Singularly Perturbed Oscillators with Exponential Nonlinearities

Abstract Singular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ > 0 small occur in many different applications. These terms have essential singularities for =0$$ = leading to very behaviour depending on sign h . In this paper, we consider two prototypical singularly perturbed oscillators such nonlinearities. We apply a suitable normalization both systems that \rightarrow 0$$ → limit is piecewise smooth system. The convergence nonsmooth system due study. By working model use blow-up approach demonstrate can be harmless some cases while other scenarios it lead further degeneracies. For our second system, deal degeneracies exponentially by extending space dimension, following Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for systems—existence (unique) cycles perturbing away from singular having desirable hyperbolicity properties.