Floquet theory

Floquet theory analyzes linear differential equations with periodic coefficients by expressing their solutions as periodic functions multiplied by exponential factors. It characterizes stability through the growth rates encoded by those factors.

Fundamental matrix: A matrix whose columns form a basis of solutions to a linear system of differential equations. Its evolution over one period contains the information Floquet theory uses to assess stability.
Linear differential equation: A differential equation in which the unknown function and its derivatives appear linearly. Floquet theory applies to linear systems whose coefficients vary periodically.
Parametric resonance: Instability produced when a system's parameters vary periodically at frequencies related to its natural oscillations. Floquet multipliers identify the parameter ranges where periodic modulation amplifies oscillations.
Gaston Floquet: A French mathematician whose work established the theory of linear differential equations with periodic coefficients. The theory bears his name and formalizes his analysis of periodic linear systems.
Constant-coefficient differential equation: A differential equation whose coefficients do not vary with the independent variable. Its stability follows directly from constant matrices, without tracking evolution over a period.
Monodromy matrix: The linear map that advances solutions of a periodic differential equation through one full period. Its eigenvalues determine the growth or decay of solutions from period to period.
Periodic function: A function that repeats its values after a fixed nonzero interval called its period. The repeating coefficients are what allow evolution to be compared cycle by cycle.
Mathieu equation: A second-order differential equation with a sinusoidally periodic coefficient. Its stability regions are a classic application of Floquet theory.
Émile Picard: A French mathematician known for foundational work in analysis and differential equations. His broader work on differential equations formed part of the mathematical setting in which Floquet theory developed.
Almost-periodic function: A function that approximates periodic repetition to arbitrary accuracy without necessarily having a fixed period. Almost-periodic coefficients lack the exact cycle required for the standard monodromy construction.

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