Singularly Perturbed Oscillators with Exponential Nonlinearities

Singular exponential nonlinearities of the form e(h(x)c-1) with is an element of > 0 small occur in many different applications. These terms have essential singularities for is an element of = 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 is an element of -> 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 Kris-tiansen (Nonlinearity 30(5): 2138-2184, 2017), and prove-for both systems-existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

Automated summary

This paper investigates two prototypical singularly perturbed oscillators with exponential nonlinearities, normalizing both systems to a piecewise smooth system in the limit is an element of -> 0, showing exponential convergence due to the nonlinearities studied. By extending spatial dimensions, degeneracies caused by exponentially small terms are tackled for the second model system, with (unique) limit cycles proven to exist for both systems by perturbing away from singular cycles with desirable hyperbolicity properties.