Fractal analysis of Hopf bifurcation for a class of completely integrable nonlinear Schr?dinger Cauchy problems

摘要

We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schr?dinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schr?dinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in $L^2(Omega)$ is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schr?dinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in $mathbb{R}^n$.