12.1 Limit cycles
Definition 12.1.1. A limit cycle is a periodic orbit \(\Gamma \) that is the \(\omega \)-limit set (or \(\alpha \)-limit set, the limit as \(t \to -\infty \)) of some point not on \(\Gamma \). It is stable if it is the \(\omega \)-limit set of every point in a neighbourhood of it.
Example 12.1.2. In polar coordinates the system \(r' = r(1 - r^2)\), \(\theta ' = 1\), which in Cartesian form is \(x' = x - y - x(x^2 + y^2)\), \(y' = x + y - y(x^2 + y^2)\), has the periodic orbit \(r = 1\). Every other orbit except the origin approaches it: \(r\) increases for \(0 < r < 1\) and decreases for \(r > 1\), and the radial equation is the logistic equation in \(r^2\), with \(r(t)^2 = 1/(1 + (r_0^{-2} - 1)e^{-2t})\). So \(r = 1\) is a stable limit cycle, and the origin an unstable focus. The orbits spiral onto the cycle from both sides.
The example shows the typical picture: an unstable equilibrium inside, a stable cycle around it, orbits from inside and outside winding onto it. Its proof used the explicit radial equation. The Poincaré–Bendixson theorem gives the same conclusion without any formula.
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