Periodic orbits and non-integrability of Armbruster-Guckenheimer-Kim potential

In this paper we study the periodic orbits of the Hamiltonian system with the Armburster-Guckenheimer-Kim potential and its \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}^{1}$\end{document} non-integrability in the sense of Liouville-Arnold.


Introduction and statement of the main results
The main goal of this work is to study the periodic orbits and the non-integrability of the Hamiltonian system with the potential energy given by the Armburster-Guckenheimer-Kim potential, see Armbruster et al. (1989), which has often been used in the study of the dynamics of galaxies. We investigate the periodic orbits using the averaging theory and the nonintegrability is studied through the existence of periodic orbits that do not have all their multipliers equal to 1.
This Hamiltonian consists of a two dimensional harmonic potential plus the following quartic terms (1) The Hamiltonian system is given bẏ x = p x , p x = −x + ax x 2 + y 2 + bxy 2 , y = p y , p y = −y + ay x 2 + y 2 + bx 2 y. (2) As usual the dot denotes derivative with respect to the independent variable t, the time. We name (2) the Armburster-Guckenheimer-Kim Hamiltonian systems, or simply the AGK systems.
In this work we use the averaging method of first order to compute periodic orbits, see Sect. 2. This method allows to find analytically periodic orbits of the AGK systems (2) at any positive values of the energy as a function of the parameters a and b. Roughly speaking this method reduces the problem of finding periodic solutions of some differential system to the one of finding zeros of some convenient finite dimensional function. This method was also used by Jiménez-Lara and Llibre (2011aLlibre ( , 2011b. We divide the plane of parameters (a, b) in the following four parts: the two straight lines and the two regions Here the closure of a subset R of R 2 is denoted by CL(R).
Our main result on the periodic orbits of the AGK system (2) is summarized as follows.
Theorem 1 is proved in Sect. 2. In particular, Theorem 1 states that if (a, b) / ∈ L 1 then the Hamiltonian system at any positive energy level has periodic orbits and we can use these particular periodic orbits to prove our second main result about the C 1 nonintegrability in the sense of Liouville-Arnold of the AGK system (2).  with Hamiltonian H given by (1) cannot have a C 1 second first integral G such that the gradients of H and G are linearly independent at each point of the periodic orbits found in Theorem 1.

Theorem 2 The
Theorem 2 is proved in Sect. 3. The proof of Theorem 1 is based on the averaging theory for computing periodic orbits, see the Sect. 2. And the proof of Theorem 2 is based on the Poincaré's Method that allows to prove the non Liouville-Arnold integrability independently of the class of differentiability of the second first integral, see Sect. 1 for more details. The main difficulty for applying Poincaré's non-integrability method to a given Hamiltonian system is to find for such system periodic orbits having multipliers different from 1. For applying the Poincaré non-integrability theory to AGK-system, we need to study some of the periodic orbits of these system and to computer their multipliers.

Proof of Theorem 1
To prove Theorem 1 we shall apply Theorem 4 to the Hamiltonian system (2). The periodic orbits of a Hamiltonian system of more than one degree of freedom are generically on cylinders fulfilled of periodic orbits in the phase space (see Abraham and Marsden 1978), then we will not be able to apply directly Theorem 4 to a Hamiltonian system because the corresponding averaged function f 1 at the equilibrium point p will be always zero. This problem will be solved by fixing an energy level where the periodic orbits can be isolated.
To apply Theorem 4 we need a small parameter ε. In system (2) we consider the change of variables (x, p x , y, p y In the new variables, system (2) becomeṡ This system again is Hamiltonian with the Hamiltonian As the change of variables is only a scale transformation, for all ε different from zero, the original and the transformed systems (2) and (3) have essentially the same phase portrait, and additionally system (3) for ε sufficiently small is close to an integrable one.
Notice that F 1 is 2π -periodic in the variable θ , the independent variable of system (9). Averaging the function F 1 with respect to the variable θ we have We must find the zeros (r * , α * ) of f 1 (r, α), and check that the Jacobian at these points is not zero, i.e. det From f 11 (r, α) = 0 we obtain that either r = 0 or, r = ± √ 2h or α = 0, π/2, −π/2, π. The solutions r = 0 and r = − √ 2h are not good, because r > 0. So, the good solutions of f 11 (r, α) = 0 are r = √ 2h and α = 0, π/2, −π/2, π. Now we look for the solutions of f 12 (r, α) = 0. We obtain nine possible solutions (r * , α * ) with r * > 0: with corresponding values of ρ given by (8) tending to √ h for the solutions s 1 , s 2 , s 3 , s 4 when ε → 0 and tending to 0 for the solutions s 5 , s 6 , s 7 , s 8 , s 9 when ε → 0. Of course in (11) for the solutions s 8 and s 9 we assume that −1 ≤ (a − 2b)/(a + b) ≤ 1. These inequalities only occur in the two closed sectors limited by the two straight lines b = 0 and b = 2a minus the origin and contained in the quadrants {(a, b) : plane (a, b).
at the nine solutions s 1 , . . . , s 9 . Then we obtain the Jacobian at the solutions s 1 and s 2 , the Jacobian at the solutions s 3 and s 4 , the Jacobian 0 at the solutions s 5 , s 6 and s 7 , and the Jacobian at the solutions s 8 and s 9 . Notice that the above Jacobian at the solutions s 5 , s 6 and s 7 are zero, then we cannot use Theorem 4 for these solutions. However we have that for h = 0 the Jacobian is nonzero at s 1 and s 2 when b(a +b) = 0, the Jacobian is non-zero at s 3 and s 4 when (2a − b)(a + b) = 0, and the Jacobian is non-zero at s 8 and s 9 when b(b − 2a) = 0.
Summarizing, from Theorem 4, the solutions s 1 and s 2 of f (r * , α * ) = 0 provide two periodic solutions of system (9) (and consequently of the Hamiltonian system (3)  Note that if a + b = 0, then we do not have any periodic solution given by s i for i = 1, 2, 3, 4, 8, 9 because either their Jacobian is zero (for i = 1, 2, 3, 4) or they are not defined (for i = 8, 9, see (11)).

