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

Abstract

In this paper we study the periodic orbits of the Hamiltonian system with the Armburster-Guckenheimer-Kim potential and its $\mathcal{C}^{1}$ non-integrability in the sense of Liouville-Arnold.

Appendices

Appendix 1: Periodic orbits and the Liouville-Arnold integrability

We recall that a Hamiltonian system with Hamiltonian H of two degrees of freedom is integrable in the sense of Liouville-Arnold if it has a first integral G independent with H (i.e. the gradient vectors of H and G are independent in all the points of the phase space except perhaps in a set of zero Lebesgue measure), and in involution with H (i.e. the parenthesis of Poisson of H and G is zero). The Liouville-Arnold Theorem describe the dynamics of the integrable Hamiltonian systems, see for more details see Abraham and Marsden (1978), Arnold et al. (2006) and the Sect. 7.1.2 of Arnold et al. (2006), respectively.

We consider the autonomous differential system

$$ \dot{x} =f(x), $$

(16)

where f:U→ℝⁿ is C ², U is an open subset of ℝⁿ and the dot denotes the derivative respect to the time t. We write its general solution as ϕ(t,x ₀) with ϕ(0,x ₀)=x ₀∈U and t belonging to its maximal interval of definition.

We say that ϕ(t,x ₀) is T-periodic with T>0 if and only if ϕ(T,x ₀)=x ₀ and ϕ(t,x ₀)≠x ₀ for t∈(0,T). The periodic orbit associated to the periodic solution ϕ(t,x ₀) is γ={ϕ(t,x ₀),t∈[0,T]}. The variational equation associated to the T-periodic solution ϕ(t,x ₀) is

$$ \dot{M}= \biggl(\frac{\partial f(x)}{\partial x} \biggm{|}_{x =\phi(t,x_0)} \biggr) M, $$

(17)

where M is an n×n matrix. The monodromy matrix associated to the T-periodic solution ϕ(t,x ₀) is the solution M(T,x ₀) of (17) satisfying that M(0,x ₀) is the identity matrix. The eigenvalues λ of the monodromy matrix associated to the periodic solution ϕ(t,x ₀) are called the multipliers of the periodic orbit.

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 equation

$$ \dot{\mathbf{x}}={\varepsilon }F_1(t,\mathbf{x})+{ \varepsilon }^2 F_2(t,\mathbf{x},{\varepsilon }), \quad \mathbf{x}(0)=\mathbf{x}_0 $$

(18)

with x∈D⊂ℝⁿ,  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 equation

$$ \dot{\mathbf{y}}={\varepsilon }f_1(\mathbf{y}),\quad \mathbf {y}(0)=\mathbf{x}_0, $$

(19)

where

$$f_1(\mathbf{y})=\dfrac{1}{T}\int_0^TF_1(t, \mathbf{y})\,dt. $$

Under certain conditions, equilibrium solutions of the averaged equation turn out to correspond with T-periodic solutions of (18).

Theorem 4

Consider the two initial value problems (18) and (19). Suppose:

• (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).

• (iii) y(t) belongs to Ω on the interval of time [0,1/ε].

Then the following statements hold.

• (a) For t∈[1,ε] we have that x(t)−y(t)=O(ε), as ε→0.

• (b) If p is a singular point of the averaged (19) and

  $$ \det \biggl(\dfrac{\partial f_1}{\partial\mathbf{y}} \biggr)\biggm{|}_{\mathbf{y}=p}\neq0, $$

  then there exists a T-periodic solution φ(t,ε) of (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 coefficients

(20)

We introduce the T-periodic matrices

From Theorem 4(c) we have

$$\displaystyle\lim_{{\varepsilon }\rightarrow0} A(t, {\varepsilon })= B(t), $$

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

$$ x \longmapsto y= \bigl(I- {\varepsilon }C(t) \bigr) x, $$

(21)

permits to write (20) as

$$ \dot{y}= {\varepsilon }B_1 y+ {\varepsilon } \bigl[A(t, { \varepsilon })- B(t) \bigr] y+ O \bigl({\varepsilon }^2 \bigr). $$

(22)

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.