Limit cycles
A limit cycle is an isolated closed trajectory in the phase plane corresponding to a periodic solution.
Periodic motion
If \(\mathbf{x}(t)\) lies on a limit cycle with period \(T\), then
\[\mathbf{x}(t+T)=\mathbf{x}(t).\]Stable limit cycles
A stable limit cycle attracts nearby trajectories. Different initial conditions can therefore converge toward the same sustained oscillation.
Unstable limit cycles
An unstable limit cycle repels nearby trajectories and may separate regions with different long-term behaviour.
Biological meaning
Limit cycles can model self-sustained rhythms such as recurrent predator–prey fluctuations or biochemical oscillations that persist without external periodic forcing.
Key idea. A limit cycle is not simply any closed curve; it is an isolated periodic orbit generated by the dynamics of the system.