Quantitative analysis of limit cycles in two-stroke oscillators with exponential functions based on Perturbation Incremental Method

In the quantitative analysis of limit cycles, dealing with the exponential function (ex) poses significant challenges. To address this, the exponential function is incorporated into the perturbation incremental method (PIM) and expanded as part of equation into Taylor series and Fourier series. This study comprehensively presents the implementation details of PIM with the exponential function. Subsequently, approximate expressions for limit cycles are derived based on four types of two-stroke oscillators characterized by distinct functions proposed by Le Corbeiller, Van der Pol, and De Figueiredo. Relationships between the positions of these characteristic functions and the resulting limit cycles are also investigated. Quantitatively, it is demonstrated that the characteristic function intersects the x-axis at exactly one point within the limit cycle range of the two-stroke oscillator, excluding the origin. Moreover, it is observed that for a nonlinear parameter where 0.5<λ<0.6, a particular type of two-stroke oscillation transitions into a four-stroke oscillation due to an increase in the number of intersections with the x-axis from one to two within the limit cycle range. Finally, the results indicate that the approximate expressions of the limit cycles obtained through PIM closely match those obtained from numerical algorithms such as the Runge–Kutta method.