A Mechanism for Stabilization of Dynamics in Nonlinear Systems with Different Time Scales

TL;DR

This paper investigates how solutions in nonlinear systems with multiple time scales and periodic resets stabilize, using analytical, numerical, and sociological model analogies to understand the underlying mechanisms.

Contribution

It introduces a mechanism for stabilization in nonlinear systems with different time scales and resets, generalizing results across various dimensions.

Findings

Solutions can stabilize through specific resetting mechanisms

Analytical and numerical methods confirm stabilization effects

Sociological models serve as effective analogues for analysis

Abstract

There are many natural, physical, and biological systems that exhibit multiple time scales. For example, the dynamics of a population of ticks can be described in continuous time during their individual life cycle yet discrete time is used to describe the generation of offspring. These characteristics cause the population levels to be reset periodically. A similar phenomenon can be observed in a sociological college drinking model in which the population is reset by the incoming class each year, as described in the 2006 work of Camacho et al. With the latter as our motivation we analytically and numerically investigate the mechanism by which solutions in certain systems with this resetting conditions stabilize. We further utilize the sociological college drinking model as an analogue to analyze certain one-dimensional and two-dimensional nonlinear systems, as we attempt to generalize our results to higher dimensions.