Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems (Q860742)

Consider a Hamiltonian system with time periodic coefficients, \(H_\varepsilon (t+T) = H_\varepsilon (t)\), obtained as a perturbation (with perturbation parameter \(\varepsilon\)) of a linear Hamiltonian system. Given a (multi-) periodic solution for \(H_0\), one would like to know about its stability under perturbations. This is controlled by the monodromy matrix \(M(T,\varepsilon)\), and more specifically by its eigenvalues \(\mu_i\), also known as (Floquet) multipliers. In particular, the eigenvalues lying on the unit circle in \({\mathbb C}\) at \(\varepsilon = 0\) are critical, and the stability of the (quasi-) periodic solution depends on how these depend on the perturbation parameter \(\varepsilon\). When \(\mu_i\) is a regular eigenvalue, it is also said to be a definite multiplier; when \(\mu_i\) is a generalized eigenvalue (i.e., the algebraic and geometric multiplicities are not equal), it is said to be an undefinite multiplier. For definite critical multipliers, the relevant variation \((d \mu_i / d \varepsilon)_{\varepsilon=0}\) is described by the Krein formula [\textit{M. G. Krejn}, Dokl. Akad. Nauka SSSR, n. Ser. 73, 445--448 (1950; Zbl 0041.05602)]. This paper under review provides an extension of Krein's formula to the case of undefinite multipliers of multiplicity two. The result is used to provide an extension of the Krein-Gelfand-Lidsky theorem to the case with undefinite multipliers; this is also applied to study a class of equations of reaction-diffusion type, giving instability criteria for solutions with periodic structure.