Periodic motion in perturbed elliptic oscillators revisited

We analytically study the Hamiltonian system in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{4}$\end{document} with Hamiltonian H=12(px2+py2)+12(ω12x2+ω22y2)−εV(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} H= \frac{1}{2} \bigl(p_{x}^{2}+p_{y}^{2} \bigr)+\frac{1}{2} \bigl(\omega_{1}^{2} x ^{2}+\omega_{2}^{2} y^{2} \bigr)- \varepsilon V(x,y) \end{aligned}$$ \end{document} being V(x,y)=−(x2y+ax3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x,y)=-(x^{2}y+ax^{3})$\end{document} with a∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\in\mathbb{R}$\end{document}, where ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon$\end{document} is a small parameter and ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega_{1}$\end{document} and ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega_{2}$\end{document} are the unperturbed frequencies of the oscillations along the x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x$\end{document} and y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y$\end{document} axis, respectively. Using averaging theory of first and second order we analytically find seven families of periodic solutions in every positive energy level of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H$\end{document} when the frequencies are not equal. Four of these seven families are defined for all a∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\in\mathbb{R}$\end{document} whereas the other three are defined for all a≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\ne0$\end{document}. Moreover, we provide the shape of all these families of periodic solutions. These Hamiltonians may represent the central parts of deformed galaxies and thus have been extensively used and studied mainly numerically in order to describe local motion in galaxies near an equilibrium point.


Introduction and statement of the main results
After equilibrium points the periodic solutions are the most simple non-trivial solutions of a differential system. Their study is of special interest because the motion in their neighborhood can be determined by their kind of stability. The stable periodic orbits explain the dynamics of bounded regular motion, while the unstable ones helps to understand the possible chaotic motion of the system. So, periodic orbits play a very important role in understanding the orbital structure of a dynamical system.
The general form of a potential for a two-dimensional dynamical system composed of two harmonic oscillators with cubic perturbing terms is U = 1 2 ω 2 1 x 2 + ω 2 2 y 2 + εV (x, y), where ω 1 and ω 2 are the unperturbed frequencies of the oscillator along the x and the y axes, respectively, ε is the small perturbation parameter and V is the cubic function containing the perturbed terms. We will use the perturbation function with a ∈ R. These perturbed oscillators are important because they describe the motion of a star under the gravity field of a galaxy, for more information see for instance the paper of Caranicolas (2004) and the references quoted there. The Hamiltonian associated to the potential V is H = H (x, y, p x , p y ) = 1 2 p 2 x + p 2 y + U (x, y), and the corresponding Hamiltonian system iṡ x = p x , y = p y , As usual the dot denotes derivative with respect to the time t ∈ R. Due to the physical meaning the frequencies ω 1 and ω 2 are both positive. We note that system (3) for ε = 0 can be solved. It has the solutions on the energy level H = h of the form x(t) = C 1 cos(tω 1 ) + C 2 sin(tω 1 ), y(t) = C 3 cos(tω 2 ) + C 4 sin(tω 2 ), p x (t) = −C 1 ω 1 sin(tω 1 ) + C 2 ω 1 cos(tω 1 ), p y (t) = −C 3 ω 2 sin(tω 2 ) + C 4 ω 2 cos(tω 2 ), Note that the solutions of system (3) for ε = 0 given in (4) are periodic if and only if ω 2 /ω 1 = p/q with p, q ∈ N and p, q coprime, where as usual N denotes the set of positive integers. The period of these periodic solutions is As far as we know there are no rigorous analytic studies of the existence of periodic solutions for the Hamiltonian system (3) with ω 1 = ω 2 . Periodic orbits when a = 0 have been studied by several authors from both analytical and numerical point of view by using different techniques, see for instance Contopoulos and Moutsoulas (1965), Contopoulos and Zikides (1980), Davoust (1983), Caranicolas and Innanen (1992) for ω 1 = ω 2 , or Contopoulos (1970a) for a numerical study for some values ω 1 = ω 2 . The perturbed potential with ω 2 = ω 1 and a = 0 has been studied analytically in Elipe et al. (1995), where the authors found six families of periodic orbits by using similar techniques than the ones in Miller (1991).
In this paper we will study the periodic orbits of the Hamiltonian system (3) with perturbed potential (1) by using averaging theory. More precisely, we will study the cases ω 2 = 2ω 1 and ω 2 = 3ω 1 with first and second order averaging, respectively. These cases together with the case ω 2 = ω 1 are the unique cases that we are able to study with these averaging techniques. We will prove the existence of families of periodic solutions parameterized by the energy in every energy level H = h > 0, and these families will be given explicitly up to first order in the small parameter ε. The case a = 0 has been studied by several authors so it is not considered in this work. See also Alfaro et al. (2013) for other uses of averaging theory.

