From Periodicity to Chaos - Stabilization, Bifurcation, and Synchronization of the Van der Pol Oscillator
This work investigates the dynamics of the van der Pol oscillator, a well-known dynamical model used to represent many naturally occurring phenomena, under both unforced and externally forced conditions, with a focus on its stability characteristics and bifurcation behavior. An approximate analytical solution is derived using the Method of Multiple Scales (MMS) and validated against numerical simulations conducted via the ode45 solver. The evolution of the system's limit cycle is examined as the strength of the nonlinear damping term increased. A range of bifurcation scenarios is explored by systematically varying relevant control parameters. The system's response to external forcing at different damping levels is analyzed to uncover transitions toward chaotic behavior and subsequent phase locking (entrainment). Finally, the relevance of the van der Pol oscillator as a model for naturally occurring rhythmic or periodic processes is discussed, highlighting its applicability in representing biological and physical systems.