2025, Vol.28, No.3, pp.208 - 219

In various physical and engineering systems in nature, we encounter different types of oscillators, many of which exhibit nonlinear behavior. One such system, the Van der Pol oscillator, is known for its self-sustained limit cycle behavior. However, when subjected to external forcing, its dynamics can change abruptly from its self-sustained oscillation. The central research question is: Does the forced Van der Pol oscillator exhibit irregular dynamics such as quasi-periodicity or chaos, and if so, what are the possible conditions? We show that for different values of the amplitude and ratios of the external forcing frequency with respect to the natural frequency of the oscillator, it can induce a range of dynamic regimes, including higher periodic and chaotic ones. Using numerical simulations in Python, we analyze the behavior of the system through time series plots, phase portraits, and bifurcation diagrams. Our results reveal that as the bifurcation parameter, the forcing amplitude is varied, having different frequency ratios, the system transitions through periodic, quasiperiodic, and chaotic states. This type of nonlinear interaction has wide applications in realworld systems, including biological rhythms, electrical circuits, and astrophysical processes.

Key words: ergodicity, chaos, Lyapunov exponent, switching dynamical systems, piecewise-smooth map, switching electronic circuit

DOI: https://doi.org/10.5281/zenodo.17240563

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