Theorem 1
The following statements hold for the Hamiltonian system (3) with Hamiltonian H given in (2) and V (x, y) in (1).
(a) Using averaging theory of first order for |ε| = 0 sufficiently small at every positive energy level H = h and with ω 2 = 2ω 1 > 0, we find for the Hamiltonian system (3), four periodic solutions (two linearly stable and two unstable) bifurcating from the periodic solutions of (4) with a period tending to 2π/ω 1 as ε → 0. The two unstable periodic orbits have a stable and an unstable manifold, each one formed by two cylinders. All these periodic solutions can be written as ( where c 1 = cos τ , c 2 = cos(2τ ), s 1 = sin τ s 2 = sin(2τ ) and τ = ω 1 t. (b) Using averaging theory of second order for |ε| = 0 sufficiently small at every positive energy level H = h and with ω 2 = 3ω 1 > 0, for each a = 0 we find for the Hamiltonian system (3), three periodic solutions (two linearly stable and one unstable) bifurcating from the periodic solutions of (4) with a period tending to 2π/ω 1 as ε → 0. The unstable periodic orbit has a stable and an unstable manifold, each one formed by two cylinders. All these periodic solutions can be written as (x(t),ỹ(t),p x (t),p y (t)) + O(ε) with (x(t),ỹ(t),p x (t),p y (t)) being respectively, for i = 1, 2, 3, where c 1 = cos τ , c 3 = cos(3τ ), s 1 = sin τ , s 3 = sin(3τ ), τ = ω 1 t and r 1 , r 2 , r 3 as well as the stability of the solutions for each r i are given in the proof of the theorem.
The proof of Theorem 1 is given in Sect. 3. We note that the two solutions are explicit periodic solutions of our Hamiltonian system for all ε. Moreover they are axial periodic solutions living on the y-axis. While the other five periodic solutions described in Theorem 1 are non-axial periodic solutions. The origin of the Hamiltonian system (3) with the potential (1) is always an equilibrium point. The periodic orbits given in Theorem 1 are near the origin for small values of the energy h > 0. Near the stable periodic orbits of Theorem 1 the KAM 2-dimensional tori will persist for small values of ε, provided that the Kolmogorov condition or the isoenergetic non-degeneracy condition holds. These tori can be studied following the arguments given in the papers of Cushman et al. (2007) and Belmonte et al. (2007).
We must mention that there are other analytical results, which do not use the averaging method, for studying the periodic orbits with some resonances ω 1 : ω 2 for Hamiltonian systems with Hamiltonians of the form where ε is a small parameter. Thus, for instance in Cushman et al. (2007) the authors studied the periodic orbits with resonance 1 : ±2 using the Birkhoff normal form. In Marchesiello and Pucacco (2013) the authors studied the periodic orbits with resonance 1 : 2 using the Lie transform normal form theory. In Pucacco and Marchesiello (2014) the periodic orbits with resonance 1 : 1 and symmetry Z 2 × Z 2 are studied using normal form theory. In Schmidt and Dullin (2010) some properties of the periodic orbits with resonance p : ±q are studied using the Birkhoff normal form. We remark that the potentials W (x, y) of those papers are different between them, and different from the potential studied in this paper. More references about these normal form techniques for studying the periodic orbits of Hamiltonian systems can be found in the references quoted in those four mentioned papers.
In Sect. 2 we present a summary of the results on the averaging theory that we shall need for proving our results.

