Floquet theory

About: Floquet theory is a research topic. Over the lifetime, 6501 publications have been published within this topic receiving 139616 citations.

Ordinary differential equations

Abstract: Foreword to the Classics Edition Preface to the First Edition Preface to the Second Edition Errata I: Preliminaries II: Existence III: Differential In qualities and Uniqueness IV: Linear Differential Equations V: Dependence on Initial Conditions and Parameters VI: Total and Partial Differential Equations VII: The Poincare-Bendixson Theory VIII: Plane Stationary Points IX: Invariant Manifolds and Linearizations X: Perturbed Linear Systems XI: Linear Second Order Equations XII: Use of Implicity Function and Fixed Point Theorems XIII: Dichotomies for Solutions of Linear Equations XIV: Miscellany on Monotomy Hints for Exercises References Index.

New Results in Linear Filtering and Prediction Theory

Abstract: A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this \"variance equation\" completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.

Floquet topological insulator in semiconductor quantum wells

Abstract: Topological phases of matter have captured our imagination over the past few years, with tantalizing properties such as robust edge modes and exotic non-Abelian excitations, and potential applications ranging from semiconductor spintronics to topological quantum computation. Despite recent advancements in the field, our ability to control topological transitions remains limited, and usually requires changing material or structural properties. We show, using Floquet theory, that a topological state can be induced in a semiconductor quantum well, initially in the trivial phase. This can be achieved by irradiation with microwave frequencies, without changing the well structure, closing the gap and crossing the phase transition. We show that the quasi-energy spectrum exhibits a single pair of helical edge states. We discuss the necessary experimental parameters for our proposal. This proposal provides an example and a proof of principle of a new non-equilibrium topological state, the Floquet topological insulator, introduced in this paper.

