Let Z = G/Q be a complex flag manifold and let Go be a real form of G. Then the representation theory of the real reductive Lie group Go is intimately connected with the geometry of Go-orbits on Z. The open orbits correspond to the discrete series representations and their analytic continuations, closed orbits correspond to the principal series, and certain other orbits give the other series of...

The orbit method conjectures a close relationship between the set of irreducible unitary representations of a Lie group G, and admissible coadjoint orbits in the dual of the Lie algebra. We define admissibility for nilpotent coadjoint orbits of p-adic reductive Lie groups, and compute the set of admissible orbits for a range of examples. We find that for unitary, symplectic, orthogonal, general...

We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump Ši...

The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and nonco...

Relaxation oscillators can usually be represented as a feedback system with hysteresis. The relay relaxation oscillator consists of relay hysteresis and a linear system in feedback. The objective of this work is to study the existence of periodic orbits and the dynamics of coupled relay oscillators. In particular, we give a complete analysis for the case of unimodal periodic orbits, and illustr...

We study N -homoclinic orbits near a heteroclinic cycle in a reversible system. The cycle is assumed to connect two equilibria of saddle-focus type. Using Lin’s method we establish the existence of infinitely many N -homoclinic orbits for each N near the cycle. In particular, these orbits exist along snaking curves, thus mirroring the behaviour one-homoclinic orbits. The general analysis is ill...

We discuss the general characteristics of the orbits of the Martian satellites, Phobos and Deimos. We provide a concise review of the various descriptions of the orbits by both analytical theories and direct numerical integrations of their equations of motion. After summarizing the observational data used to determine the orbits, we discuss the results of our latest orbits obtained from a least...

The orbits of Weyl groups W (An) of simple An type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming points of the orbits of W (An) into points of subalgebra orbits are listed for all cases n ≤ 8 and for the infinite series of algebra-subalgebra pairs An ⊃ An−k−1 × Ak × U1, A2n ⊃ Bn, A2n−1 ⊃ Cn, A2n−1 ⊃ Dn. Numerous ...

An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using existing algorithms. Here we consider forcing relations between homoclinic orbits. We outline a general procedure for computing the forcing relation, and apply this to compute the equivalence and forcing relations f...

In this paper we present regions of motion and periodic orbits in the spatial elliptic restricted three body problem (ER3BP). Periodic orbits and regions of motion are fundamental keys to understand any dynamical system; for this reason the Hill’s surfaces or the families of halo orbits have been extensively studied in the frame of the circular restricted three body problem. It is our opinion t...