Singularly Perturbed Oscillators with Exponential Nonlinearities

TL;DR

This paper investigates singularly perturbed oscillators with exponential nonlinearities, demonstrating exponential convergence to nonsmooth limits and analyzing degeneracies, ultimately proving the existence of unique limit cycles.

Contribution

It introduces a normalization approach for exponential nonlinearities in oscillators, extending the analysis to degeneracies and proving limit cycle existence.

Findings

Exponential convergence to nonsmooth systems is established.

Degeneracies due to small exponential terms are characterized.

Unique limit cycles are proven to exist in the models.

Abstract

Singular exponential nonlinearities of the form $e^{h(x)\epsilon^{-1}}$ with $\epsilon>0$ small occur in many different applications. These terms have essential singularities for $\epsilon=0$ leading to very different behaviour depending on the sign of $h$. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $\epsilon\rightarrow 0$ limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen [20], and prove - for both systems - existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.