Floquet Theory - Consequences and Applications

Consequences and Applications

This mapping gives rise to a time-dependent change of coordinates, under which our original system becomes a linear system with real constant coefficients . Since is continuous and periodic it must be bounded. Thus the stability of the zero solution for and is determined by the eigenvalues of .

The representation is called a Floquet normal form for the fundamental matrix .

The eigenvalues of are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps . A Floquet exponent (sometimes called a characteristic exponent), is a complex such that is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since, where is an integer. The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise.

• Floquet theory is very important for the study of dynamical systems.
• Floquet theory shows stability in Hill differential equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field.