× S1
b2 > 4ac h ≤ h2 (0, x1] × S1
h ∈ (h2, 0) {(0, x3] ∪ [x2, x1]} × S1
h = 0 {(0, r2] ∪ [r1, ∞)} × S1
h ∈ (0, h1) {(0, x2] ∪ [x1, ∞)} × S1
h ≥ h1 R× S1
4. Energy levels Ih
From the definition of Ih, we have
Ih =⋃
(r,θ)∈Rh
Er,θ , (4.1)
where
Er,θ = {(r, θ, pr , pθ ) ∈ E : p2r + p2
θ
r2= 2
r3(hr3 − ar2 − br − c)}.
Clearly, for each (r, θ) given, the set Er,θ can be either an ellipse, a point or the empty set depending if (r, θ) belongsto the interior of Rh, the boundary of Rh or does not belong to Rh, respectively.
With the above classification of Rh together with Eq. (4.1) and some topology, it follows easily the topology ofIh according to the different values of h, a, b and c. In fact, we need to study only 11 different kinds of topology
24 C. Valls / Physica D 157 (2001) 16–39
objects, because the study of the remaining ones are topologically equivalents. We will explain how to derive theseobjects by means of the information we got on the Hill regions, as soon as they will appear. We want to remark thatcase c = 0 was already treated by Llibre et al. [9].
1. If c = 0, a < 0, b > 0; c > 0, a = 0, b < 0; c > 0, a < 0, b ∈ R, then Ih is diffeomorphic to: ∅, if h < h0;S
1, if h = h0; S3, if h ∈ (h0, 0); S3 \ S1, if h ≥ 0, where h0 = −a2/4b if c = 0, h0 = 4b3/27c2 if a = 0 andh0 = [c2(−b(9ac − 2b2) − (2(b2 − 3ac)3/2]/27c2, if c > 0, a < 0 and b ∈ R.
The third statement follows from the fact that there exists A, B ∈ R with A �= B,
Rh ≈ [A, B] × S1,
and that
π−1([A, B] × {θ}) ≈ S2,
for each fixed θ ∈ S1. Then, Ih ≈ S3 by using the Hopf fibration of S3.The fourth statement follows from the third one removing a circle. This is due to the fact that in the r variable,
we have a half-closed interval, instead of a closed interval. We remark that S3 \ S1 is diffeomorphic to an opensolid torus T3 of R3.
2. If c = 0, a < 0, b ≤ 0; c = 0, a = 0, b < 0; c < 0, a = 0, b ≤ 0; c < 0, a < 0, b = 0; c, a, b < 0;c, a < 0, b > 0 with b2 ≤ 3ac, then Ih is diffeomorphic to S3 \ S1, if h < 0; S3 \ {S1 ∪ S1}, if h ≥ 0.
In the first statement the two removed copies of S1 are disjoint and correspond to the endpoints of an openinterval in the r variable. We note that S3 \ {S1 ∪ S1} is diffeomorphic to R+ × S1 × S1.
3. If c = 0, a = 0, b > 0; c = 0, a > 0, b ≥ 0; c > 0, a = 0, b ≥ 0; c > 0, a > 0, b = 0; c, a, b > 0 orc, a > 0, b < 0 with b2 ≤ 3ac then Ih is diffeomorphic to ∅, if h ≤ 0; S3 \ S1, if h > 0.
4. If a = b = c = 0, then Ih is diffeomorphic to ∅, if h < 0; S3 \ {S1 ∪ S1}, if h > 0.5. If c = 0, a > 0, b < 0; c < 0, a = 0, b > 0; c < 0, a > 0, b ∈ R, then Ih is diffeomorphic to S3 \ S1, if
h ≤ 0; {S3 \ S1} ∪ {S3 \ S1}, if h ∈ (0, h0); Y , if h = h0; S3 \ {S1 ∪ S1}, if h > h0; where h0 = −a2/4b ifc = 0, h0 = 4b3/27c2 if a = 0 and h0 = [c2(−b(9ac − 2b2)+ (2(b2 − 3ac)3/2]/27c2, if c < 0, a > 0, b ∈ R.
The second statement follows from the fact that, for any A, B ∈ R with A �= B, then
Rh ≈ {(0, A] ∪ [B, ∞)} × S1 ≈ {(0, A] × S1} ∪ {[B, ∞) × S1}.
In the third statement, we have denoted by Y the union of two open solid tori identifying point to point thepoints of two circles of each torus which cannot be contracted into a single point inside the corresponding torus.This is due to the fact that, in that case, Rh ≈ {(0, A] ∪ [A, ∞)} × S1, being A some real number different forany of the above mentioned situations.
6. If c, a > 0, b < 0 and b2 ∈ (3ac, 4ab), then Ih is diffeomorphic to ∅, if h ≤ 0; S3 \ S1, if h ∈ (0, h2) orh > h1; Y , if h = h2; S3 ∪ {S3 \ S1}, if h ∈ (h2, h1); Y , if h = h1; S3 \ S1, if h > h1, where
h1,2 = −b(9ac − 2b2) ± 2(b2 − 3ac)3/2
27c2.
In the third statement we have denoted by Y the union of one open solid tori and a closed one identifyingpoint to point the points of two circles of each torus which cannot be contracted into a single point inside thecorresponding torus. This is due to the fact that, in that case, Rh ≈ {(0, x2] ∪ [x2, x1]} × S1.
The fourth statement follows from the fact that, if x3 �= x1, then Rh ≈ {[x2, x3] ∪ [x1, ∞)} × S1 ≈{[x2, x3] × S1} ∪ {[x1, ∞) × S1}.
