Non-integrability, stability and periodic solutions for a quartic galactic potential in a rotating reference frame

Abstract

This work aims to study the influence of the rotation of the galaxy which it is modelled as a bi-symmetrical potential consists of a two-dimensional harmonic oscillator with quartic perturbing terms on some dynamics aspects for the problem of the motion of stars. We prove analytically the non-integrability of the motion (i.e., the motion is chaotic) when the parameters meet certain conditions. Poincaré surface of section is introduced as a numerical method that is employed to confirm the obtained analytical results. We present the equilibrium points and examine their stability. We also clarify the force resulting from the rotating frame serves as a stabilizer for the maximum equilibrium points. We illustrate graphically the size of stability zones depends on the value of the angular velocity for the frame. Based on the Lyapunov theorem, the periodic solutions are constructed near the equilibrium point. Additionally, we prove the existence of one or two families of periodic solutions according to the equilibrium point is either saddle or stable, respectively. The permitted zones of possible motion are delimited and they are graphically explained for different values of the parameters.

Appendix A: Kowalevski exponents

An autonomous dynamical system which takes the form

$$ \frac{dz_{j}}{dt}=\mathcal{K}_{j}(z_{1}, z_{2}, \dots , z_{m}), j=1, 2, \dots , m, $$

(51)

is similarity-invariant system if it is invariant under the transformation

$$ t\rightarrow \alpha ^{-1}t,\quad z_{j}\rightarrow \alpha ^{g_{j}}z_{j}, j=1, 2, \dots , m, $$

(52)

where $\alpha$ and $g_{i}$ are arbitrary constants. The similarity invariant system (51) is generally has the particular solution

$$ z_{j}=c_{j}t^{-g_{j}}, j=1, 2, \dots , m, $$

(53)

where $c_{i}$ are non-zero solutions of the equations

$$ \mathcal{K}_{j}(c_{1}, c_{2}, \dots , c_{m})+g_{j}c_{j}=0, j=1, 2, \dots , m. $$

(54)

The Kowalevski matrix $K=[k_{ij}]$ is determined by

$$ K_{ji}=\frac{\partial \mathcal{K}_{j}}{\partial z_{j}}+\delta _{ji}g_{j}, j,i=1, 2, \dots , m. $$

(55)

The Kowalevski exponents are the eigenvalues of the Kowalevski matrix and we refer to them by a symbol $\rho$ (for more details, see, e.g. Yoshida 1983a; Yoshida 1983b).

Theorem 5

Let us assume there is at least one Kowalevski exponents for the similarity invariant system is irrational or complex number, then the similarity invariant system is not algebraically integrable.

For a non-similarity dynamical system, we search for an invariant transformation converting it to a similarity invariant system when the constant $\alpha$ tends to zero or infinity.