12.5 Stability of periodic orbits
Let \(\gamma (t)\) be a periodic solution of period \(T\). Linearising along it gives the variational equation \(Y' = Df(\gamma (t))Y\), a linear system with \(T\)-periodic coefficients, and Floquet theory (Section 7) applies to it. Since \(\gamma '(t)\) is itself a periodic solution of the variational equation (differentiate \(\gamma ' = f(\gamma )\)), one Floquet multiplier is always \(1\), corresponding to a shift along the orbit. The other multipliers decide the stability of the orbit.
Theorem 12.5.1. For a periodic orbit \(\gamma \) of period \(T\) of a planar system, the non-trivial Floquet multiplier is \[ \rho = \exp \int _0^T\operatorname {div}f\bigl (\gamma (t)\bigr )\,dt . \] If \(\rho < 1\) the orbit is an orbitally asymptotically stable limit cycle (nearby orbits approach it); if \(\rho > 1\) it is unstable.
Proof. By Theorem 7.3.2(i), the product of the two multipliers is \(\exp \int _0^T\tr Df(\gamma )\,dt\), and \(\tr Df = \operatorname {div}f\). One multiplier is \(1\), so the other is \(\rho \). The stability statement is the Poincaré map version of Theorem 7.3.2: on a transversal \(\Sigma \) through \(\gamma (0)\), the return map \(P\) has the fixed point \(\gamma (0)\) and derivative \(\rho \) there, so \(|\rho | < 1\) makes the fixed point attracting, and nearby orbits approach \(\gamma \). □
For Example 12.1.2, \(\operatorname {div}f = 2 - 4r^2\), which is \(-2\) on \(r = 1\), so \(\rho = e^{-4\pi }\), strongly stable. For van der Pol at \(\mu = 1\), Software 12.6.2 evaluates the integral numerically. The multiplier equals the factor by which a small displacement off the cycle is reduced in one revolution.
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