C. Valls / Physica D 157 (2001) 16–39 25
7. If c, a > 0, b < 0 and b2 = 4ac, then Ih is diffeomorphic to ∅, if h < 0; S1, if h = 0; S3 ∪ {S3 \ S1}, ifh ∈ (0, 4
√a3c3/27c2); Y , if h = 4
√a3c3/27c2; S3 \ S1, if h > 4
√a3c3/27c2.
8. If c, a > 0, b < 0 and b2 > 4ac. In that case, Ih is diffeomorphic to ∅, if h < h2; S1, if h = h2; S3, ifh ∈ (h2, 0]; S3 ∪ {S3 \ S1}, if h ∈ (0, h1); Y , if h = h1; S3 \ S1, if h > h1.
9. If c, a < 0, b > 0 and b2 ∈ (3ac, 4ac), then Ih is diffeomorphic to S3\S1, if h < h2; Y , if h = h2; {S3\S1}∪S3,if h ∈ (h2, h1); Y , if h = h1; S3 \ S1, if h ∈ (h1, 0); S3 \ {S1 ∪ S1}, if h ≥ 0.
10. If c, a < 0, b > 0 and b2 = 4ac. In that case, Ih is diffeomorphic to S3 \ S1, if h < −4√
a3c3/27c2; Y , ifh = −4
√a3c3/27c2; {S3 \ S1} ∪ S3, if h ∈ (−4
√a3c3/27c2, 0); Y , if h = 0; S3 \ {S1 ∪ S1} if h > 0.
11. If c, a < 0, b > 0 and b2 > 4ac. In that case, Ih is diffeomorphic to S3 \ S1, if h < h2 or h ∈ [h1, 0); Y , ifh = h2; {S3 \ S1} ∪ S3, if h ∈ (h2, 0); ∪2(S3 \ S1) if h ∈ [0, h1); Y , if h = h1; S3 \ {S1 ∪ S1} if h > h1.
5. The invariant sets Ihm
In this section, we compute the invariant sets Ihm. We note that,
Ihm = Ih ∩ {pθ = m} ≈ (g−1(h) ∩ {pθ = m}) × S1, (5.1)
where we have used
Ih = {(r, θ, pr , pθ ) ∈ E : g(r, pr , pθ ) = h} ≈ g−1(h) × S1 (5.2)
with
g(r, pr , pθ ) = 1
2
(p2
r + p2θ
r2
)+ a
r+ b
r2+ c
r3.
If h ∈ R is a regular value of the map g : R+ ×R2 → R and g−1(h) �= 0, then g−1(h) is a surface of R+ ×R2. Itis easy to verify that the intersection of
g−1(h) with {r = r0 = cte}
is an ellipse, a point or the empty set according with hr30 − ar2
0 − br0 − c is positive, zero, or negative, respectively.So, by studying the union of ellipses or points of the form
g−1(h) ∩ {r = r0}
moving r0 > 0, we can obtain the sets g−1(h). Therefore, from (5.2), we can also calculate in a different way withrespect to the previous section, the topology of the energy levels Ih.
In this section, we focus on the description of the foliation of Ih by Ihm when h varies. In fact we need to studyin detail only 14 foliations of Ih by Ihm because the study of the remaining cases are topologically equivalent.
We remind that in [9], working with c = 0, the authors obtain the first 11 foliations of this section (Tables 5–7).In the forthcoming arguments, we have used the notation: m1 =
√−(a2 + 4bh)/2h, m2 = √−2b, m3 =√
−a2/2h, m4 =√
3(2c2h)1/3 − 2b, m5 = (− 12q0 + √
D0)1/3 + (− 12q0 − √
D0)1/3 − (a2/6h)1/2 and m6 =√m2
5 − 2b, where
D0 = 19683c4h2 + 81a4c2
h2− a9c
h4− 2187a3c3 and q0 = a6
108h3− 5a3c
h− 54c2h.
26 C. Valls / Physica D 157 (2001) 16–39
Table 5The invariant set Ih and its foliation by Ihm for c = 0, a > 0, b, h ∈ Rb h Ih Ihm Ih/S1
b > 0 h > 0 S3 \ S1
S1 × R, for m ∈ R 1
h ≤ 0 ∅ ∅b = 0 h > 0 S
3 \ S1S
1 × R, for m ∈ R 1h ≤ 0 ∅ ∅
b < 0 h > −a2/4b S3 \ {S1 ∪ S1} ⋃
2S1 × R, if |m| < m1 2⋃S1
2 S1 × R, if |m| = m1 2⋃
2S1 × R, if m1 < |m| < m2 2
S1 × R, if m2 ≤ |m| 2
h = −a2/4b Y⋃S
1
2 S1 × R, if m = 0 3⋃
2S1 × R, if 0 < |m| < m2 3
S1 × R, if m2 ≤ |m| 3
0 < h < −a2/4b⋃
2(S3 \ S1)⋃
2S1 × R, if |m| < m2 4
S1 × R, if |m| ≥ m2 4
h ≤ 0 S3 \ S1
S1 × R, if |m| < m2 1
Moreover, once h ∈ R we set
h1,2 = 1
27c2
−
(b + m2
2
)(9ac − 2
(b + m2
2
)2)±((
b + m2
2
)2
− 3ac
)3/2 ,
and then, we define m7 and m8 as the values of m for which h1 = h and h2 = h, respectively.Let us describe the foliation of Ih by means of Ihm for any value of a, b, c ∈ R.
