FloquetMultiplikatoren

Floquet multipliers are a fundamental concept in the study of linear differential equations with periodic coefficients, particularly those described by Floquet's theorem. When a linear system of ordinary differential equations exhibits periodic coefficients, its solutions can be expressed in a specific form involving exponential terms and periodic functions. Floquet multipliers are the roots of the characteristic equation associated with this particular solution structure.

More precisely, consider a system of differential equations dx/dt = A(t)x, where A(t) is a periodic matrix with period T, i.e., A(t+T) = A(t). Floquet's theorem states that the general solution can be written as x(t) = P(t)e^(λt), where P(t) is a periodic function with period T and λ is a constant. The matrix exponential e^(λT) evaluated at t=T, which is equivalent to e^(λT), plays a crucial role. The eigenvalues of the matrix P(T) or, equivalently, the eigenvalues of e^(λT), are known as the Floquet multipliers.

The nature of these multipliers provides critical information about the stability of the system. If the magnitude of all Floquet multipliers is less than 1, the system is asymptotically stable. If any multiplier has a magnitude greater than 1, the system is unstable. If all multipliers have a magnitude less than or equal to 1 and at least one has a magnitude equal to 1, the stability depends on further analysis, often related to the presence of simple or multiple eigenvalues on the unit circle. Floquet multipliers are therefore essential tools for analyzing the long-term behavior and stability of periodically forced dynamical systems in various fields, including physics, engineering, and mathematics.