Floquet theory

Branch of ordinary differential equations Given a system in which the forces are periodic—such as a pendulum under a periodic driving force, or an oscillating circuit driven by alternating current—the overall behavior of the system is not necessarily fully periodic. For instance, consider a child being pushed on a swing: although the motion is driven by regular, periodic pushes, the swing can gradually reach greater heights while still oscillating to and fro. This results in a combination of underlying periodicity and growth. Floquet theory provides a way to analyze such systems. Its essential insight is similar to the swing example: the solution can be decomposed into two parts—a periodic component (reflecting the repeated motion) and an exponential factor (reflecting growth, decay, or neutral stability). This decomposition allows for the analysis of long-term behavior and stability in time-periodic systems. Formally, Floquet theory is a branch of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form x ˙ = A ( t ) x , {\displaystyle {\dot {x}}=A(t)x,} with x ∈ R n {\displaystyle x\in {R^{n}}} and A ( t ) �