7 Periodic Coefficients and Floquet Theory

Many systems are driven periodically: a child on a swing who pumps by raising and lowering the body, a pendulum whose support vibrates, an electrical circuit with a periodically varying capacitance, a satellite whose orbit takes it periodically through a varying gravitational field, a population with seasonal birth rates. Linearising about a periodic orbit of any non-linear system also produces a linear system with periodic coefficients. Floquet (1883) showed that such systems, although they have no closed-form solution in general, are equivalent to constant-coefficient systems by a periodic change of variables, and that their stability is governed by the eigenvalues of a single matrix. The theory explains parametric resonance: a system can be destabilised, or stabilised, by varying a parameter at the right frequency.

Throughout, \(A(t)\) is continuous and \(T\)-periodic, \(A(t + T) = A(t)\), and \(\Phi (t)\) is the fundamental matrix of \(x' = A(t)x\) with \(\Phi (0) = I\).

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