We summarize recent developments in the use of spectral methods for analyzing large numbers of orbits in N-body simulations to obtain insights into the global phase space structure of dark matter halos. The fundamental frequencies of oscillation of orbits can be used to understand the physical mechanism by which the shapes of dark matter halos evolve in response to the growth of central baryoni...

Gutzwiller's trace formula allows interpreting the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations, and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex " ghost orbits ". By stu...

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1) transversal homoclinic orbits, (2) Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite-dimensional autonomous systems, the author was able to prove t...

Using spectral methods, we analyse the orbital structure of dark matter (DM) in N -body simulations in an effort to understand the physical processes that drive the evolution of dark matter halo shapes caused by growing central masses. A longstanding issue is whether the change in the shapes of DM halos is the result of chaotic scattering of the major family of box orbits that serves as the bac...

In this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits, hyperbolic fixed points or combinations of hyperbolic fixed points and/or periodic orbits in a class of two-degree-offreedom, integrable Hamiltonian systems subject to arbitrary Hamiltonian perturbations. Our methods differ from previous methods in that the invariant sets (periodic or...

Curves in a family derived from powers of the polar coordinate formula for ellipses are found to provide good fits to bound orbits in a range of power-law potentials. This range includes the well-known 1/r (Keplerian) and logarithmic potentials. These approximate orbits, called p-ellipses, retain some of the basic geometric properties of ellipses. They satisfy and generalize Newton’s apsidal pr...

We study the families of periodic orbits of the spatial isosceles 3-body problem (for small enough values of the mass lying on the symmetry axis) coming via the analytic continuation method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular momentum, we reduce the dimension of the phase space of the problem by two units. Since periodic orbits of the r...