Manifolds Of Unstable Periodic Orbits Research Articles

Transversal heteroclinic and homoclinic orbits in singular perturbation problems

In this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations. By using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem. We derive analytical conditions for transversality. We show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters. We describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics.

Multifractality, multifractal phase transitions, and symmetry-increasing bifurcations in ac-driven phase-slip centers.

Chaos at and beyond onset is studied for nonequilibrium current-carrying dissipative states in quasi-one-dimensional dirty superconductors. For the case of ac-driving currents, phase-slip center solutions of the generalized time-dependent Ginzburg-Landau equations show a universal transition at the onset of chaos. For currents below the onset, the pervasive feature is the nonhyperbolicity: homoclinic tangency between stable and unstable manifolds of unstable periodic orbits. Pointwise dimensions \ensuremath{\alpha}(x) evaluated on the attractors show abnormally low values, indicating regions made of an overlapping of stable and unstable manifolds of saddle orbits. These regions cause a break of self-similarity and a phaselike transition in the multifractal probability measure of the attractor. Finally, one of these tangencies causes a symmetry-increasing bifurcation.

Chaotic behavior of a classical coupled Morse system around the escape energy region

In the geometrical optics approximation, stable and unstable manifolds of periodic orbits, invariant tori, and hyperbolic invariant manifolds are shown to exist and produce trapping of bundles of light rays near the axis of a translation-invariant, axisymmetric optical fiber whose squared refractive index is a parabolic function of squared radius. Periodic symmetry-breaking perturbations in the refractive index are shown to destroy this ray trapping and to produce homoclinic tangles, through which nearby trapped rays may escape, and untrapped rays may become at least temporarily trapped. Melnikov's technique is used to prove that the perturbations cause the stable and unstable manifolds of the unperturbed periodic orbits, invariant tori, and hyperbolic invariant manifolds to develop transverse intersections and therefore, to form homoclinic tangles. These tangles imply either homoclinic chaos by the Smale horseshoe mechanism and the Poincaré-Birkhoff-Smale theorem, or Arnol'd diffusion. In both situations, lobe dynamics will dominate phase space transport, seen here as a flux of (initially) untrapped rays passing through the trapping region.