Introduction to floquet theory wiley online library. We develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties, a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous. Introduction of the floquetmagnus expansion in solid. Exponents arising in the study of solutions of a linear ordinary differential equation invariant with respect to a discrete abelian group cf. The subscripts i, j indicate the site, hi, jiimplies the sum over nearest neighboring sites. Floquet theory and stability of nonlinear integrodifferential equations article pdf available in acta mathematica hungarica 1094 november 2005 with 283 reads how we measure reads.
Floquet theory for parabolic differential equations i. Introduction the theory of electromagnetic fields in periodic structures has important applications in the field of microwave electronics, and it is this aspect of the subject that has supplied the motivation for much of the investigation described in this report. The formalism presents similarities to the standard formalism of the quantum mechanics of the conservative systems and also, it enables to develop a perturbation technique analogous to those used in the theory of the conservative manyparticle systems. Floquettheorybased smallsignal stability analysis of single. A new wave of interest in this subject has been stimulated during the last two decades by the development of the inverse scattering method for. Linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations 17, 94, 156, 177, 178, 272, 389. Derivation of a floquet formalism within a natural framework.
Semiclassical manymode floquet theory sciencedirect. They are for the use of students in my graduate ode class. The expansion into floquet modes with the proper keldysh description models the external time. Floquet topological phases anomalous hall state without chern number floquet topological phases in noninteracting systems time glide symmetry tenfold way classification floquet topological phases in interacting systems group cohomology classification 1d and 2d models morimoto, po, vishwanath, prb. Floquetbloch decomposition for the computation of dispersion. Floquetbloch theory and its application to the dispersion. This is a topic of research for dynamic systems, or it is also named as geometric theory of differential equations. Show full abstract system to an effective twosite model of hubbard type, and applying floquet theory, leads to a detailed understanding of this effect. The versatility of this technique is illustrated on systems of increasing complexity with and without damper centralizing springs. Threedimensional floquet stability analysis of the wake in cylinder arrays by n. It is important because it allows us to identify the wave functions that are the same if an integer number of drive periods passes.
Mathematicalproblemsinengineering 3 traction free surfaces propagating shape y 2b t r n t x z t in a symmetric mode antisymmetric mode b 0 510 15 0. Floquet theory for integral and integrodifferential equations. Li october 19, 2009 these notes are based on the lecture notes of professor james s. The calculation of light shifts by perturbation theory fails for scenarios that arise, for example, when the trapping light is near an excitedstate transition or for polychromatic fields. Floquet theory for systems of ordinary differential equations, has its origin in f, and in the basic result about the existence of a monodromy matrix see, e. Floquet theory generates a canonical transformation to modal variables if the periodic matrix can be made symplectic at the initial time. Homogenization approach and blochfloquet theory for bandgap. It is shown that the singlemode floquet formalism of shirley can be extended to a generalized manymode floquet theory, yielding a practical nonperturbative technique for the semiclassical treatment of the interaction of a quantum system several monochromatic oscillating fields. A usual derivation of timeindependent semigroup generator makes use of. It is noted that this type of stability is not in lyapunov sense. Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form. In this paper, we are interested to study floquet theory for qdifference equa.
Jul 12, 2008 here, i describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use. Miller we consider the steady propagation of a light beam in a planar waveguide whose width and depth are periodically modulated in the direction of propagation. Floquet theory for partial differential equations p. Floquet theory is very useful to study stability of a given periodic solution, noted that not equilibrium here. The simplest example is a periodic ordinary differential equation. Two coupled oscillators with periodic parametric excitation. Introduction the theory of electromagnetic fields in periodic structures has important applications in the field of microwave electronics, and it is this aspect of the subject that has supplied the motivation for. We numerically solve the floquetkeldysh dmft 40,51 with a second order iterative perturbation theory ipt, where the the local selfenergy sab is derived by four bubble diagrams, see fig. Ahlbrandt and ridenhour have studied floquet theory on periodic time scales 1. Floquet exponentsmultipliers are analogous to the eigenvalues of jacobian matrices of equilibrium points. On the application of floquet theorem in development of time. Yields knowledge of whether all solutions are stable. In 32 the floquet multipliers were studied and in 42 an analytical approach was developed.
