7.1 The monodromy matrix
Lemma 7.1.1. \(\Phi (t + T) = \Phi (t)M\) for all \(t\), where \(M = \Phi (T)\) is the monodromy matrix. A solution with \(x(0) = x_0\) satisfies \(x(t + kT) = \Phi (t)M^kx_0\).
Proof. \(\Psi (t) = \Phi (t + T)\) satisfies \(\Psi '(t) = A(t + T)\Phi (t + T) = A(t)\Psi (t)\), so it is a fundamental matrix, and \(\Psi (t) = \Phi (t)C\) for a constant matrix \(C\) (Section 4.1). At \(t = 0\), \(C = \Phi (T)\). By induction \(\Phi (t + kT) = \Phi (t)M^k\). □
The monodromy matrix maps the state at time \(0\) to the state one period later, and its powers describe the state after many periods. It is the Poincaré map, or stroboscopic map, of the linear system: looking at the solution only at the times \(0, T, 2T, \dots \) reduces the continuous problem to the iteration \(x_{k+1} = Mx_k\).
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