a self-sustained oscillator
A child on a swing keeps going by pumping their legs; a heart beats steadily for a lifetime; a violin string sings a constant note under a steadily drawn bow. None of these need a periodic push from outside — they each draw on a steady source of energy and convert it into a rhythm of their own making, with their own amplitude and frequency. A self-sustained oscillator is exactly such a system: left alone with a power source, it settles into a stable, repeating cycle all by itself.
The defining mechanism is unusual damping that changes sign. An ordinary damped oscillator always loses energy and winds down to rest. A self-sustained oscillator instead PUMPS energy in when its amplitude is small and bleeds energy out when its amplitude is large. The classic example is the van der Pol equation x'' - mu(1 - x^2) x' + x = 0: the damping coefficient -mu(1 - x^2) is negative (energy in) for small x and positive (energy out) for large x. The two effects balance at one preferred amplitude, and the oscillation locks onto it.
In the phase plane this preferred oscillation is a limit cycle — a closed loop that nearby trajectories spiral toward, whether they start inside it or outside. That is the signature property: the oscillator FORGETS its starting conditions and converges to the same rhythm regardless, which is why your heartbeat returns to its steady cycle after a scare, and why an electronic oscillator produces a clean tone no matter how it was switched on. The amplitude is set by the system, not by how you started it.
This robustness is exactly why self-sustained oscillators are everywhere in biology and electronics: heartbeats, neuron firing, circadian clocks, radio-frequency oscillators, lasers. But the same independence from initial conditions means you cannot dial the amplitude by gently nudging the start — to change the rhythm you must change the system itself. And the underlying equations are genuinely nonlinear; the clean limit cycle is a feature you must earn, not assume.
In the van der Pol oscillator with mu = 1, start the state anywhere — a tiny wiggle near the origin or a wild swing far out — and it spirals onto one and the same closed loop, then circles it forever. A small start grows up to the loop; a large start decays down to it. The loop is the oscillator's chosen amplitude.
All trajectories converge to one limit cycle — the amplitude is the system's choice, not yours.
A self-sustained oscillation is NOT the same as resonance. Resonance needs an external periodic forcing tuned to the natural frequency; a self-sustained oscillator generates its own rhythm from a steady energy source, with the amplitude fixed by the system rather than by any driving signal.