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Marking 100 years of the Van der Pol equation: Overview of the dynamics of the classical Van der Pol oscillator and its generalizations
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DOI：10.1016/j.jsv.2026.120004.png)
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Abstract
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In 1926, Balthazar Van der Pol published an article presenting an autonomous nonlinear differential equation governing the oscillator that exhibits self-exciting oscillations. This equation has been utilized to this day to model a range of self-oscillatory problems across multiple disciplines. This review article is written to mark the anniversary of this well-known equation. First, a historical context is established concerning the author of the equation, its origin, and its links to other prominent equations, such as the Rayleigh equation, the Liénard equations, and the Fitzhugh-Nagumo mathematical model. Second, a summary of qualitative studies of the classical free Van der Pol equation is provided. Third, its dynamic responses are illustrated in two specific scenarios characterized by weak and strong non-linearity, along with two distinctive phenomena that arise from them: a limit cycle and relaxation oscillations. Subsequently, a systematic overview of the generalizations of the Van der Pol oscillator is presented with respect to the forms of the restoring force and/or the damping force, highlighting the variations in their responses and associated phenomena compared to the classical Van der Pol oscillators.
Keywords:
Van der Pol oscillator
Limit cycle
Relaxation oscillations
Nullcline
Local bifurcation
Global bifurcation
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4.9
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1.7W
Citations:
4.8W
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