1. Case 1: c = 0, a > 0, b ≥ 0, h > 0; c = 0, a = 0, b > 0, h > 0; c = 0, a < 0, b > 0, h ≥ 0; c > 0, a =0, b ≥ 0, h > 0; c > 0, a = 0, b < 0, h > 0; c > 0, a < 0, b ∈ R, h ≥ 0; c > 0, a > 0, b = 0, h > 0;c > 0, a > 0, b > 0, h > 0; c > 0, a > 0, b < 0, b2 ≤ 3ac, h > 0; c > 0, a > 0, b < 0, b2 ∈ (3ac, 4ac), h ∈(0, h2) or h > h1; c > 0, a > 0, b < 0, b2 = 4ac, h > 4
√a3c3/27c2; c > 0, a > 0, b < 0, b2 > 4ac, h > h1.
Under these assumptions the manifold Ih is homeomorphic to S3 \S1, i.e., the solid torus without the boundaryobtained from rotating Fig. 1 (with r0 = +∞) around the e-axis. In this picture, one can see how the cylindersIhm foliate the solid torus for m ∈ R. We also can see how the flow moves on the cylinders Ihm.
Table 6The invariant set Ih and its foliation by Ihm for c = 0, a = 0
b h Ih Ihm Ih/S1
b > 0 h > 0 S3 \ S1
S1 × R, for m ∈ R 1
h ≤ 0 ∅ ∅b = 0 h > 0 S
3 \ {S1 ∪ S1} ⋃2S
1 × R, if m = 0 6
S1 × R, if m ∈ R \ {0} 6
h = 0 R× S1S
1 × R, if m = 0
h < 0 ∅ ∅b < 0 h ≥ 0 ∅ ∅
h < 0 S3 \ S1
S1 × R, for |m| < m2 1
C. Valls / Physica D 157 (2001) 16–39 27
Table 7The invariant set Ih and its foliation by Ihm for c = 0, a < 0, b, h ∈ Rb h Ih Ihm Ih/S1
b > 0 h ≥ 0 S3 \ S1
S1 × R, for m ∈ R 1
−a2/4b < h < 0 S3
S1, if |m| = m1 7S
1 × S1, if |m| < m1 7h = −a2/4b S
1S
1, if m = 0h < −a2/4b ∅ ∅
b = 0 h ≥ 0 S3 \ {S1 ∪ S1} ⋃
2R× S, if m = 0 6R× S, if |m| �= 0 6
h < 0 S3 \ S1
S1, if |m| = m3 8S
1 × S1, if 0 < |m| < m3 8S
1 × R, if m = 0 8b < 0 h ≥ 0 S
3 \ {S1 ∪ S1} ⋃2R× S, if |m| < m2 5R× S, if |m| > m2 5
h < 0 S3 \ S1
S1, if |m| = m1 9
S1 × S1, if m2 < |m| < m1 9
R× S1, if |m| ≤ m2 9
2. Case 2: c = 0, a > 0, b < 0 and h > −a2/4b. Under these assumptions, the manifold Ih is homeomorphic toS
3 \ {S1 ∪ S1}, i.e., the solid torus of dimension 3 without the boundary and without the central circular axis.This topological space can be obtained rotating Fig. 2 around the e-axis. In this picture we can see how is thefoliation of Ih by the cylinders Ihm.• A cylinder Ihm for |m| ≥ m2.• Two cylinders Ihm for m1 < |m| < m2.• The invariant set Ihm for m = m1 (respectively m = −m1) formed by the periodic orbit α (respectively β)
together with four invariant cylinders: two of them form the stable manifold of α (respectively β) and theother two, the unstable manifold of α (respectively β).
Fig. 1. Manifold Ih/S1 for either one of the following cases: c = 0, a > 0, b ≥ 0, h > 0; c = 0, a = 0, b > 0, h > 0;c = 0, a < 0, b > 0, h ≥ 0; c > 0, a = 0, b ≥ 0, h > 0; c > 0, a = 0, b < 0, h ≥ 0; c > 0, a > 0, b = 0, h > 0; c > 0, a < 0, b ∈ R, h ≥ 0;c > 0, a > 0, b > 0, h > 0; c > 0, a > 0, b < 0, b2 = 3ac, h > 0; c > 0, a > 0, b < 0, b2 ∈ (3ac, 4ac), h ∈ (0, h2)
or h > h1; c > 0, a > 0, b < 0, b2 = 4ac, h > 4√
a3c3/27c2; c > 0, a > 0, b < 0, b2 > 4ac, h > h1;c > 0, a = 0, b < 0, b2 = 4ac, h > 4
√a3c3/27c2; c > 0, a > 0, b < 0, b2 > 4ac, h > h1; c > 0, a = 0, b < 0, h ≤ 0;
c = 0, a > 0, b < 0, h ≤ 0; c = 0, a = 0, b < 0, h < 0; c < 0, a = 0, b > 0, h ≤ 0; c < 0, a = 0, b = 0, h < 0; c < 0, a = 0, b < 0, h < 0;c < 0, a < 0, b = 0, h < 0; c < 0, a > 0, b ∈ R, h ≤ 0; c < 0, a < 0, b < 0, h < 0; c < 0, a < 0, b > 0, b2 ≤ 3ac, h < 0;c < 0, a < 0, b > 0, b2 ∈ (3ac, 4ac), h < h2 or h ∈ (0, h1); c < 0, a < 0, b > 0, b2 = 4ac, h < −4
√a3c3/27c2;
c < 0, a < 0, b > 0, b2 > 4ac, h < h2. For the first fifteen cases, r0 = +∞ and for the rest, r0 = 0.
