Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Structure of trajectories of complex-matrix eigenvalues in the Hermitian-non-Hermitian transition

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

EVOLUTIONARY TRAJECTORIES OF ULTRACOMPACT "BLACK WIDOW" PULSARS WITH VERY LOW MASS COMPANIONS

The existence of millisecond pulsars with planet-mass companions in close orbits is challenging from the stellar evolution point of view. We calculate in detail the evolution of binary systems self-consistently, including mass transfer, evaporation, and irradiation of the donor by X-ray feedback, demonstrating the existence of a new evolutionary path leading to short periods and compact donors as required by the observations of PSR J1719-1438. We also point out the alternative of an exotic nature of the companion planet-mass star.