Orbital structure in oscillating galactic potentials

Abstract

This paper focuses on symmetric potentials subjected to periodic driving. Four unperturbed potentials V_0(r) were considered, namely the Plummer potential and Dehnen potentials with \gamma=0.0, 0.5, and 1.0, each subjected to a time-dependence of the form V(r,t)=V_0(r)(1+m_0\sin(\omega t)). In each case, the orbits divide clearly into regular and chaotic, distinctions which appear absolute. In particular, transitions from regularity to chaos are seemingly impossible. Over finite time intervals, chaotic orbits subdivide into what can be termed `sticky' chaotic orbits, which exhibit no large scale secular changes in energy and remain trapped in the phase space region where they started; and `wildly chaotic' orbits, which do exhibit systematic drifts in energy as the orbits diffuse to inhabit different phase space regions. This latter distinction is not absolute, apparently corresponding instead to orbits penetrating a `leaky' phase space barrier. The three different orbit types can be identified simply in terms of the frequencies for which their Fourier spectra have the most power. An examination of the statistical properties of orbit ensembles as a function of driving frequency \omega allows one to identify the specific resonances that determine orbital structure. Attention focuses also on how, for fixed amplitude m_0, such quantities as the mean energy shift, the relative measure of chaotic orbits, and the mean value of the largest Lyapunov exponent vary with driving frequency \omega and how, for fixed \omega, the same quantities depend on m_0.