Floquet–Hill analysis of time-periodic flows

Floquet exponents

The stability of a base flow that is periodic in time, with period \(2\pi/\omega\), is governed by a linear system with periodic coefficients. By Floquet’s theorem its solutions are \(x(t) = e^{\mu t}\,\hat x(t)\) with \(\hat x\) periodic; the Floquet exponents \(\mu\) are defined modulo \(i\omega\) (adding \(i k\omega\) only relabels the harmonics), and a real operator has \(\mu\) and \(\bar\mu\) as exponents. Conatus always reports the folded value and says how it folded it.

Hill’s method

Instead of integrating over one period (a monodromy matrix), the periodic part is expanded in Fourier harmonics in time,

\[ x(y, t) = e^{\mu t} \sum_{n=-N_h}^{N_h} x_n(y)\, e^{i n \omega t}, \]

and the base flow \(U(y, t) = \sum_m U_m(y)\,e^{i m\omega t}\) couples harmonic \(n\) to \(n \pm m\). Each harmonic is discretised in space with the same sparse spectral operators as a steady problem. The result is a block-banded matrix \(M(\mu)\), block tridiagonal when the base flow has a single harmonic, and the exponents are the values where it is singular: either the roots of \(\det M(\mu)\), found by a secant method from a declared seed, or the eigenvalues of the corresponding generalised problem near a shift. Block elimination along the harmonics keeps the cost linear in the number of harmonics.

A root is never accepted on a residual alone (the residual of a nearly singular matrix is small everywhere). root.certify(residual_tol=..., convergence_tol=...) marks it certified only if the secant converged, the residual is below the declared tolerance, and the root moves by less than convergence_tol between two resolutions. Two further checks are run on request and recorded when passed to certify: the contrast \(\sigma_{\min}/\sigma_2\) of \(M(\mu)\) (contrast(mu), small problems) and the argument principle (winding(center, radius, n_points), the winding of \(\det M\) around the root); root.explain() lists which checks were run.

References

  • G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. ENS 12 (1883).

  • G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math. 8 (1886).

  • C. Davis, The stability of time-periodic flows, Annu. Rev. Fluid Mech. 8 (1976) 57–74.