Proof of Theorem 2
We assume that we are under the assumptions of Theorem 1, and that one of the six founded periodic solutions corresponding to the solutions s 1 , s 2 , s 3 , s 4 , s 8 and s 9 exist, and that their associated Jacobians (13), (14) and (15) are nonzero. So the corresponding multipliers are not all equal to 1. Hence, under the assumptions of Theorem 1, from Theorem 3 it follows Theorem 2.
For an autonomous differential system, one of the multipliers is always 1, and its corresponding eigenvector is tangent to the periodic orbit.
A periodic solution of an autonomous Hamiltonian system always has two multipliers equal to one. One multiplier is 1 because the Hamiltonian system is autonomous, and another is 1 due to the existence of the first integral given by the Hamiltonian.
Theorem 3 If a Hamiltonian system with two degrees of freedom and Hamiltonian H is Liouville-Arnold integrable, and G is a second first integral such that the gradients of H and G are linearly independent at each point of a periodic orbit of the system, then all the multipliers of this periodic orbit are equal to 1.
Theorem 3 is due to Poincaré (1899). It gives us a tool to study the non Liouville-Arnold integrability, independently of the class of differentiability of the second first integral. The main problem for applying this theorem is to find periodic orbits having multipliers different from 1.

Appendix 2: Averaging theory of first order
Now we shall present the basic results from averaging theory that we need for proving the results of this paper.
The next theorem provides a first order approximation for the periodic solutions of a periodic differential system, for the proof see Theorems 11.5 and 11.6 of Verhulst (1991).
Consider the differential equatioṅ with x ∈ D ⊂ R n , t ≥ 0. Moreover we assume that both F 1 (t, x) and F 2 (t, x, ε) are T -periodic in t. Separately we consider in D the averaged differential equatioṅ where Under certain conditions, equilibrium solutions of the averaged equation turn out to correspond with T -periodic solutions of (18). (18) and (19). Suppose:

Theorem 4 Consider the two initial value problems
(i) F 1 , its Jacobian ∂F 1 /∂x, its Hessian ∂ 2 F 1 /∂x 2 , F 2 and its Jacobian ∂F 2 /∂x are defined, continuous and bounded by a constant independent of ε in [0, ∞) × D and ε ∈ (0, ε 0 ]. (ii) F 1 and F 2 are T -periodic in t (T independent of ε  (18) which is close to p such that ϕ(0, ε) → p as ε → 0. (c) The stability or instability of the limit cycle ϕ(t, ε) is given by the stability or instability of the singular point p of the averaged system (19). In fact, the singular point p has the stability behavior of the Poincaré map associated to the limit cycle ϕ(t, ε).
In the following we use the ideas of the proof of Theorem 4(c). For more details see the Sects. 6.3 and 11.8 of Verhulst (1991). Suppose that ϕ(t, ε) is a periodic solution of (18) corresponding to y = p an equilibrium point of the averaged system (19). Linearizing (18) in a neighborhood of the periodic solution ϕ(t, ε) we obtain a linear equation with T -periodic coefficientṡ x = εA(t, ε)x, A(t, ε) = ∂ ∂x F 1 (t, x) − F 2 (t, x, ε) x=ϕ (t,ε) .
We introduce the T -periodic matrices and it is clear that B 1 is the matrix of the linearized averaged equation. The matrix C has average zero. The near identity transformation permits to write (20) aṡ y = εB 1 y + ε A(t, ε) − B(t) y + O ε 2 .
Notice that A(t, ε) − B(t) → 0 as ε → 0, and also the characteristic exponents of (22) depend continuously on the small parameter ε. It follows that, for ε sufficiently small, if the determinant of B 1 is not zero, then 0 is not an eigenvalue of the matrix B 1 and then it is not a characteristic exponent of (22). By the near-identity transformation we obtain that system (20) has not multipliers equal to 1.