arXiv subjects

Jason J. Bramburger

Publications and source records attributed to Jason J. Bramburger.

39 records · Page 3Linked to original sources

arXiv2019

Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators

It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generator given by a directed graph Laplacian operator, through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficient conditions to extend ultracontractive properties of undirected graph Laplacians to those of the directed variety. Such directed graph Laplacians can often be observed in the study of coupled oscillators, where recent work made explicit the link between synchronous patterns to systems of identically coupled oscillators and ultracontractive properties of undirected graph semigroups. Therefore, in this work we demonstrate the applicability of our results on directed graph semigroups by extending the aforementioned investigation beyond the idealized case of identically coupled oscillators.

math.DS↗

arXiv2018

Rotating Wave Solutions to Lattice Dynamical Systems II: Persistence Results

This work comes as the second part in a series of investigations into the dynamics of rotating waves as solutions to lattice dynamical systems. Such nonlinear waves as solutions to mathematical equations are of great interest throughout the physical sciences due to their association with many electrophysiological pathologies and this investigation aims to further the understanding of rotating waves from a mathematical perspective. Here we focus on so-called Lambda-Omega differential equations, a well-studied generalization of the celebrated Ginzburg-Landau equation, to show that there exists an interval of sufficiently small coupling values for which a rotating wave solution persists. This result is achieved using a wide range of functional analytic tools, primarily in an effort to apply a non-standard Implicit Function Theorem. This work initiates subsequent studies into the dynamics and bifurcations of rotating waves away from the reaction-diffusion equation setting to differential equations for which traditional symmetry based centre manifold reductions cannot be applied.

math.DS↗

arXiv2018

Rotating Wave Solutions to Lattice Dynamical Systems I: The Anti-Continuum Limit

Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the model exhibiting a continuous Euclidean symmetry, this work is aimed at understanding these nonlinear waves in the absence of such symmetries. Here we will consider a spatially discrete lattice dynamical system of Ginzburg-Landau type and prove the existence of rotating waves in the anti-continuum limit. This result is achieved by providing a link between the work on phase systems stemming from the study of identically coupled oscillators on finite lattices to carefully track the solutions as the size of the lattice grows. It is shown that in the infinite square lattice limit of these phase systems that a rotating wave solution exists, which can be extended to the Ginzburg-Landau system of study here. The results of this work provide a necessary first step in the investigation of rotating waves as solutions to lattice dynamical systems in an effort to understand the dynamics of such solutions outside of the idealized situation where the underlying symmetry of a differential equation can be exploited.

math.DS↗