Floquet theory

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

with a piecewise continuous periodic function with period 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 with that transforms the periodic system to a traditional linear system with constant, real coefficients.

In solid-state physics, the analogous result is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists such that is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using . The solution of the linear differential equation with the initial condition is where is any fundamental matrix solution.

Floquet's theoremEdit

Let   be a linear first order differential equation, where   is a column vector of length   and   an   periodic matrix with period   (that is   for all real values of  ). Let   be a fundamental matrix solution of this differential equation. Then, for all  ,

 

Here

 

is known as the monodromy matrix. In addition, for each matrix   (possibly complex) such that

 

there is a periodic (period  ) matrix function   such that

 

Also, there is a real matrix   and a real periodic (period- ) matrix function   such that

 

In the above  ,  ,   and   are   matrices.

Consequences and applicationsEdit

This mapping   gives rise to a time-dependent change of coordinates ( ), under which our original system becomes a linear system with real constant coefficients  . Since   is continuous and periodic it must be bounded. Thus the stability of the zero solution for   and   is determined by the eigenvalues of  .

The representation   is called a Floquet normal form for the fundamental matrix  .

The eigenvalues of   are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps  . A Floquet exponent (sometimes called a characteristic exponent), is a complex   such that   is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since  , where   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.

ReferencesEdit

  • 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)]. 19. Berlin: Springer-Verlag. pp. x+247. ISBN 3-540-50613-6. MR 1051888.CS1 maint: ref=harv (link)
  • Floquet, Gaston (1883), "Sur les équations différentielles linéaires à coefficients périodiques" (PDF), Annales 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 linksEdit