28 C. Valls / Physica D 157 (2001) 16–39
Fig. 2. Manifold Ih/S1 for c = 0, a > 0, b < 0, h > −a2/4b.
• Two cylinders Ihm for 0 ≤ |m| < m1.We also see how the flow moves on the surfaces Ihm.
3. Case 3: c = 0, a > 0, b < 0 and h = −a2/4b. Now, the manifold Ih ≈ Y is homeomorphic to two copies of asolid torus without the boundary which have their central circular axis identified. The two copies of the solidtorus without the boundary can be obtained rotating Fig. 3 around the e-axis, and Ih is obtained identifying thecentral circular axis of both tori. Thus, the foliation of Ih by Ihm is of the following form:• The invariant set Ihm for m = 0, formed by a periodic orbit with four invariant manifolds. This periodic orbit
correspond to the central circular axis, and has two unstable and stable invariant manifolds. Each of thesefour invariant manifolds is diffeomorphic to a cylinder, and every solid torus contains one unstable and onestable invariant manifold.
• Two cylinders Ihm for 0 < |m| < m2, each contained into a different solid torus.• One cylinder Ihm for |m| ≥ m2, these cylinders are contained into the same solid torus.The foliation of Ih by all these invariant sets Ihm, is shown in Fig. 3. We also show how is the movement of theflow on the surfaces Ihm.
4. Case 4: c = 0, a > 0, b < 0 and 0 < h < −a2/4b. In this case, the manifold Ih is homeomorphic to twocopies of a solid torus without the boundary. It can be obtained rotating Fig. 4 around the e-axis. In this picture,we can see that the foliation of Ih by Ihm is given by• Two cylinders Ihm for |m| < m2 (each contained into a different solid torus).• One cylinder Ihm for |m| ≥ m2 (these cylinders are contained into the same solid torus).We also provide the movement of the flow on the cylinders Ihm.
Fig. 3. Manifold Ih/S1 for c = 0, a > 0, b < 0, h = −a2/4b.
C. Valls / Physica D 157 (2001) 16–39 29
Fig. 4. Manifold Ih/S1 for c = 0, a > 0, b < 0 and 0 < h < −a2/4b.
5. Case 5: c = 0, a < 0, b < 0, h ≥ 0; c < 0, a = 0, b < 0, h = 0. In this case, we obtain that Ih is diffeomorphicto two copies of a solid torus without the boundary. The foliation of Ih by means of Ihm is given by• Two cylinders Ihm for |m| ≤ m2 (each contained into a different solid torus).• One cylinder for |m| > m2 (these cylinders are contained into the same solid torus).Rotating Fig. 5 around the e-axis, we obtain Ih. Moreover, this figure shows how the flow moves on the cylindersIhm.
6. Case 6: c = 0, a > 0, b < 0, h ≤ 0; c = 0, a = 0, b < 0, h < 0 and c < 0, a = 0, b < 0, h = 0. Under theseassumptions, the manifold Ih is homeomorphic to a solid torus without the boundary. It can be obtained rotatingFig. 1 around the e-axis. We point out that, for the first two cases, r0 = 0 and for the last one r0 = +∞. In thispicture, we can see that the foliation of Ih by Ihm consists in one cylinder Ihm for every m such that |m| < m2.The dynamics of the flow on the cylinders Ihm is also obtained.
7. Case 7: c = 0, a = 0, b = 0, h > 0; c = 0, a < 0, b = 0, h ≥ 0 and c < 0, a = 0, b = 0, h = 0. In this casethe manifold Ih is homeomorphic to a solid torus without the boundary and the central circular axis. It can beobtained rotating Fig. 6 around the e-axis. Hence, the foliation of Ih by Ihm is as follows:• Two cylinders Ihm for m = 0.• One cylinder Ihm for every m ∈ R \ {0}.We remark that the movement of the flow on the cylinders Ihm can be also seen from Fig. 6.
Fig. 5. Manifold Ih/S1 for c = 0, a < 0, b < 0, h ≥ 0; c < 0, a = 0, b < 0, h = 0.
30 C. Valls / Physica D 157 (2001) 16–39
Fig. 6. Manifold Ih/S1 for either one of the following situations c = 0, a = 0, b = 0, h > 0; c = 0, a < 0, b = 0, h ≥ 0;c < 0, a = 0, b = 0, h = 0.
8. Case 8: c = 0, a < 0, b > 0, −a2/4b < h < 0; c > 0, a = 0, b < 0, 4b3/27c2 < h < 0; c > 0, a <
0, b ∈ R, h2 < h < 0 and c > 0, a > 0, b < 0, b2 > 4ac, h ∈ (h2, 0]. The manifold Ih is homeomorphic toS
3. In this case S3 is foliated as in the Hopf foliation, i.e., Ihm is a periodic orbit(topologically a circle) when|m| = md and a two-dimensional torus when |m| < md . Under the hypothesis of this case, md can take eitherof the following values: m1, if c = 0, m4 if a = 0, m5 if b = 0 and m6 if ac < 0, b �= 0. This foliation ispresented in Fig. 7, where we obtain the sphere S3 identifying the points (of the two surfaces of the cones gluedby their bases) which are symmetric with respect to the plane containing the common bases. In Fig. 7 one canalso see how the flow moves on the torus Ihm for |m| < md .
9. Case 9: c = 0, a < 0, b = 0, h < 0. Under these assumptions, the manifold Ih is homeomorphic to a solidtorus without the boundary. It can be obtained rotating Fig. 8 around the e-axis. In this picture we can see• A cylinder Ihm for m = 0.• A tow-dimensional torus Ihm for 0 < |m| < m3.• A periodic orbit (topologically a circle) Ihm for |m| = m3.The movement of the flow on the surfaces Ihm is also given.
