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| '''Floquet theory''' is a branch of the theory of [[ordinary differential equations]] relating to the class of solutions to [[linear differential equation]]s of the form
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| :<math>\dot{x} = A(t) x,\,</math>
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| with <math>\displaystyle A(t)</math> a [[Piecewise#Continuity|piecewise continuous]] periodic function with period <math>T</math>.
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| The main theorem of Floquet theory, '''Floquet's theorem''', due to {{harvs|txt|authorlink=Gaston Floquet|first=Gaston |last=Floquet|year=1883}}, gives a [[canonical form]] for each [[Fundamental solution|fundamental matrix solution]] of this common [[linear system]]. It gives a [[Change of coordinates|coordinate change]] <math>\displaystyle y=Q^{-1}(t)x</math> with <math>\displaystyle Q(t+2T)=Q(t)</math> that transforms the periodic system to a traditional linear system with constant, real [[coefficients]].
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| In [[solid-state physics]], the analogous result (generalized to three dimensions) is known as [[Bloch wave|Bloch's theorem]].
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| Note that the solutions of the linear differential equation form a vector space. A matrix <math>\phi\,(t)</math> is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix <math>\Phi(t)</math> is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists <math>t_0</math> such that <math>\Phi(t_0)</math> is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using <math>\Phi(t)=\phi\,(t){\phi\,}^{-1}(t_0)</math>. The solution of the linear differential equation with the initial condition <math>x(0)=x_0</math> is <math>x(t)=\phi\,(t){\phi\,}^{-1}(0)x_0</math> where <math>\phi \,(t)</math> is any fundamental matrix solution.
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| == Floquet's theorem == <!-- [[Floquet theorem]] redirects to this section -->
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| Let <math>\dot{x}= A(t) x</math> be a linear first order differential equation,
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| where <math>x(t)</math> is a column vector of length <math>n</math> and <math>A(t)</math> an <math>n \times n</math> periodic matrix with period <math>T</math> (that is <math>A(t + T) = A(t)</math> for all real values of <math>t</math>). Let <math>\phi\, (t) </math> be a fundamental matrix solution of this differential equation. Then, for all <math>t \in \mathbb{R}</math>,
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| :<math> \phi(t+T)=\phi(t) \phi^{-1}(0) \phi (T).\ </math>
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| Here
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| :<math>\phi^{-1}(0) \phi (T)\ </math>
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| is known as the [[monodromy matrix]].
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| In addition, for each matrix <math>B</math> (possibly complex) such that
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| :<math>e^{TB}=\phi^{-1}(0) \phi (T),\ </math>
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| there is a periodic (period <math>T</math>) matrix function <math>t \mapsto P(t)</math> such that
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| :<math>\phi (t) = P(t)e^{tB}\text{ for all }t \in \mathbb{R}.\ </math>
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| Also, there is a ''real'' matrix <math>R</math> and a ''real'' periodic (period-<math>2T</math>) matrix function <math>t \mapsto Q(t)</math> such that
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| :<math>\phi (t) = Q(t)e^{tR}\text{ for all }t \in \mathbb{R}.\ </math>
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| In the above <math>B</math>, <math>P</math>, <math>Q</math> and <math>R</math> are <math>n \times n</math> matrices.
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| == Consequences and applications ==
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| This mapping <math>\phi \,(t) = Q(t)e^{tR}</math> gives rise to a time-dependent change of coordinates (<math>y = Q^{-1}(t) x</math>), under which our original system becomes a linear system with real constant coefficients <math>\dot{y} = R y</math>. Since <math>Q(t)</math> is continuous and periodic it must be bounded. Thus the stability of the zero solution for <math>y(t)</math> and <math>x(t)</math> is determined by the eigenvalues of <math>R</math>.
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| The representation <math>\phi \, (t) = P(t)e^{tB}</math> is called a ''Floquet normal form'' for the fundamental matrix <math>\phi \, (t)</math>.
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| The [[eigenvalue]]s of <math>e^{TB}</math> are called the [[characteristic multiplier]]s of the system. They are also the eigenvalues of the (linear) Poincaré maps <math>x(t) \to x(t+T)</math>. A '''Floquet exponent''' (sometimes called a characteristic exponent), is a complex <math>\mu</math> such that <math>e^{\mu T}</math> is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since <math>e^{(\mu + \frac{2 \pi i k}{T})T}=e^{\mu T}</math>, where <math>k</math> 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 stability|Lyapunov stable]] if the Lyapunov exponents are nonpositive and unstable otherwise.
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| * Floquet theory is very important for the study of [[dynamical systems]].
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| * 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]].
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| * [[Bond softening]] and [[bond hardening]] in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.
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| == Floquet's theorem applied to Mathieu equation==
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| Mathieu's equation is related to the wave equation for the elliptic cylinder.
