Periodic orbits in a three-dimensional galactic potential model via averaging theory

Abstract

Using averaging theory, the existence of periodic solutions is established for an axially symmetric galaxy model described by a Hamiltonian endowed with a three-dimensional potential of two Miyamoto disks galaxies rotating around its z-axis.

Appendix: Averaging theory

In this section, we shall present a basic result from averaging theory that we need for proving the results in this paper. The key tool is to provide an algorithm that computes periodic orbits based on averaging theory for Hamiltonian differential systems.

To attain this aim, we focus our attention to the problem of existence of T-periodic orbits for the differential system

$$\begin{aligned} \dot{\mathbf{x}}=F_0(t,\mathbf{x})+\varepsilon F_1(t,\mathbf{x})+\varepsilon ^2 F_2(t,\mathbf{x},\varepsilon ), \end{aligned}$$

(A.1)

for $\varepsilon$ positive and small enough, where $F_0,F_1:{\mathbb {R}}\times \Omega \rightarrow {\mathbb {R}}^n$ and $F_2:{\mathbb {R}}\times \Omega \times (-\varepsilon _0,\varepsilon _0)\rightarrow {\mathbb {R}}^n$ are ${\mathcal {C}}^2$-functions in their variables, T-periodic in the first variable and $\Omega$ is an open set in ${\mathbb {R}}^n$. Let $\mathbf{x}(t,\mathbf{z},\varepsilon )$ be the solution of system (A.1) such that $\mathbf{x}(0,\mathbf{z},\varepsilon )=\mathbf{z}$.

We assume that for the unperturbed system

$$\begin{aligned} \dot{\mathbf{x}}=F_0(t,\mathbf{x}), \end{aligned}$$

(A.2)

there exists an open subset of periodic orbits and that the linearization along a periodic orbit $\mathbf{x}(t,\mathbf{z},0)$ for (A.2) is given by

$$\begin{aligned} \dot{\mathbf{y}}=D_\mathbf{x}F_0(t,\mathbf{x}(t,\mathbf{z},0))\mathbf{y}. \end{aligned}$$

(A.3)

Let us denote by $M_{\mathbf{z}}(t)$ some fundamental matrix of the linear differential system (A.3).

Also, we assume that there exists an open set W with ${\overline{W}}\subset \Omega$ so that for each point $\mathbf{z}\in {\overline{W}}$, the orbit $\mathbf{x}(t,\mathbf{z},0)$ through $\mathbf{z}$ is T-periodic. It is said that the set ${\overline{W}}$ is isochronous, that is, all of its points are associated with periodic orbits having the same period T. The following result gives us conditions under which we are certain of the existence of bifurcations of T-periodic orbits from the periodic solutions $\mathbf{x}(t,\mathbf{z}, 0)$ associated with ${\overline{W}}.$

Let $\Pi :{\mathbb {R}}^{k}\times {\mathbb {R}}^{n-k}\rightarrow {\mathbb {R}}^{k}$ be the projection of ${\mathbb {R}}^{n}$ onto the first k coordinates, that is, $\Pi (x_1,\dots ,x_n)=(x_1,\dots ,x_k)$.

The next averaging result was stated and proved by Malkin [9] and Roseau [11]. This deals with nonlinear periodic differential systems depending on a small parameter, such that the unperturbed system has an invariant manifold of periodic solutions. More precisely, it provides sufficient conditions for which some of the periodic orbits of this invariant manifold persist after the perturbation.

Theorem 3

(Perturbations of isochronous systems) Let $\beta _0:{\overline{V}}\rightarrow {\mathbb {R}}^{n-k}$ be a ${\mathcal {C}}^2$-function, where V is an open and bounded subset in ${\mathbb {R}}^{k}$. Define the set ${\mathcal {Z}}=\left\{ \mathbf{z}_\alpha =(\alpha ,\beta _0(\alpha )),\alpha \in {\overline{V}}\right\} \subset \Omega$ so that for any $\mathbf{z}_\alpha \in {\mathcal {Z}}$ the solution $\mathbf{x}(t,\mathbf{z}_\alpha ,0)$ of (A.2) is T-periodic, and for each $\mathbf{z}_\alpha \in {\mathcal {Z}}$ there exists a fundamental matrix $M_{\mathbf{z}_\alpha }(t)$ of (A.3) such that the $n\times n$-matrix $M_{\mathbf{z}_\alpha }^{-1}(0)-M_{\mathbf{z}_\alpha }^{-1}(T)$ has the form $\begin{bmatrix} A_{k\times k} &{} \mathbf{0}_{k\times (n-k)} \\ B_{(n-k)\times k} &{} \Delta _\alpha \end{bmatrix}$ where $\Delta _\alpha$ is a $(n-k)\times (n-k)$-matrix with $\det \Delta _\alpha \ne 0$.

Consider the function ${\mathcal {F}}:{\overline{V}}\rightarrow {\mathbb {R}}^k$, defined as

$$\begin{aligned} {\mathcal {F}}(\alpha )=\Pi \left( \int _{0}^{T}M_{\mathbf{z}_\alpha }^{-1}(t)F_1(t,\mathbf{x}(t,\mathbf{z}_\alpha ))\mathrm{d}t\right) . \end{aligned}$$

Assuming that there exists $a\in V$ so that ${\mathcal {F}}(a)=\mathbf{0}$ and

$$\begin{aligned} \det \left( \frac{\mathrm{d}{\mathcal {F}}}{\mathrm{d}\alpha }(a)\right) \ne 0, \end{aligned}$$

(A.4)

then there exists a periodic solution $\gamma (t,\varepsilon )$ of system (A.1) such that $\gamma (0,\varepsilon )\rightarrow a$ as $\varepsilon \rightarrow 0.$