The averaging theory of first and second order
In this section we summarize the averaging theory of second order, it provides sufficient conditions for the existence of periodic solutions for a periodic differential system depending on a small parameter. See Buicȃ and Llibre (2004) for additional details and for the proofs of the results stated in this section.
Theorem 2 Consider the differential systeṁ where where Then for |ε| > 0 sufficiently small, there exists a T -periodic solution ϕ(t, ε) of the system such that ϕ(0, ε) → a when ε → 0. The kind of stability or instability of the limit cycle ϕ(t, ε) is given by the eigenvalues of the Jacobian matrix

an open and bounded set and for each
Note that a sufficient condition for showing that the Brouwer degree of a function f at a fixed point a is nonzero, is that the Jacobian of the function f at a (when it is defined) is non-zero, see Lloyd (1978).
Under the assumption (ii.1) Theorem 2 provides the averaging theory of first order, and it provides the averaging theory of second order when assumption (ii.2) holds.
Case 2.2: p = 5q. As in the previous case the averaging theory does not provide any information on the periodic solutions of (14).
Case 2.2: p = q. This case corresponds to the case ω 1 = ω 2 studied in Elipe et al. (1995). So it is not considered in this work.
Case 2.3: p = 3q. Since p and q are coprime, we take q = 1. We seek solutions of system f 2 (x) = 0 for which the Jacobian of f 2 evaluated at the solution be non-zero.
Equation (16) has the solutions of (15) and probably new ones. By doing the change of variables t = r 2 in (16) we obtain a new cubic polynomial equation g(t) = 0 with positive discriminant a 2 405a 2 + 13 2 5843390625a 6 + 604158750a 4 unless a positive real constant. So the polynomial g(t) when a = 0 has three real roots, for more details about the discriminant of a cubic polynomial see Abramowitz and Stegun (1972). Using the Descartes rule on the signs of a polynomial we can see that these roots cannot be negative, and of course they are non-zero, so they are positive and we denote them as t 1 (a, h) > t 2 (a, h) > t 3 (a, h) for a = 0 and h > 0. When a = 0 the polynomial g(t) has two roots t = 8h/27 and t = 2h. Notice that the last root does not provide solutions of equation f 22 (r, j π) = 0 because r = ± √ 2h. Now we study which of these solutions provide solutions of equation f 22 (r, j π) = 0. It is not difficult to check that the factors K 1 = (3h − 2r 2 ) and K 2 = (8h − 27(25a 2 + 1)r 2 ) in f 22 (r, j π) do not change their sing on the solutions r = ± √ t i (a, h) for all i = 1, 2, 3. In particular, Analyzing the signs of the two summands of f 22 (r, j π) we conclude that f 22 (r, j π) = 0 has the solutions: r = − √ t 1 (a, h), r = √ t 2 (a, h) and r = − √ t 3 (a, h) when either j = 0 and a > 0 or j = π and a < 0; and r = √ t 1 (a, h), r = − √ t 2 (a, h) and r = √ t 3 (a, h) when either j = 1 and a > 0 or j = 0 and a < 0. This proves statements (a) and (b).
We compute the resultant between the polynomial g(t) and g 1 (t) with respect to the variable t and we obtain a polynomial P (a, h), in the variables a and h, with the property that if the polynomials g(t) and g 1 (t) have a common root, this occurs for values of (a, h) such that P (a, h) = 0, for more information about the resultant of two polynomials see for instance Lang (1993), Olver (1999). The polynomial P (a, h) can be factorized as −26214400π 12 a 8 h 18 P 1 (a)P 2 (a)P 3 (a), where P 1 (a), P 2 (a), and P 3 (a) are polynomials in the variable a of degrees 4, 6, and 10 respectively whose coefficients are all positive. Since P (a, h) is zero if and only if a = 0 (recall that h > 0), there are no solutions of system g(t) = 0, g 1 (t) = 0 with a = 0, and consequently there are no solutions of system f 2 (x) = 0 with a = 0 having Jacobian equal to zero. On the other hand it is easy to check that the solution f 22 (r, j π) = 0 for a = 0, r = 2 3 2 3 √ h has Jacobian equal to zero. This completes the proof of the lemma.
We are interested in the sign of m(r, a, h) on the solutions of f 22 (r, j π). By proceeding as in Lemma 3 we see that there is no (a, h) with a = 0 and h > 0 such that m(r, a, h) evaluated on the solutions of f 22 (r, j π) be 0.
The first two periodic solutions in (17) are stable and the third one is unstable, whereas the first and the third periodic solutions in (18) are stable and the second one is unstable. Clearly these three solutions are different, so this completes the proof of statement (b) Theorem 1.