Delay Equations: Functional-, Complex-, and Nonlinear Analysis

Abstract: 0 Introduction and preview.- 0.1 An example of a retarded functional differential equation.- 0.2 Solution operators.- 0.3 Synopsis.- 0.4 A few remarks on history.- I Linear autonomous RFDE.- I.1 Prelude: a motivated introduction to functions of bounded variation.- I.2 Linear autonomous RFDE and renewal equations.- I.3 Solving renewal equations by Laplace transformation.- I.4 Estimates for det ?(z) and related quantities.- I.5 Asymptotic behaviour for t ? ?.- I.6 Comments.- II The shift semigroup.- II.1 Introduction.- II.2 The prototype problem.- II.3 The dual space.- II.4 The adjoint shift semigroup.- II.5 The adjoint generator and the sun subspace.- II.6 The prototype system.- II.7 Comments.- III Linear RFDE as bounded perturbations.- III.1 The basic idea, followed by a digression on weak* integration.- III.2 Bounded perturbations in the sun-reflexive case.- III.3 Perturbations with finite dimensional range.- III.4 Back to RFDE.- III.5 Interpretation of the adjoint semigroup.- III.6 Equivalent description of the dynamics.- III.7 Complexification.- III.8 Remarks about the non-sun-reflexive case.- III.9 Comments.- IV Spectral theory.- IV.1 Introduction.- IV.2 Spectral decomposition for eventually compact semigroups.- IV.3 Delay equations.- IV.4 Characteristic matrices, equivalence and Jordan chains.- IV.5 The semigroup action on spectral subspaces for delay equations.- IV.6 Comments.- V Completeness or small solutions?.- V.l Introduction.- V.2 Exponential type calculus.- V.3 Completeness.- V.4 Small solutions.- V.5 Precise estimates for ??(z)-1?.- V.6 Series expansions.- V.7 Lower bounds and the Newton polygon.- V.8 Noncompleteness, series expansions and examples.- V.9 Arbitrary kernels of bounded variation.- V.10 Comments.- VI Inhomogeneous linear systems.- VI.1 Introduction.- VI.2 Decomposition in the variation-of-constants formula.- VI.3 Forcing with finite dimensional range.- VI.4 RFDE.- VI.5 Comments.- VII Semiflows for nonlinear systems.- VII.1 Introduction.- VII.2 Semiflows.- VII.3 Solutions to abstract integral equations.- VII.4 Smoothness.- VII.5 Linearization at a stationary point.- VII.6 Autonomous RFDE.- VII.7 Comments.- VIII Behaviour near a hyperbolic equilibrium.- VIII.1 Introduction.- VIII.2 Spectral decomposition.- VIII.3 Bounded solutions of the inhomogeneous linear equation.- VIII.4 The unstable manifold.- VIII.5 Invariant wedges and instability.- VIII.6 The stable manifold.- VIII.7 Comments.- IX The center manifold.- IX.1 Introduction.- IX.2 Spectral decomposition.- IX.3 Bounded solutions of the inhomogeneous linear equation.- IX.4 Modification of the nonlinearity.- IX.5 A Lipschitz center manifold.- IX.6 Contractions on embedded Banach spaces.- IX.7 The center manifold is of class Ck.- IX.8 Dynamics on and near the center manifold.- IX.9 Parameter dependence.- IX.10 A double eigenvalue at zero.- IX.11 Comments.- X Hopf bifurcation.- X.l Introduction.- X.2 The Hopf bifurcation theorem.- X.3 The direction of bifurcation.- X.4 Comments.- XI Characteristic equations.- XI.1 Introduction: an impressionistic sketch.- XI.2 The region of stability in a parameter plane.- XI.3 Strips.- XI.4 Case studies.- XI.5 Comments.- XII Time-dependent linear systems.- XII.1 Introduction.- XII.2 Evolutionary systems.- XII.3 Time-dependent linear RFDE.- XII.4 Invariance of X?: a counterexample and a sufficient condition.- XII.5 Perturbations with finite dimensional range.- XII.6 Comments.- XIII Floquet Theory.- XIII.1 Introduction.- XIII.2 Preliminaries on periodicity and a stability result.- XIII.3 Floquet multipliers.- XIII.4 Floquet representation on eigenspaces.- XIII.5 Comments.- XIV Periodic orbits.- XIV.1 Introduction.- XIV.2 The Floquet multipliers of a periodic orbit.- XIV.3 Poincare maps.- XIV.4 Poincare maps and Floquet multipliers.- XIV.5 Comments.- XV The prototype equation for delayed negative feedback: periodic solutions.- XV.1 Delayed feedback.- XV.2 Smoothness and oscillation of solutions.- XV.3 Slowly oscillating solutions.- XV.4 The a priori estimate for unstable behaviour.- XV.5 Slowly oscillating solutions which grow away from zero, periodic solutions.- XV.6 Estimates, proof of Theorem 5.5(i) and (iii).- XV.7 The fixed-point index for retracts in Banach spaces, Whyburn'0 Introduction and preview.- 0.1 An example of a retarded functional differential equation.- 0.2 Solution operators.- 0.3 Synopsis.- 0.4 A few remarks on history.- I Linear autonomous RFDE.- I.1 Prelude: a motivated introduction to functions of bounded variation.- I.2 Linear autonomous RFDE and renewal equations.- I.3 Solving renewal equations by Laplace transformation.- I.4 Estimates for det ?(z) and related quantities.- I.5 Asymptotic behaviour for t ? ?.- I.6 Comments.- II The shift semigroup.- II.1 Introduction.- II.2 The prototype problem.- II.3 The dual space.- II.4 The adjoint shift semigroup.- II.5 The adjoint generator and the sun subspace.- II.6 The prototype system.- II.7 Comments.- III Linear RFDE as bounded perturbations.- III.1 The basic idea, followed by a digression on weak* integration.- III.2 Bounded perturbations in the sun-reflexive case.- III.3 Perturbations with finite dimensional range.- III.4 Back to RFDE.- III.5 Interpretation of the adjoint semigroup.- III.6 Equivalent description of the dynamics.- III.7 Complexification.- III.8 Remarks about the non-sun-reflexive case.- III.9 Comments.- IV Spectral theory.- IV.1 Introduction.