Floquet theory

Floquet theory This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (July 2015) (Learn how and when to remove this template message) 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 {displaystyle {dot {x}}=A(t)x,} with {displaystyle displaystyle A(t)} a piecewise continuous periodic function with period {displaystyle T} and defines the state of the stability of solutions.

The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet (1883), gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change {displaystyle displaystyle y=Q^{-1}(t)x} with {displaystyle displaystyle Q(t+2T)=Q(t)} that transforms the periodic system to a traditional linear system with constant, real coefficients.

When applied to physical systems with periodic potentials, such as crystals in condensed matter physics, the result is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix {displaystyle phi ,(t)} is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix {displaystyle Phi (t)} is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists {displaystyle t_{0}} such that {displaystyle Phi (t_{0})} is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using {displaystyle Phi (t)=phi ,(t){phi ,}^{-1}(t_{0})} . The solution of the linear differential equation with the initial condition {displaystyle x(0)=x_{0}} is {displaystyle x(t)=phi ,(t){phi ,}^{-1}(0)x_{0}} where {displaystyle phi ,(t)} is any fundamental matrix solution.

Contents 1 Floquet's theorem 2 Consequences and applications 3 References 4 External links Floquet's theorem Let {displaystyle {dot {x}}=A(t)x} be a linear first order differential equation, where {displaystyle x(t)} is a column vector of length {displaystyle n} and {displaystyle A(t)} an {displaystyle ntimes n} periodic matrix with period {displaystyle T} (that is {displaystyle A(t+T)=A(t)} for all real values of {displaystyle t} ). Let {displaystyle phi ,(t)} be a fundamental matrix solution of this differential equation. Then, for all {displaystyle tin mathbb {R} } , {displaystyle phi (t+T)=phi (t)phi ^{-1}(0)phi (T).} Here {displaystyle phi ^{-1}(0)phi (T)} is known as the monodromy matrix. In addition, for each matrix {displaystyle B} (possibly complex) such that {displaystyle e^{TB}=phi ^{-1}(0)phi (T),} there is a periodic (period {displaystyle T} ) matrix function {displaystyle tmapsto P(t)} such that {displaystyle phi (t)=P(t)e^{tB}{text{ for all }}tin mathbb {R} .} Also, there is a real matrix {displaystyle R} and a real periodic (period- {displaystyle 2T} ) matrix function {displaystyle tmapsto Q(t)} such that {displaystyle phi (t)=Q(t)e^{tR}{text{ for all }}tin mathbb {R} .} In the above {displaystyle B} , {displaystyle P} , {displaystyle Q} and {displaystyle R} are {displaystyle ntimes n} matrices.

Consequences and applications This mapping {displaystyle phi ,(t)=Q(t)e^{tR}} gives rise to a time-dependent change of coordinates ( {displaystyle y=Q^{-1}(t)x} ), under which our original system becomes a linear system with real constant coefficients {displaystyle {dot {y}}=Ry} . Since {displaystyle Q(t)} is continuous and periodic it must be bounded. Thus the stability of the zero solution for {displaystyle y(t)} and {displaystyle x(t)} is determined by the eigenvalues of {displaystyle R} .

The representation {displaystyle phi ,(t)=P(t)e^{tB}} is called a Floquet normal form for the fundamental matrix {displaystyle phi ,(t)} .

The eigenvalues of {displaystyle e^{TB}} are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps {displaystyle x(t)to x(t+T)} . A Floquet exponent (sometimes called a characteristic exponent), is a complex {displaystyle mu } such that {displaystyle e^{mu T}} is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since {displaystyle e^{(mu +{frac {2pi ik}{T}})T}=e^{mu T}} , where {displaystyle k} is an integer. The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise.

Floquet theory is very important for the study of dynamical systems. 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. Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem. References C. Chicone. Ordinary Differential Equations with Applications. Springer-Verlag, New York 1999. Ekeland, Ivar (1990). "One". Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 19. Berlin: Springer-Verlag. pp. x+247. ISBN 3-540-50613-6. MR 1051888. Floquet, Gaston (1883), "Sur les équations différentielles linéaires à coefficients périodiques" (PDF), Annales Scientifiques de l'École Normale Supérieure, 12: 47–88, doi:10.24033/asens.220 Krasnosel'skii, M.A. (1968), The Operator of Translation along the Trajectories of Differential Equations, Providence: American Mathematical Society, Translation of Mathematical Monographs, 19, 294p. W. Magnus, S. Winkler. Hill's Equation, Dover-Phoenix Editions, ISBN 0-486-49565-5. N.W. McLachlan, Theory and Application of Mathieu Functions, New York: Dover, 1964. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. M.S.P. Eastham, "The Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973. ISBN 978-0-7011-1936-2. External links "Floquet theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994] hide Authority control National libraries France (data)GermanyIsraelUnited States Other Faceted Application of Subject Terminology Categories: Dynamical systemsDifferential equations