Manifolds Of Periodic Solutions Research Articles

Periodic Motion and Stability of Gravitational Planar Triple Systems

The stability of gravitational triple systems is a well known problem in celestial mechanics. The basic model used is the general three body problem. Many criteria estimated from the integrals of motion and zero velocity curves or from purely numerical simulations have been given in literature. In this paper we propose a different approach for the study of stability of triple systems based on the numerical computation of manifolds of periodic orbits and their linear stability. Such an approach has been used for the study of two-planet exosolar systems but here, applying the method of continuation with respect to the masses, we refer to systems where all bodies can have similar mass values. In the present work we apply the proposed approach by starting from the circular family of periodic orbits, which is known to exist for the planetary type problem, and we restrict our computations to the case of two equal masses. By considering that the system has a hierarchical structure, the constructed manifold of periodic solutions can be projected on a plane defined by the relative distance and the relative mass of the system. On such a plane a stability map can be constructed showing the stability limits on the manifold of periodic orbits.

Poincaré analysis of wave motion in ultrarelativistic electron-ion plasmas

Based on a relativistic Maxwell-fluid description, the existence of ultrarelativistic laser-induced periodic waves in an electron-ion plasma is investigated. Within a one-dimensional propagation geometry nonlinear coupling of the electromagnetic and electrostatic components occurs that makes the fourth-order problem nonintegrable. A Hamiltonian description is derived, and the manifolds of periodic solutions are studied by Poincaré section plots. The influence of ion motion is investigated in different intensity regimes. For ultrarelativistic laser intensities the phase-space structures change significantly compared to the weakly relativistic case. Ion motion becomes very important such that finally electron-ion plasmas in the far-ultrarelativistic regime behave similarly to electron-positron plasmas. The characteristic new types of periodic solutions of the system are identified and discussed.

Continuation of Periodic Orbits of a Third Order Oscillator under Autonomous Perturbation

We consider systems of nonlinear third-order differential equations depending on a small parameter such that the unperturbed system has a three dimension manifold of periodic solutions. We impose some conditions on the system in order to reduce the third order perturbed system to the second one. By using Poincare-Andronov-Hopf theorem we prove the existence of many periodic solutions the perturbed system.

Bifurcation of Periodic Solutions of Integrodifferential Systems with Applications to Time Delay Models in Population Dynamics

A Fredholm alternative is proved for a general linear system of Stieltjes integrodifferential equations. This result is used to derive necessary and sufficient conditions for the bifurcation of nontrivial periodic solutions of a nonlinear perturbation of the system containing n parameters. The results are applied to several models from mathematical ecology which describe the dynamics of various species interactions (included are models of mutualistic, competitive and predator-prey interactions) under the influence of time delays. These applications illustrate how, for such models, the existence of multi-dimensional manifolds of periodic solutions of various periods in the presence of unstable equilibria can occur as a result of the presence of time delays at least for birth rates near certain ciritical values.