Lemma 8.4 If C is a n n × matrix with 0 det ≠ C , then, there exists a n n × (complex) matrix B such that C e = . Proof: For any matrix C , there exists an invertible matrix P , s.t. 1 P CP J − = , where J is a Jordan matrix. If C e = , then, 1 1 1 P B P B e P e P P CP J − − − = = = . Therefore, it is suffice to prove the result when C is in a canonical form. Suppose that 1 ( , , ) s C diag C C...

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which es...

A fractional generalization of the Floquet theorem is suggested for Schr\"odinger equations (FTSE)s with time-dependent periodic Hamiltonians. The obtained result, called (fFT), formulated in form Mittag-Leffler function, which considered as eigenfunction Caputo derivative. formula makes it possible to reduce FTSE standard quantum mechanics Hamiltonian, where valid. Two examples related resonan...

The existence and uniqueness of a periodic solution of the system of differential equations d dt x(t) = A(t)x(t − ) are proved. In particular the Krasnoselskii’s fixed point theorem and the contraction mapping principle are used in the analysis. In addition, the notion of fundamental matrix solution coupled with Floquet theory is also employed.  

One of the classical topics in the qualitative theory of differential equations is the Floquet theory. It provides a means to represent solutions and helps in particular for stability analysis. In this paper first we shall study Floquet theory for integro-differential equations (IDE), and then employ it to address stability problems for linear and nonlinear equations.

In this paper, the dynamic stability analysis of a simply supported beam carrying a sequence of moving masses is investigated. Many applications such as motion of vehicles or trains on bridges, cranes transporting loads along their span, fluid transfer pipe systems and the barrel of different weapons can be represented as a flexible beam carrying moving masses. The periodical traverse of masses...

Recent advances in theory of solid state nuclear magnetic resonance (NMR) such as Floquet-Magnus expansion and Fer expansion, address alternative methods for solving a time-dependent linear differential equation which is a central problem in quantum physics in general and solid-state NMR in particular. The power and the salient features of these theoretical approaches that are helpful to descri...

Floquet theory provides rigorous foundations for the theory of periodically driven quantum systems. In the case of non-periodic driving, however, the situation is not so well understood. Here, we provide a critical review of the theoretical framework developed for quasi-periodically driven quantum systems. Although the theoretical footing is still under development, we argue that quasiperiodica...

We study the geometric phase phenomenon in the context of the adiabatic Floquet theory (the so-called the (t, t) Floquet theory). A double integration appears in the geometric phase formula because of the presence of two time variables within the theory. We show that the geometric phases are then identified with horizontal lifts of surfaces in an abelian gerbe with connection, rather than with ...

We give the precise conditions under which a periodic discrete-time linear state-space system can be transformed into a time-invariant one by a change of basis. Thus our theory is the discrete-time counterpart of the classical theory of Floquet transforms developed by Floquet and Lyapunov in the 1800s for continuous-time systems. We state and prove a necessary and sufficient condition for a “di...