Van der Pol oscillator

A nonlinear dynamical system with amplitude-dependent damping that produces self-sustained periodic oscillations, approaching a stable limit cycle from a wide range of initial conditions.

Van der Pol equation: The differential equation describing an oscillator whose damping depends on its state and oscillation amplitude. It expresses the oscillator’s dynamics directly, including the nonlinear damping that sustains its motion.
Nonlinear system: A system whose governing equations are not linear in its state variables. Amplitude-dependent damping makes the oscillator nonlinear, so superposition does not apply.
Balthasar van der Pol: A Dutch physicist who studied nonlinear electrical oscillations and introduced the oscillator bearing his name. His work on vacuum-tube circuits produced the model.
Cardiac pacemaker: A biological or electronic system that initiates rhythmic heart contractions. The oscillator has been used to model self-sustained rhythmic activity in pacemaker cells.
Harmonic oscillator: An oscillator whose restoring force is proportional to displacement, producing sinusoidal motion at a fixed frequency. Unlike the Van der Pol oscillator, it has linear dynamics and no self-sustaining amplitude regulation.
Limit cycle: An isolated periodic orbit in a dynamical system that nearby trajectories approach or leave. The oscillator’s stable periodic motion is represented by an attracting limit cycle.
Dynamical system: A rule specifying how a system’s state changes over time. The oscillator is studied through the evolution of its position and velocity.
Triode: A vacuum tube with a control grid between its cathode and anode. Van der Pol used triode circuits to investigate self-sustained electrical oscillations.
Neural oscillator: A model of neural activity that produces rhythmic changes in neuronal or population states. Van der Pol dynamics can represent rhythmic neural activity and its synchronization.
Rayleigh oscillator: A nonlinear oscillator with velocity-dependent damping that can sustain a stable periodic orbit. It shares self-excitation with Van der Pol dynamics but places the nonlinearity differently.

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