TITLE:
Regularity and the Limit Cycle of the Van der Pol System: A Singular Perturbation Analysis
AUTHORS:
Emmanuel O. Ohwadua, Stephen U. Egarievwe
KEYWORDS:
Van der Pol Oscillator, Limit Cycle, Regularity, Singular Perturbation, Relaxation Oscillations, Matched Asymptotic, Geometric Singular Perturbation Theory
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
23,
2026
ABSTRACT: The Van der Pol oscillator, a definitive model for self-sustained nonlinear oscillations, exhibits a rich dynamical structure governed by the nonlinear damping parameter, μ. While its limit cycle behaviour is well-established numerically, a comprehensive analytical characterization of the solution’s regularity across the full spectrum of μ, particularly in relation to the singular perturbation nature of the system, remains a subject of deep interest. This paper presents a unified singular perturbation analysis to rigorously examine the interplay between the regularity of solutions and the topological and geometric properties of the stable limit cycle. We decompose the problem into two asymptotic regimes. For the weakly nonlinear regime where the nonlinear damping parameter less than one but greater than zero, we employ the method of multiple scales to derive the asymptotic form of the limit cycle, proving its smooth dependence on the nonlinear damping parameter and establishing its amplitude and phase regularity. In the strongly nonlinear, relaxation oscillation regime where the nonlinear damping parameter is significantly greater than 1, we treat the system as a singularly perturbed dynamical system. Utilizing geometric singular perturbation theory and matched asymptotic expansions, we analysed the trajectory’s structure—comprising slow drift along cubic critical manifolds and fast jumps. We explicitly characterize the loss of differentiability at the fold points (turning points) and demonstrate how this singularity is resolved in the composite solution, leading to a continuous but piecewise-smooth limit cycle in the infinite limit of μ. Our central result bridges these regimes, providing a continuous description of how the limit cycle’s regularity evolves from infinitely differentiable to piecewise differentiable as the nonlinear damping parameter tends to zero and infinity respectively. This work not only clarifies the intrinsic mathematical structure of the Van der Pol system but also provides a template for analysing regularity transitions in a broad class of singularly perturbed nonlinear oscillators.