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| Given <math>a \in \mathbb{R}, q \in \mathbb{C}</math>, the [[Mathieu equation]] is given by
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| : <math>\frac {d^2 y} {dw^2} +(a-2q \cos 2w )y=0.</math>
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| The Mathieu equation is a linear second-order differential equation with periodic coefficients.
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| One of the most powerful results of Mathieu's functions is the Floquet's Theorem [1, 2].
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| It states that solutions of Mathieu equation for any pair (''a'', ''q'') can be expressed in the form
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| : <math>y(w)=F_{\nu}(w)=e^{iw \nu} P(w) \,</math>
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| or
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| : <math>y(w)=F_{\nu}(-w)=e^{-iw \nu} P(-w) \,</math>
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| where <math> \nu</math> is a constant depending on ''a'' and ''q'' and ''P''(.) is <math> \pi </math>-periodic in ''w''.
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| The constant <math> \nu</math> is called the ''characteristic exponent''.
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| If <math> \nu</math> is an integer, then <math>F_{\nu}(w)</math> and <math>F_{\nu}(-w)</math> are linear dependent solutions. Furthermore,
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| : <math>y(w+k \pi) =e^{i \nu k \pi}y(w)\text{ or }y(w+k \pi) =e^{-i \nu k \pi}y(w), \,</math>
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| for the solution <math>F_{\nu}(w)</math> or <math>F_{\nu}(-w)</math>, respectively.
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| We assume that the pair (''a'', ''q'') is such that <math>| \cosh (i \nu \pi) | <1</math> so that the solution <math> y(w)</math> is bounded on the real axis. General solution of Mathieu's equation (<math>q \in \mathbb{R}</math>, <math> \nu</math> non-integer) is the form
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| : <math>y(w) =c_1 e^{i w \nu}P(w)+ c_2e^{-i w \nu}P(-w), \,</math>
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| where <math>c_1</math> and <math>c_2</math> are arbitrary constants.
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| All bounded solutions −those of fractional as well as integral order− are described by an infinite series of [[harmonic oscillation]]s whose amplitudes decrease with increasing frequency.
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| Another very important property of Mathieu's functions is the orthogonality [3]:
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| If <math>a( \nu +2p,q)</math> and <math>a( \nu +2s,q)</math> are simple roots of
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| : <math> \cos(\pi\nu) - y(\pi = 0) = 0, \, </math>
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| then:
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| : <math>\int_0^\pi F_{\nu+2p} (w) F_{\nu+2s}(-w) \, dw = 0,\qquad p \ne s,</math>
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| i.e.,
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| : <math>\langle F_{\nu +2p} (w),F_{\nu +2s} (w)\rangle = 0, \qquad p \ne s,</math>
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| where <·,·> denotes an [[inner product]] defined from 0 to ''π''.
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| == References ==
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| *C. Chicone. ''Ordinary Differential Equations with Applications.'' Springer-Verlag, New York 1999.
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| * {{cite book|last=Ekeland|first=Ivar|authorlink=Ivar Ekeland|chapter=One|title=Convexity methods in Hamiltonian mechanics|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]|volume=19|publisher=Springer-Verlag|location=Berlin|year=1990|pages=x+247|isbn=3-540-50613-6|mr=1051888|ref=harv}}
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| * {{citation|first=Gaston|last= Floquet|title=Sur les équations différentielles linéaires à coefficients périodiques|journal=Annales de l'École Normale Supérieure|volume=12|pages= 47–88 |year=1883|url= http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1883_2_12_/ASENS_1883_2_12__47_0/ASENS_1883_2_12__47_0.pdf}}
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| * {{Citation
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| | surname = Krasnosel'skii
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| | given = M.A.
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| |authorlink=Mark Krasnosel'skii | title = The Operator of Translation along the Trajectories of Differential Equations
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| | publisher=[[American Mathematical Society]]
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| | place = [[Providence, Rhode Island|Providence]]
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| | year=1968}}, Translation of Mathematical Monographs, 19, 294p.
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| *W. Magnus, S. Winkler. ''Hill's Equation'', Dover-Phoenix Editions, ISBN 0-486-49565-5.
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| *N.W. McLachlan, ''Theory and Application of Mathieu Functions'', New York: Dover, 1964.
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| *{{cite book
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| | surname = Teschl
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| | given = Gerald
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| |authorlink=Gerald Teschl
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| | title = Ordinary Differential Equations and Dynamical Systems
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| | publisher=[[American Mathematical Society]]
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| | place = [[Providence, Rhode Island|Providence]]
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| | year = 2012
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| | isbn= 978-0-8218-8328-0
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| | url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
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| *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.
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| ==External links==
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| * {{springer|title=Floquet theory|id=p/f040640}}
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| [[Category:Dynamical systems|*]]
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| [[Category:Differential equations|*]]
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