10. Case 10: c = 0, a < 0, b < 0, h < 0. Under these assumptions the manifold Ih is diffeomorphic to a solidtorus without the boundary. It can be obtained rotating Fig. 9 around the e-axis. In this picture, we can see thatthe foliation of Ih by means of Ihm is as follows:
Fig. 7. Manifold Ih (modulo identifications) for either one of the following situations c = 0, a < 0, b > 0, −a2/4b < h < 0;c > 0, a = 0, b < 0, 4b3/27c2 < h < 0; c > 0, a < 0, b ∈ R, h2 < h < 0; c > 0, a > 0, b < 0, b2 > 4ac, h ∈ (h2, 0].
C. Valls / Physica D 157 (2001) 16–39 31
Fig. 8. Manifold Ih/S1 for c = 0, a < 0, b = 0, h < 0.
• A cylinder Ihm for 0 ≤ |m| ≤ m2.• A two-dimensional torus Ihm for m2 < |m| < m1.• A periodic orbit Ihm for |m| = m1.We also provide how are the dynamics of the flow on the surfaces Ihm.
11. Case 11: c < 0, a = 0, b > 0, h ≤ 0; c < 0, a = 0, b = 0, h < 0; c < 0, a = 0, b < 0, h < 0;c < 0, a < 0, b = 0, h < 0; c < 0, a > 0, b ∈ R, h ≤ 0; c < 0, a < 0, b < 0, h < 0; c < 0, a < 0, b >
0, b2 ≤ 3ac, h < 0; c < 0, a < 0, b > 0, b2 ∈ (3ac, 4ac), h < h2 or h ∈ (h1, 0); c < 0, a < 0, b > 0, b2 =4ac, h < −4
√a3c3/27c2; c < 0, a < 0, b > 0, b2 > 4ac, h < h2. Under these assumptions the manifold Ih
is homeomorphic to S3 \ S1, i.e. the solid torus without the boundary, which can be obtained rotating Fig. 1(with r0 = 0) around the e-axis. In this picture, we provide the foliation of Ih by means of the cylinders Ihm
and the movement of the flow on the cylinders.12. Case 12: c < 0, a = 0, b > 0, h > 4b3/27c2; c < 0, a = 0, b = 0, h > 0; c < 0, a = 0, b < 0, h > 0;
c < 0, a < 0, b = 0, h ≥ 0; c < 0, a > 0, b ∈ R, h > h1; c < 0, a < 0, b < 0, h ≥ 0; c < 0, a < 0, b >
0, b2 < 4ac, h ≥ 0; c < 0, a < 0, b > 0, b2 = 4ac, h > 0; c < 0, a < 0, b > 0, b2 > 4ac, h > h1. Underthese assumptions, the manifold Ih is homeomorphic to S3 \ {S1 ∪ S1}, i.e., the three-dimensional solid toruswithout the boundary and without the central circular axis. This topological space can be obtained rotatingFig. 10 around the e-axis. In this picture, we can see• Two cylinders Ihm for |m| > md .
Fig. 9. Manifold Ih/S1 for c = 0, a < 0, b < 0, h < 0.
32 C. Valls / Physica D 157 (2001) 16–39
Fig. 10. Manifold Ih/S1 for either one of the following situations: c < 0, a = 0, b > 0, h > 4b3/27c2 (with md = m4);c < 0, a = 0, b = 0, h > 0 (with md = m4); c < 0, a = 0, b < 0, h > 0 (with md = m4); c < 0, a < 0, b = 0, h ≥ 0 (withmd = m5); c < 0, a > 0, b ∈ R, h > h1 (with md = m6); c < 0, a < 0, b < 0, h ≥ 0; c < 0, a < 0, b > 0, b2 < 4ac, h ≥ 0;c < 0, a < 0, b > 0, b2 = 4ac, h > 0; c < 0, a < 0, b > 0, b2 > 4ac, h > h1 (for the last four cases, md = m7).
• The invariant set Ihm for m = md (respectively m = −md ) formed, by the periodic orbit α (respectively β)together with four invariant cylinders: two of them form the stable manifold of α (respectively β) and theother two form the unstable manifold of α (respectively β).
• Two cylinders Ihm for |m| < md .We point out that md is either m4 if a = 0, m6 if ac < 0 and m7 if ac > 0.
We also show, in Fig. 10, how the flow moves on the surfaces Ihm.13. Case 13: c < 0, a = 0, b > 0, h = 4b3/27c2; c < 0, a > 0, b ∈ R, h = h1; c < 0, a < 0, b > 0, b2 =
4ac, h = 0; c < 0, a < 0, b > 0, b2 > 4ac, h = h1. Now, the manifold Ih ≈ Y is homeomorphic to twocopies of a solid torus without the boundary and with their central circular axis identified. This object can beobtained rotating Fig. 11 around the e-axis, and Ih is obtained identifying the central circular axis of both tori.Thus, Ih is foliated in the following way:• The invariant set Ihm for m = 0 is formed by a periodic orbit with four invariant manifolds. This periodic
orbit correspond to the central circular axis and has two unstable and stable invariant manifolds. Each ofthese invariant manifolds is diffeomorphic to a cylinder, and every solid torus contains one unstable and onestable invariant manifold.
