Oscillations
How systems swing, settle, and sustain rhythm — from a single bob on a spring to whole populations of synchronizing oscillators.
Oscillations
An oscillation is motion that returns. Pull a mass on a spring aside and let go; nudge a pendulum; strike a tuning fork — each repeats a pattern in time rather than running off to infinity or grinding to a halt. Oscillation is the simplest non-trivial thing a Dynamical System can do, and it is everywhere: in clocks and bridges, hearts and lasers, planetary librations and AC circuits. This section builds the idea up one layer at a time.
We begin with the frictionless ideal, the Simple Harmonic Oscillator, whose sinusoidal motion is the template for everything that follows. The Pendulum shows how that template is only an approximation — a small-angle shadow of a richer nonlinear system. Adding friction gives the Damped Oscillator; pushing back against that friction with an external force gives the Driven Oscillator, and tuning the drive to the system's natural rhythm produces Resonance. Finally we leave the linear world entirely: a Limit Cycle is an oscillation a system generates on its own, and Coupled Oscillators reveal how separate rhythms lock together into collective motion.
The arc of this section
- Simple Harmonic Oscillator — the linear restoring force and its sine-wave solution.
- Pendulum — small-angle harmonic motion versus the full nonlinear swing.
- Damped Oscillator — friction, decay, and the under/critical/over-damped trichotomy.
- Driven Oscillator — external forcing, transients, and steady state.
- Resonance — the dramatic amplitude peak when drive meets natural frequency.
- Limit Cycle — self-sustained oscillation with no external clock.
- Coupled Oscillators — normal modes, beating, and synchronization.