Limit Processes and Bifurcation Theory of Quasi-Diffusive Perturbations

TL;DR

This paper develops a new theoretical framework for understanding fluctuations near bifurcation points in dynamical systems with diffusion-like perturbations, focusing on one-dimensional cases.

Contribution

It introduces a limit process approach for analyzing bifurcations under quasi-diffusive perturbations, extending classical bifurcation theory.

Findings

Describes limit processes near bifurcation points

Focuses on one-dimensional dynamical systems

Provides groundwork for future multi-dimensional analysis

Abstract

The bifurcation theory of ordinary differential equations (ODEs), and its application to deterministic population models, are by now well established. In this article, we begin to develop a complementary theory for diffusion-like perturbations of dynamical systems, with the goal of understanding the space and time scales of fluctuations near bifurcation points of the underlying deterministic system. To do so we describe the limit processes that arise in the vicinity of the bifurcation point. In the present article we focus on the one-dimensional case.