Floquet theory: a useful tool for understanding nonequilibrium dynamics

Abstract

Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers.

Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops

“…Stochastic simulations showed that noise can extend the Hopf bifurcation point in both single-(F 31 ) and dual-feedback systems (F 31,32 ), producing oscillations at parameter values that does not lead to oscillations in a deterministic setting. Comparing the bifurcation points in F 31 and F 31,32 at different levels of the inner loop showed that the points were not altered, suggesting that addition of the second loop in the nested arrangement had no effect on the stochastic bifurcation points. Further understanding concerning the interplay between noise and oscillations in coupled-feedback settings should be among the focuses of future systems biology studies.…”

“…To facilitate analytical treatment, we instead employ the Routh -Hurwitz theorem [29,30] which states that the eigenvalues, essentially the roots of the characteristic polynomial jJ 2 lIj 31,21 . To verify that (2.4) also constitute the sufficient condition for oscillations in these systems, we employed analysis based on the Floquet theory that globally assesses the stability of the obtained periodic solutions [31][32][33]. The mathematical and computational background of the Floquet analysis is given in the electronic supplementary material.…”

“…This method was numerically implemented in MATHEMATICA v. 8.0 (code can be provided on request). In the following sections, we will present first the analytical analysis based on (2.4) for the single-loop system F 31 followed by the exemplified dual-loop motif F 31,32 . Detailed derivations for the remaining motifs are provided in the electronic supplementary material.…”

Instability of a fluctuating membrane driven by an ac electric field

“…However, since the forcing electric terms are periodic in time and our system is linear, we may use techniques derived from the Floquet theorem to compute the growth rate s of any perturbation. We will briefly outline these techniques here, mostly following Klausmeier [50] (see also [36,39,51]). …”

Various oscillation patterns in phase models with locally attractive and globally repulsive couplings

“…Even under large perturbations the system always returns to the original periodic orbit after a finite time interval, thus the global stability of the oscillation is very strong. We can distinguish the locally unstable but globally attractive oscillation from the large-amplitude oscillation by calculating the largest Lyapunov exponent of the oscillation [17]. The dashed curve LM indicates the boundary between the orange-triangle and the yellow-pentagon regions for N = 20 and r = 0.4.…”