3. The Floquet Coefficient for Nonautonomous Linear Hamiltonian Systems: Atkinson Problems

Let a family of linear Hamiltonian systems determined by a coefficient matrix H be perturbed as to obtain \(H +\lambda J^{-1}\varGamma\), where \(\lambda \in \mathbb{C}\), \(J = \left [\begin{matrix}\scriptstyle 0_{n}&\scriptstyle -I_{n} \\ \scriptstyle I_{n}&\scriptstyle \ \ 0_{n} \end{matrix}\right ]\), and Γ is a positive semidefinite matrix-valued function satisfying an Atkinson nondegeneracy condition. Such a condition ensures the exponential dichotomy property for \(\lambda \notin \mathbb{R}\), as well as the existence of the corresponding Weyl functions, which are determined by the initial data of the solutions bounded as \(t \rightarrow \pm \infty \). These properties can be exploited to define an analytic to define an analytic function \(w_{\varGamma }(\lambda )\) for \(\lambda \notin \mathbb{R}\) on the upper complex half-plane, which is called the Floquet coefficient, whose real part \(-\beta _{\varGamma }(\lambda )\) agrees with the negative Lyapunov index and whose imaginary part \(\alpha _{\varGamma }(\lambda )\) provides an extension of the rotation number. To prove these facts is the goal of this chapter. Among the many consequences of the analysis presented here, it is appropriate to highlight two: first, the rotation number vanishes for λ in an nonempty real interval if and only of all systems corresponding to those values of λ have exponential dichotomy; and second, the rotation number provides a labelling for the gaps of the intervals of the spectrum of the family of operators \(\mathcal{L} = J(d/dt - H)\).