Occurrence of periodic Lam\'e functions at bifurcations in chaotic Hamiltonian systems

We investigate cascades of isochronous pitchfork bifurcations of straight-line librating orbits in some two-dimensional Hamiltonian systems with mixed phase space. We show that the new bifurcated orbits, which are responsible for the onset of chaos, are given analytically by the periodic solutions of the Lam\'e equation as classified in 1940 by Ince. In Hamiltonians with C_${2v}$ symmetry, they occur alternatingly as Lam\'e functions of period 2K and 4K, respectively, where 4K is the period of the Jacobi elliptic function appearing in the Lam\'e equation. We also show that the two pairs of orbits created at period-doubling bifurcations of touch-and-go type are given by two different linear combinations of algebraic Lam\'e functions with period 8K.Comment: LaTeX2e, 22 pages, 14 figures. Version 3: final form of paper, accepted by J. Phys. A. Changes in Table 2; new reference [25]; name of bifurcations "touch-and-go" replaced by "island-chain

Improved approximate inspirals of test-bodies into Kerr black holes

We present an improved version of the approximate scheme for generating inspirals of test-bodies into a Kerr black hole recently developed by Glampedakis, Hughes and Kennefick. Their original "hybrid" scheme was based on combining exact relativistic expressions for the evolution of the orbital elements (the semi-latus rectum p and eccentricity e) with approximate, weak-field, formula for the energy and angular momentum fluxes, amended by the assumption of constant inclination angle, iota, during the inspiral. Despite the fact that the resulting inspirals were overall well-behaved, certain pathologies remained for orbits in the strong field regime and for orbits which are nearly circular and/or nearly polar. In this paper we eliminate these problems by incorporating an array of improvements in the approximate fluxes. Firstly, we add certain corrections which ensure the correct behaviour of the fluxes in the limit of vanishing eccentricity and/or 90 degrees inclination. Secondly, we use higher order post-Newtonian formulae, adapted for generic orbits. Thirdly, we drop the assumption of constant inclination. Instead, we first evolve the Carter constant by means of an approximate post-Newtonian expression and subsequently extract the evolution of iota. Finally, we improve the evolution of circular orbits by using fits to the angular momentum and inclination evolution determined by Teukolsky based calculations. As an application of the improved scheme we provide a sample of generic Kerr inspirals and for the specific case of nearly circular orbits we locate the critical radius where orbits begin to decircularise under radiation reaction. These easy-to-generate inspirals should become a useful tool for exploring LISA data analysis issues and may ultimately play a role in source detection.Comment: 25 pages, 14 figures, some typos corrected, short section on conservative corrections added, minor changes for consistency with published versio

Linear stability analysis of resonant periodic motions in the restricted three-body problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$, $0\leq \mu \leq 1/2$, that circle each other with period equal to $2\pi$. When $\mu=0$, the problem admits orbits for the massless particle that are ellipses of eccentricity $e$ with the primary of mass 1 located at one of the focii. If the period is a rational multiple of $2\pi$, denoted $2\pi p/q$, some of these orbits perturb to periodic motions for $\mu > 0$. For typical values of $e$ and $p/q$, two resonant periodic motions are obtained for $\mu > 0$. We show that the characteristic multipliers of both these motions are given by expressions of the form $1\pm\sqrt{C(e,p,q)\mu}+O(\mu)$ in the limit $\mu\to 0$. The coefficient $C(e,p,q)$ is analytic in $e$ at $e=0$ and C(e,p,q)=O(e^{\abs{p-q}}). The coefficients in front of e^{\abs{p-q}}, obtained when $C(e,p,q)$ is expanded in powers of $e$ for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass $1-\mu$