Although limit cycles are widely explored in fast-slow systems, the addition of exponential nonlinearities \(e^{h(y)}\) significantly increases the complexity of quantitative analysis. To overcome this challenge, we employ perturbation incremental method (PIM), which derives analytical approximate expressions for the limit cycles by expanding the exponential nonlinear terms into Fourier series and embedding them within the framework. First, fast-slow systems containing exponential nonlinearities are introduced, with the Hester fast-slow system and Le Corbeiller fast-slow system used as examples to develop the details. Then, this method is applied to fast-slow systems containing exponential nonlinearities, thereby expanding the scope of application. Through numerical simulations, the limit cycles of the Hester and Le Corbeiller fast-slow systems are quantitatively analyzed, and the accuracy of the analytical approximation expressions is verified. The results show that the PIM can effectively and accurately solve the limit cycles compared with traditional numerical methods such as the Runge–Kutta method.
