Introduction over the last two decades considerable new interest in the theory of delay di. 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. Kuchment p a 1979 representations of solutions of linear partial differential equations with constant or periodic coefficients the theory operator equations voronezh gos. Finally, an apparent discontinuity in the eigenproblem is investigated and explained by its physical and numerical relevance. Floquet theory for partial differential equations iopscience. Furthermore, it is usually only for these simpler equations that a numerical method can be fully analyzed. Floquet theory for partial differential equations 2 the functionals. In 32 the floquet multipliers were studied and in 42 an analytical approach was developed. The outcomes are compared against known linear stability results in pipe flows. Floquet theory applicable to linear ordinary di erential equations with periodic coe cients and periodic boundary conditions. Every component of a solution of 1 is a linear combination of functions of the form of the floquet solutions.
The theory of parametric stability and instability for integral and integrodifferential equations is not a mere adaptation of the classical floquet theory, but instead it involves new complications, raise new problems, and lead to new conditions, that have no counterpart in the theory of parametric stability and parametric resonance for odes. Read, highlight, and take notes, across web, tablet, and phone. An analogue of the floquet theory for functional di. In the case when all the characteristic exponents are distinct or if there are multiple ones among them, but they correspond to simple. Home floquet theory for partial differential equations. Suppose that a linearized onepoint collocation method is used to solve the linear nonautonomous differential equation with.
Floquet theory for parabolic differential equations i. Floquet theory for partial differential equations springerlink. Kuchment, floquet theory for partial differential equations. Idea if a linear di erential equation has periodic coe cients and periodic boundary conditions, then the solutions will generally be a. Pdf analytical approach for the floquet theory of delay. That means that the unknown, or unknowns, we are trying to determine are functions.
Pdf floquet theory and stability of nonlinear integrodifferential. Floquet theory for partial differential equations pdf free download. How floquet theory applies to index 1 differential. To name a few, the following recent papers should be mentioned. Floquets seminal paper dealt with the solution of 1d partial differential equations with periodic coefficients. Floquet theory for parabolic differential equations. Two coupled oscillators with periodic parametric excitation. In this chapter we will consider the basic elements of the theory of partial di erential equations that are relevant to the subsequent development. Floquet boundary value problem of fractional functional. This investigation lays the groundwork for a validation study. Equations with deviating argument comments references.
How floquet theory applies to index 1 differential algebraic. Its central result is the following theorem sometimes called floquetlyapunov theorem 120. Floquet theory for partial differential equations book. Floquet theory and proceed to assess the linearstability of these flows.
List of dynamical systems and differential equations. Dynamics of numerics of nonautonomous equations with. The main prerequisite is a familiarity with the subjects usually gathered under the rubic real analysis. Floquetbloch theory and its application to the dispersion. Recently, there are some papers focused on initial value problem of fractional functional di. Periodic ordinary and partial differential equations. Springer, basel linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations 17, 94.
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. Bressloff, will emphasize partial differential equations. Floquet theory for integral and integrodifferential equations. Floquets theorem student theses faculty of science and. 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. Some familiarity with the elementary theory of inner vector spaces would be an asset but is not expected. R do we have a satisfactory understanding of the qualitative behavior of the solutions. Pdf floquet theory for linear differential equations.
Yields knowledge of whether all solutions are stable. The theory of averaging is treated from a fresh perspective that is intended to introduce the modern approach to this classical subject. Before we dive into floquet theory, first some basic concepts from ordinary differential equations and linear algebra are described. The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular fourier analysis, distribution theory, and sobolev spaces. Linear differential equations with periodic coefficients constitute a well developed part.
Generators and decomposition of state spaces for linear systems 65 3. Floquet theory for integral and integrodifferential. The problem of reducing x atx, where ait is a quasiperiodic n x n matrix, to a system with constant coefficients is studied by means of an associated linear partial differential equation. So you dont get modes in the usual sense found in waveguide theory, i. Then, for fixed and, the method admits a unique periodic solution that. Floquet theory for linear differential equations with meromorphic solutions. Its central result is the following theorem sometimes called floquet lyapunov theorem 120. This is a list of dynamical system and differential equation topics, by wikipedia page. Floquet theory for partial differential equations kuchment, p. Book floquet theory for partial differential equations. The theorem of floquet on the representation of the fundamental matrix 4, as well as the reduction theorem of lyapunov 8wx wx see, e. Basic definitions and examples to start with partial di. There is a sjgnificant distinction between the cases of ordinary and partial differential periodic equations.
Floquet theory for partial differential equations p. The results may be interpreted as the analogs for certain partial differential equations of floquets theory for ordinary differential equations or as. Then, clearly, and the differential equation has a unique, asymptotically stable periodic solution. You might find it helpful to research floquets theorem first. If is a negative constant, could be scaled in such a way that. The resulting floquet theory leads to a homogeneous vector valued re. Theory of ordinary differential equations by earl a. Partial differential equations with periodic coefficients and bloch. A complete proof of the averaging theorem is presented, but the main theme of the chapter is partial averaging at a resonance. Ordinary differential equations and dynamical systems. Kuchment, floquet theory for partial differential equations, uspekhi mat. Relaxation of the conditions on the smoothness of the coefficients 8.
See also list of partial differential equation topics, list of equations. 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. Theory and applications of partial functional differential. Further, this solution is the greens function for equation 1. The ams has granted the permisson to make an online edition available as pdf 4. One of the classical topics in the qualitative theory of di. This chapter is the place where the analytic tools developed in chapters 1 and 2 start to work for periodic partial differential equations. Floquet theory for partial differential equations nasaads. Floquet theory for hypoelliptic equations and systems in. We relate the values of v to the real parts of the floquet multipliers for such linear periodic systems, and thereby prove all floquet subspaces are at most twodimensional. An important group of processes described by integral pde is connected with. Contributions to the control theory of some partial functional integrodifferential equations in banach spaces abstract this thesis is a contribution to control theory of some partial. Permission is granted to retrieve and store a single copy for personal use only.
640 1413 285 845 1320 527 341 323 1192 838 1474 453 1076 490 1104 878 681 63 891 1348 719 722 93 852 282 762 748 1223 592 102 1082