Fig. 11. Manifold Ih/S1 for either one of the following situations: c < 0, a = 0, b > 0, h = 4b3/27c2 (with r0 = −3c/2b);c < 0, a > 0, b ∈ R, h = h1; c < 0, a < 0, b > 0, b2 = 4ac, h = 0; c < 0, a < 0, b > 0, b2 > 4ac, h = h1 (the last three caseswith r0 = (−b + √
b2 − 3ac)/a).
C. Valls / Physica D 157 (2001) 16–39 33
Fig. 12. Manifold Ih/S1 for either one of the following situations: c < 0, a = 0, b > 0, 0 < h < 4b3/27c2; c < 0, a > 0, b ∈ R, 0 < h < h1;c < 0, a < 0, b > 0, b2 > 4ac, h ∈ [0, h1).
• The invariant set Ihm when m ∈ R \ {0}. This set is formed by two cylinders each contained into a differentsolid torus.
14. Case 14: c < 0, a = 0, b > 0, 0 < h < 4b3/27c2; c < 0, a > 0, b ∈ R, 0 < h < h1; c < 0, a <
0, b > 0, b2 > 4ac, h ∈ [0, h1). Under these assumptions, the manifold Ih is diffeomorphic to two copies of asolid torus without the boundary, which can be obtained rotating Fig. 12 around the e-axis. In this picture, werepresent the two cylinders (each contained into a different solid torus) that foliate Ih for m ∈ R. We can alsosee how the flow moves on the cylinders Ihm.
15. Case 15: c > 0, a > 0, b < 0, b2 ∈ (3ac, 4ac), h ∈ (h2, h1); c > 0, a > 0, b < 0, b2 = 4ac, h ∈(0, 4
√a3c3/27c2); c > 0, a > 0, b < 0, b2 > 4ac, h ∈ (0, h1); c < 0, a < 0, b > 0, b2 ∈ (3ac, 4ac), h ∈
(h2, h1); c < 0, a < 0, b > 0, b2 = 4ac, h ∈ (−4√
a3c3/27c2, 0); c < 0, a < 0, b > 0, b2 > 4ac, h ∈(h2, 0). Under these assumptions, the manifold Ih is homeomorphic to S3 ∪ {S3 \ S1}. This topological spacecan be obtained rotating Fig. 13 around the e-axis. In this picture, we can see• The cylinder Ihm for |m| > md .
Fig. 13. Manifold Ih/S1 for either one of the following situations: c > 0, a > 0, b < 0, b2 ∈ (3ac, 4ac), h ∈ (h2, h1);c > 0, a > 0, b < 0, b2 = 4ac, h ∈ (0, 4
√a3c3/27c2); c > 0, a > 0, b < 0, b2 > 4ac, h ∈ (0, h1);
c < 0, a < 0, b > 0, b2 ∈ (3ac, 4ac), h ∈ (h2, h1); c < 0, a < 0, b > 0, b2 = 4ac, h ∈ (−4√
a3c3(27c2), 0);c < 0, a < 0, b > 0, b2 > 4ac, h ∈ (h2, 0). We point out that md = m7 and r0 = (b − √
b2 − 3ac)/a if c, a < 0 and md = m8 andr0 = (b + √
b2 − 3ac)/a if c, a > 0.
34 C. Valls / Physica D 157 (2001) 16–39
Fig. 14. Manifold Ih/S1 for either one of the following situations: c > 0, a > 0, b < 0, b2 ∈ (3ac, 4ac), h = h2 or h = h1;c > 0, a > 0, b < 0, b2 = 4ac, h = 4
√a3c3/27c2; c > 0, a > 0, b < 0, b2 > 4ac, h = h1; c < 0, a < 0, b > 0, b2 ∈ (3ac, 4ac), h = h2
or h = h1; c < 0, a < 0, b > 0, b2 = 4ac, h = −4√
a3c3/27c2; c < 0, a < 0, b > 0, b2 > 4ac, h = h2. We point out thatr1 = −((a3/27h3)+(ab/h2)+(c/2h))1/3. Moreover, r0 = (b+√
b2 − 3ac)/a and, if h = h1, or h = 4√
a3c3/27c2 and r0 = (b−√b2 − 3ac)/a
and, if h = h2 or h = −4√
a3c3/27c2.
• The Ihm for m = md (respectively −md ) formed by the periodic orbit α (respectively β) together with twoinvariant cylinders and two invariant two solid tori: one cylinder and one two-torus form the stable invariantmanifold of α (respectively β) and the other cylinder and two-torus form the unstable one.
We point out that md = m7 if c, a < 0 and md = m8 if c, a > 0.From Fig. 13 we can also see how is the movement of the flow on the surfaces Ihm.
16. Case 16: c > 0, a > 0, b < 0, b2 ∈ (3ac, 4ac), h = h2 or h = h1; c > 0, a > 0, b < 0, b2 = 4ac, h =4√
a3c3/27c2, c > 0, a > 0, b < 0, b2 > 4ac, h = h1; c < 0, a < 0, b > 0, b2 ∈ (3ac, 4ac), h = h2 orh = h1; c < 0, a < 0, b > 0, b2 = 4ac, h = −4
√a3c3/27c2; c < 0, a < 0, b > 0, b2 > 4ac, h = h2.
Now, the manifold Ih ≈ Y is homeomorphic to two solid tori (one without the boundary) which have theircentral circular axis identified. This object can be obtained rotating Fig. 14 around the e-axis, and Ih is obtainedidentifying the central circular axis of both tori. Thus, Ih is foliated in the following way:• The invariant set Ihm for m = 0, formed by a periodic orbit with four invariant manifolds. This periodic orbit
correspond to the central circular axis and has two unstable and stable invariant manifolds. One of theseunstable invariant manifolds is diffeomorphic to a cylinder and the other to a solid torus. The same holdsfor the two stable invariant manifolds. The solid torus without the boundary contains the stable and unstableinvariant manifolds diffeomorphic to a cylinder and the solid torus contains the stable and unstable invariantmanifolds diffeomorphic to the two tori.
