7.3 Floquet multipliers and stability
Definition 7.3.1. The eigenvalues \(\rho _1, \dots , \rho _n\) of the monodromy matrix are the Floquet multipliers. Numbers \(\mu _j\) with \(\rho _j = e^{\mu _jT}\) are Floquet exponents; they are the eigenvalues of \(B\), and are determined only modulo \(2\pi i/T\).
- (i)
- \(\rho _1\rho _2\cdots \rho _n = \exp \int _0^T\tr A(s)\,ds\).
- (ii)
- The zero solution is asymptotically stable if and only if every multiplier satisfies \(|\rho _j| < 1\); it is unstable if some \(|\rho _j| > 1\); if all \(|\rho _j| \leq 1\), it is stable if and only if the multipliers on the unit circle have Jordan blocks of size one.
- (iii)
- There is a non-zero solution with \(x(t + T) = \rho x(t)\) for every \(t\) if and only if \(\rho \) is a multiplier. In particular there is a \(T\)-periodic solution if and only if \(1\) is a multiplier, and a \(2T\)-periodic (but not \(T\)-periodic) solution if \(-1\) is a multiplier.
Proof. (i) \(\det M = \det \Phi (T) = \exp \int _0^T\tr A\) by Liouville’s formula. (ii) In the variables \(y = P(t)^{-1}x\), with \(P\) and \(P^{-1}\) bounded because they are continuous and periodic, stability of \(x' = A(t)x\) is stability of \(y' = By\), and Theorem 5.3.2 applies with eigenvalues \(\mu _j\): \(\Real \mu _j = \frac 1T\log |\rho _j|\), so \(\Real \mu _j < 0\) if and only if \(|\rho _j| < 1\), and the Jordan structures of \(B\) and \(M = e^{BT}\) correspond. (iii) If \(Mv = \rho v\), then \(x(t) = \Phi (t)v\) satisfies \(x(t + T) = \Phi (t)Mv = \rho x(t)\). Conversely if \(x(t + T) = \rho x(t)\), then at \(t = 0\), \(Mx(0) = \rho x(0)\). □
A warning that the theorem makes precise: the stability of \(x' = A(t)x\) is not determined by the eigenvalues of \(A(t)\) at each instant. There are periodic \(A(t)\) whose eigenvalues have negative real parts for every \(t\) but whose solutions grow (Problem 7.7.6), and periodic \(A(t)\) with eigenvalues of positive real part that are stable. Only the multipliers decide.
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