12 Periodic Orbits and Limit Cycles

The heartbeat, the firing of a neuron, the oscillation of an electronic circuit, the cycles of a predator–prey system with self-limitation and the rhythm of a chemical clock all repeat themselves with a definite amplitude and period, whatever the initial state. Linear systems cannot do this: their periodic solutions come in families, with an amplitude fixed by the initial condition, as at a centre. Self-sustained oscillations of a definite size are limit cycles, isolated periodic orbits that attract (or repel) their neighbours, and they are a genuinely non-linear phenomenon. In the plane there is a complete theory of their existence (Poincaré–Bendixson), of their absence (Dulac), and of their stability (the divergence integral), and this section develops it.

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