- IV.2 Spectral decomposition for eventually compact semigroups.- IV.3 Delay equations.- IV.4 Characteristic matrices, equivalence and Jordan chains.- IV.5 The semigroup action on spectral subspaces for delay equations.- IV.6 Comments.- V Completeness or small solutions?.- V.l Introduction.- V.2 Exponential type calculus.- V.3 Completeness.- V.4 Small solutions.- V.5 Precise estimates for ??(z)-1?.- V.6 Series expansions.- V.7 Lower bounds and the Newton polygon.- V.8 Noncompleteness, series expansions and examples.- V.9 Arbitrary kernels of bounded variation.- V.10 Comments.- VI Inhomogeneous linear systems.- VI.1 Introduction.- VI.2 Decomposition in the variation-of-constants formula.- VI.3 Forcing with finite dimensional range.- VI.4 RFDE.- VI.5 Comments.- VII Semiflows for nonlinear systems.- VII.1 Introduction.- VII.2 Semiflows.- VII.3 Solutions to abstract integral equations.- VII.4 Smoothness.- VII.5 Linearization at a stationary point.- VII.6 Autonomous RFDE.- VII.7 Comments.- VIII Behaviour near a hyperbolic equilibrium.- VIII.1 Introduction.- VIII.2 Spectral decomposition.- VIII.3 Bounded solutions of the inhomogeneous linear equation.- VIII.4 The unstable manifold.- VIII.5 Invariant wedges and instability.- VIII.6 The stable manifold.- VIII.7 Comments.- IX The center manifold.- IX.1 Introduction.- IX.2 Spectral decomposition.- IX.3 Bounded solutions of the inhomogeneous linear equation.- IX.4 Modification of the nonlinearity.- IX.5 A Lipschitz center manifold.- IX.6 Contractions on embedded Banach spaces.- IX.7 The center manifold is of class Ck.- IX.8 Dynamics on and near the center manifold.- IX.9 Parameter dependence.- IX.10 A double eigenvalue at zero.- IX.11 Comments.- X Hopf bifurcation.- X.l Introduction.- X.2 The Hopf bifurcation theorem.- X.3 The direction of bifurcation.- X.4 Comments.- XI Characteristic equations.- XI.1 Introduction: an impressionistic sketch.- XI.2 The region of stability in a parameter plane.- XI.3 Strips.- XI.4 Case studies.- XI.5 Comments.- XII Time-dependent linear systems.- XII.1 Introduction.- XII.2 Evolutionary systems.- XII.3 Time-dependent linear RFDE.- XII.4 Invariance of X?: a counterexample and a sufficient condition.- XII.5 Perturbations with finite dimensional range.- XII.6 Comments.- XIII Floquet Theory.- XIII.1 Introduction.- XIII.2 Preliminaries on periodicity and a stability result.- XIII.3 Floquet multipliers.- XIII.4 Floquet representation on eigenspaces.- XIII.5 Comments.- XIV Periodic orbits.- XIV.1 Introduction.- XIV.2 The Floquet multipliers of a periodic orbit.- XIV.3 Poincare maps.- XIV.4 Poincare maps and Floquet multipliers.- XIV.5 Comments.- XV The prototype equation for delayed negative feedback: periodic solutions.- XV.1 Delayed feedback.- XV.2 Smoothness and oscillation of solutions.- XV.3 Slowly oscillating solutions.- XV.4 The a priori estimate for unstable behaviour.- XV.5 Slowly oscillating solutions which grow away from zero, periodic solutions.- XV.6 Estimates, proof of Theorem 5.5(i) and (iii).- XV.7 The fixed-point index for retracts in Banach spaces, Whyburn's lemma.- XV.8 Proof of Theorem 5.5(ii) and (iv).- XV.9 Comments.- XVI On the global dynamics of nonlinear autonomous differential delay equations.- XVI.1 Negative feedback.- XVI.2 A limiting case.- XVI.3 Chaotic dynamics in case of negative feedback.- XVI.4 Mixed feedback.- XVI.5 Some global results for general autonomous RFDE.- Appendices.- I Bounded variation, measure and integration.- I.1 Functions of bounded variation.- I.2 Abstract integration.- II Introduction to the theory of strongly continuous semigroups of bounded linear operators and their adjoints.- II. 1 Strongly continuous semigroups.- II.2 Interlude: absolute continuity.- II.3 Adjoint semigroups.- II.4 Spectral theory and asymptotic behaviour.- III The operational calculus.- III.1 Vector-valued functions.- III.2 Bounded operators.- III.3 Unbounded operators.- IV Smoothness of the substitution operator.- V Tangent vectors, Banach manifolds and transversality.- V.1 Tangent vectors of subsets of Banach spaces.- V.2 Banach manifolds.- V.3 Submanifolds and transversality.- VI Fixed points of parameterized contractions.- VII Linear age-dependent population growth: elaboration of some of the exercises.- VIII The Hopf bifurcation theorem.- References.- List of symbols.- List of notation.

Topological characterization of periodically driven quantum systems

Abstract: Topological properties of physical systems can lead to robust behaviors that are insensitive to microscopic details. Such topologically robust phenomena are not limited to static systems but can also appear in driven quantum systems. In this paper, we show that the Floquet operators of periodically driven systems can be divided into topologically distinct (homotopy) classes and give a simple physical interpretation of this classification in terms of the spectra of Floquet operators. Using this picture, we provide an intuitive understanding of the well-known phenomenon of quantized adiabatic pumping. Systems whose Floquet operators belong to the trivial class simulate the dynamics generated by time-independent Hamiltonians, which can be topologically classified according to the schemes developed for static systems. We demonstrate these principles through an example of a periodically driven two-dimensional hexagonal lattice tight-binding model which exhibits several topological phases. Remarkably, one of these phases supports chiral edge modes even though the bulk is topologically trivial.