Floquet theory applicable to linear ordinary di erential equations with periodic coe cients and periodic boundary conditions. Floquet theory is very important for the study of dynamical systems. The waveguide is modeled as a uniform elastic string periodically supported by a discontinuous winkler elastic foundation and, as a result, a hill equation is found. A first computation has been made to compare our results with those presented by wu et al. Using methods of soliton theory, a class of periodic potentials. Subsequently, the first detailed theory about timedependent, periodic systems was developed by emile mathieu in 1868, particularly devoted to the analysis of surface waves in ellipticshaped lakes. Dynamical mean field theory in the nonequilibrium the generalized hubbard model for the correlated system, u 6 0, in the nonequilibrium, equation 1, is numerically solved by a singlesite dynamical mean field theory dmft 37,4659. Additionally, it has n such j and together they satisfy e1t e2t ent exp z. The floquet theory is a methodology to solve ordinary differential equations of the form. Pdf we present an analytical approach to deal with nonlinear delay.
Osa floquet theory for atomic lightshift engineering. The floquet topological insulator is defined through the topological properties of the timeindependent floquet operator h. Idea if a linear di erential equation has periodic coe cients and periodic boundary conditions, then the solutions will generally be a. Most importantly, we show below that the fti is not only a mathematical concept. Introduction of the floquetmagnus expansion in solidstate. The purpose of this article is to introduce the fme. Pdf we develop a magnus expansion well suited for floquet theory of linear ordinary differential equations with periodic coefficients. Stability of periodic systems and floquet theory panardo ilaria 6 1. These response actions serve as manybody topological invariants for topological floquet unitaries.
Pdf floquet theory and stability of nonlinear integro. Behavior of floquet topological quantum states in optically. The section of applying floquet theory to mathieu functions is not in latex, and somewhat harder to read. Transfer of floquet theory to the case of spatially periodic partial differential equations is possible, but nontrivial. R do we have a satisfactory understanding of the qualitative behavior of the solutions. This theory shows that timeperiodic perturbations only weakly affect the system thus preserving its stability for a long time. We apply floquet theory of periodic coefficient secondorder odes to an elastic waveguide. Department of chemistry and volen center for complex systems, ms 015. Floquet theory is the study of the stability of linear periodic systems in continuous time. Threedimensional floquet stability analysis of the wake. Floquet topological phases protected by dynamical symmetry. In our notation, see equation 1, c,c are the creator annihilator of an electron.
Miller australian photonics cooperative research centre, optical sciences centre, australian national university, canberra, act 0200, australia received march 6, 1997 we calculate exactly the two bound floquet modes of a periodic linear waveguide induced in a medium. Floquet theory for discontinuously supported waveguides. Here, i describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use. The fundamental solutions, the stability regions, and the dispersion curves are determined and then plotted. Floquet, laid the foundation of his most widely known theory about the.
In the case when all the characteristic exponents are distinct or if there are multiple ones among them, but they correspond to simple. The proposed method is applied for computing the waves dispersion curves of the undamped system along the. It is shown that this symplectic normalization can always be carried out, again with careful attention to the degenerate case. We numerically solve the floquet keldysh dmft 40,51 with a second order iterative perturbation theory ipt, where the the local selfenergy sab is derived by four bubble diagrams, see fig. They arise in many physical and technical applications 177, 178, 272. One of the classical topics in the qualitative theory of di. Muldowney, the books of hale, copple, coddington and levinson, and perko.
Floquet theory is widely used in the analysis of stability of dynamical systems, including the mathieu equation and hills differential equation for approximating the motion of the moon. Floquetbloch decomposition for the computation of dispersion of twodimensional periodic, damped mechanical systems. A method to rigorously enclose eigenpairs of complex interval matrices. We carry out this program for chiral topological floquet systems anomalous floquet anderson insulators in two spatial dimensions, and the group cohomology models of topological floquet unitaries. Blochfloquet theory for the determination of band gaps in locally resonant metamaterials. Periodically driven systems we will now learn about a new generalization of topology, namely how it applies to the quantum evolution of systems with a timedependent hamiltonian. Analytical approach for the floquet theory of delay differential equations.
Floquet theory for qdifference equations 359 example 3. To bridge this gap, we wish to translate the floquet formalism into a natural framework. Floquet topological insulator in semiconductor quantum wells. Majorana end modes and floquet topological insulators. Basic theory consider a linear system of n di erential. Floquet theory shows stability in hill differential equation introduced by george william hill approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field. The study of floquet theory can be found in kelley and peterson 6, hartman 4, and cronin 3 for r, and for z in kelley and peterson 5. In one sense it is trivial to show this, since any constant hamiltonian is also periodic, but presumably you want some more physical examples, so heres two. A software for prediction of periodic response of non. For this reason the gap between the description of the evolution of the biological systems and the floquet theory remains. This paper discusses numerical methods of computing a particular normal form, called a floquet normal form, of solutions for linear, timeperiodic differential equations. Every component of a solution of 1 is a linear combination of functions of the form of the floquet solutions. Todays topic, floquet topological insulators, is introduced by mark rudner from the niels bohr institute at copenhagen. An analogue of the floquet theory for functional di.