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There are many natural, physical, and biological systems that exhibitmultiple time scales. For example, the dynamics of a population of ticks can bedescribed in continuous time during their individual life cycle yet discretetime is used to describe the generation of offspring. These characteristicscause the population levels to be reset periodically. A similar phenomenon canbe observed in a sociological college drinking model in which the population isreset by the incoming class each year, as described in the 2006 work of Camachoet al. With the latter as our motivation we analytically and numericallyinvestigate the mechanism by which solutions in certain systems with thisresetting conditions stabilize. We further utilize the sociological collegedrinking model as an analogue to analyze certain one-dimensional andtwo-dimensional nonlinear systems, as we attempt to generalize our results tohigher dimensions.

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