• One cylinder Ihm, when m ∈ R \ {0}, contained in the solid torus without the boundary.
We finish this section by summarizing the foliations of Ih by Ihm for all values of a, b and c.Hence, for any a, b, c ∈ R, we have given a complete description of the flow of the phase space E by the invariant
sets Ih. The invariant sets Ih by the invariant sets Ihm and the invariant sets Ihm by the flow of the Hamiltoniansystem (1.2).
Acknowledgements
The author wish to thank A. Larsson for her dedication and help in the elaboration of the graphics contained inthis paper. The author has been supported by the DGYCIT Grant PB94-0215.
C. Valls / Physica D 157 (2001) 16–39 35
Appendix A
To obtain Tables 8–12 given in this paper, we have strongly used the following rules. Let A, B, C ∈ R withc �= 0. Then, the roots of
z3 + Az2 + Bz + C = 0 (A.1)
are given by
z1 = U + V − A
3, z2,3 = −
(U + V
2
)− A
3± i
√3
(U − V
2
),
where
U =(
−Q
2+
√∆
)1/3
, V =(
−Q
2+
√∆
)1/3
and ∆ =(
−P
3
)3
+(
Q
2
)2
with
P = 3B − A2
3, Q = 2A3
27− AB
3+ C.
Table 8The invariant set Ih and its foliation by Ihm for a = 0, c, b, h ∈ Rc b h Ih Ihm Ih/S1
c > 0 b ≥ 0 h > 0 S3 \ S1
R× S1, for m ∈ R 1
h ≤ 0 ∅ ∅b < 0 h > 0 S
3 \ S1R× S1, for m ∈ R 1
h = 0 S3 \ S1
R× S1, if |m| < m2 1
4b3/27c2 < h < 0 S3
S1, if |m| = m4 7
S1 × S1, if |m| < m4 7
h = 4b3/27c2S
1S
1, if m = 0
h < 4b3/27c2 ∅ ∅c < 0 b > 0 h > 4b3/27c2
S3 \ (S1 ∪ S1)
⋃2R× S1, if |m| �= m4 10⋃S
1
2 R× S1, if |m| = m4 10
h = 4b3/27c2 Y⋃
2R× S1, if |m| > 0 11⋃S
1
2 R× S1, if m = 0 11
0 < h < 4b3/27c2 ⋃2(S3 \ S1)
⋃2R× S1, if m ∈ R 12
h ≤ 0 S3 \ S1
R× S1, for m ∈ R 1
b = 0 h > 0 S3 \ (S1 ∪ S1)
⋃2R× S1, if |m| �= m4 10⋃S
1
2 R× S1, if |m| = m4 10
h = 0 S3 \ (S1 ∪ S1)
⋃2R× S1, if m = 0 6
R× S1, if m ∈ R \ {0} 6
h < 0 S3 \ S1
R× S1, if m ∈ R 1
b < 0 h > 0 S3 \ (S1 ∪ S1)
⋃2R× S1, if |m| �= m4 10⋃S
1
2 R× S1, if |m| = m4 10
h = 0 S3 \ (S1 ∪ S1)
⋃2R× S1, if |m| ≤ m2 5
R× S1, if |m| > m2 5
h < 0 S3 \ S1
R× S1, for m ∈ R 1
36 C. Valls / Physica D 157 (2001) 16–39
Table 9The invariant set Ih and its foliation by Ihm for b = 0, c, a, h ∈ R and b �= 0, c, a < 0, h ∈ Rc a b h Ih Ihm Ih/S1
c > 0 a < 0 b ∈ R h < h2 ∅ ∅h = h2 S
1S
1, if m = 0
h2 < h < 0 S3
S1, if |m| = m6 7
S1 × S1, if |m| < m6 7
h ≥ 0 S3 \ S1
S1 × R, for m ∈ R 1
a > 0 b = 0 h ≤ 0 ∅ ∅h > 0 S
3 \ S1S
1 × R, for m ∈ R 1
c < 0 a < 0 b = 0 h < 0 S3 \ S1
R× S1, for m ∈ R 1
h ≥ 0 S3 \ {S1 ∪ S1} ⋃
2R× S1, if |m| �= m6 10⋃S
1
2 R× S1, if |m| = m6 10
a > 0 b ∈ R h ≤ 0 S3 \ S1
S1 × R, for m ∈ R 1
0 < h < h2⋃
2S3 \ S1 ⋃
2S1 × R, if m ∈ R 12
h = h2 Y⋃S
1
2 R× S1, if m = 0 11⋃2R× S1, if m ∈ R \ {0} 11
h > h2 S3 \ {S1 ∪ S1} ⋃
2R× S1, if |m| �= m6 10⋃S
1
2 R× S1, if |m| = m6 10
Moreover,
1. If ∆ > 0, the solutions of (A.1) are one real and two complex.2. If ∆ = 0, the solutions of (A.1) are all real, one simple and two multiple.3. If ∆ < 0, the solutions of (A.1) are all real and simple.
Moreover, to decide the sign of the roots we point out that
3∑i=1
zi = −A,
3∏i=1
zi = −C, z1z2 + z1z3 + z2z3 = B.
Hence,
1. If ∆ > 0, sg(z1) = −sg(C).2. If ∆ = 0, sg(z1) = −sg(C). Furthermore, if A = 0 or B = 0, then, for j = 2, 3, sg(zj ) = sg(C). On the other
hand, if A, B �= 0, then we have to take into account• If sg(B) < 0, then sg(zj ) = sg(C). Otherwise, sg(zj ) = −sg(A).Moreover, if ∆ = 0, z1 > z2 if and only if Q < 0.
3. If ∆ < 0, then• If C, A, −B < 0, all the solutions are positive.• If C, A < 0 and −B > 0 or C < 0 and A > 0, there exist only one positive solution and the rest are negative.• If C, A > 0 and −B < 0, all the solutions are negative.• If C, A, −B > 0 or C > 0 and A < 0, there exists two positive solutions and the other is negative.
C. Valls / Physica D 157 (2001) 16–39 37
Table 10The manifold Ih and its foliation by Ihm when a, c > 0, b �= 0 and h, m ∈ Rb h Ih Ihm Ih/S1
b > 0 h ≤ 0 ∅ ∅h > 0 S
3 \ S1R× S1, for m ∈ R 1
b < 0b2 ≤ 3ac h ≤ 0 ∅ ∅
h > 0 S3 \ S1
R× S1, for m ∈ R 1b2 ∈ (3ac, 4ac) h ≤ 0 ∅ ∅
h ∈ (0, h2) S3 \ S1
R× S1, for m ∈ R 1h = h2 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14h ∈ (h2, h1) S
3 ∪ {S3 \ S1} R× S1, if |m| > m8 13
R× S1 ⋃S1S
1 × S1, if m = m8 13R× S1 ⋃
S1 × S1, if |m| < m8 13
h = h1 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14h > h1 S
3 \ S1R× S1, for m ∈ R 1
b2 = 4ac h < 0 ∅ ∅h = 0 S
1S
1, if m = 0h ∈ (0, 4
√a3c3/27c2) S
3 ∪ {S3 \ S1} R× S1, if |m| > m8 13
R× S1 ⋃S1S
1 × S1, if m = m8 13R× S1 ⋃
S1 × S1, if |m| < m8 13
h = 4√
a3c3/27c2 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14h > 4
√a3c3/27c2
S3 \ S1
R× S1, for m ∈ R 1b2 > 4ac h < h2 ∅ ∅
h = h2 S1
S1, if m = 0
h ∈ (h2, 0] S3
S1, if |m| = m8 7S
1 × S1, if |m| < m8 7h ∈ (0, h1) S
3 ∪ {S3 \ S1} R× S1, if |m| > m8 13
R× S1 ⋃S1S
1 × S1, if m = m8 13R× S1 ⋃
S1 × S1, if |m| < m8 13
h = h1 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14h > h1 S
3 \ S1R× S1, for m ∈ R 1
Table 11The manifold Ih and its foliation by Ihm when a, c < 0, b �= 0, h ≥ 0 and m ∈ Rb h Ih Ihm Ih/S1
b < 0, b > 0, b2 < 4ac h ≥ 0 S3 \ {S1 ∪ S1} ⋃
2R× S1, if |m| �= m7 10⋃S
1
2 R× S1, if |m| = m7 10b > 0, b2 = 4ac h = 0 Y
⋃2R× S1, if |m| �= 0 11⋃S
1
2 R× S1, if m = 0 11h > 0 S
3 \ {S1 ∪ S1} ⋃2R× S1, if |m| �= m7 10⋃S
1
2 R× S1, if |m| = m7 10b > 0, b2 > 4ac h ∈ [0, h1)
⋃2(S3 \ S1)
⋃2R× S1, for m ∈ R 12
h = h1 Y⋃
2R× S1, if |m| �= 0 11⋃S
1
2 R× S1, if m = 0 11h > h1 S
3 \ {S1 ∪ S1} ⋃2R× S1, if |m| �= m7 10⋃S
1
2 R× S1, if |m| = m7 10
38 C. Valls / Physica D 157 (2001) 16–39
Table 12The manifold Ih and its foliation by Ihm when a, c < 0, b �= 0, h < 0 and m ∈ Rb h Ih Ihm Ih/S1
b < 0, b > 0, b2 ≤ 3ac h < 0 S3 \ S1
R× S1, for m ∈ R 1
b > 0, b2 ∈ (3ac, 4ac) h < h2 S3 \ S1
R× S1, for m ∈ R 1
h = h2 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14
h ∈ (h2, h1) S3 ∪ {S3 \ S1} R× S1, for |m| > m7 13
R× S1 ⋃S1S
1 × S1, if m = m7 13
R× S1 ⋃S
1 × S1, if |m| < m7 13
h = h1 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14
h > h1 S3 \ S1
R× S1, for m ∈ R 1
b > 0, b2 = 4ac h < −4√
a3c3/27c2S
3 \ S1R× S1, for m ∈ R 1
h = −4√
a3c3/27c2 Y R× S1, if |m| > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14
h > −4√
a3c3/27c2S
3 ∪ {S3 \ S1} R× S1, if |m| > m7 13
R× S1 ⋃S1S
1 × S1, if m = m7 13
R× S1 ⋃S
1 × S1, if |m| < m7 13
b > 0, b2 > 4ac h < h2 S3 \ S1
R× S1, for m ∈ R 1
h = h2 Y R× S1, if m > 0 14
R× S1 ⋃S1S
1 × S1, if m = 0 14
h > h2 S3 ∪ {S3 \ S1} R× S1, if |m| > m7 13
R× S1 ⋃S1S
1 × S1, if m = m7 13
R× S1 ⋃S
1 × S1, if |m| < m7 13
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