Floquet theory
Encyclopedia
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
with
a piecewise continuous periodic function with period 
.
The main theorem of Floquet theory, Floquet's theorem, due to , 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 (generalized to three dimensions) 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.
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 
,
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.
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 multiplier
s 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.
Given
, the Mathieu equation is given by
The Mathieu equation is a linear second-order differential equation with periodic coefficients.
One of the most powerful results of Mathieu's functions is the Floquet's Theorem [1, 2].
It states that periodic solutions of Mathieu equation for any pair (a, q) can be expressed in the form
or
where
is a constant depending on a and q and P(.) is 
-periodic in w.
The constant
is called the characteristic exponent.
If
is an integer, then 
and 
are linear dependent solutions. Furthermore,
for the solution
or 
, respectively.
We assume that the pair (a, q) is such that
so that the solution 
is bounded on the real axis. General solution of Mathieu's equation (
, 
non-integer) is the form
where
and 
are arbitrary constants.
All bounded solutions --those of fractional as well as integral order-- are described by an infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency.
Another very important property of Mathieu's functions is the orthogonality [3]:
If
and 
are simple roots of
then:
i.e.,
where <.,.> denotes an inner product defined from 0 to π.
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
s of the form
with
a piecewise continuous periodic function with period .The main theorem of Floquet theory, Floquet's theorem, due to , gives a canonical form
Canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
for each fundamental matrix solution
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
of this common linear system
Linear system
A linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....
. It gives a coordinate change
with that transforms the periodic system to a traditional linear system with constant, real coefficients.In solid-state physics
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
, the analogous result (generalized to three dimensions) is known as Bloch's theorem
Bloch wave
A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
.
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 theorem
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 ,In addition, for each matrix
(possibly complex) such thatthere is a periodic (period
) matrix function such thatAlso, there is a real matrix
and a real periodic (period-) matrix function such thatIn the above
, , and are matrices.Consequences and applications
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 multiplierCharacteristic multiplier
In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent...
s 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 stableLyapunov stability
Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...
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 HillGeorge William HillGeorge William Hill , was an American astronomer and mathematician.Hill was born in New York City, New York to painter and engraver John William Hill. and Catherine Smith Hill. He moved to West Nyack with his family when he was eight years old. After attending high school, Hill graduated from...
) approximating the motion of the moonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
as a harmonic oscillatorHarmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
in a periodic gravitational fieldGravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
.
Floquet's theorem applied to Mathieu equation
Mathieu's equation is related to the wave equation for the elliptic cylinder.Given
, the Mathieu equation is given by
The Mathieu equation is a linear second-order differential equation with periodic coefficients.
One of the most powerful results of Mathieu's functions is the Floquet's Theorem [1, 2].
It states that periodic solutions of Mathieu equation for any pair (a, q) can be expressed in the form
or
where
is a constant depending on a and q and P(.) is -periodic in w.The constant
is called the characteristic exponent.If
is an integer, then and are linear dependent solutions. Furthermore,
for the solution
or , respectively.We assume that the pair (a, q) is such that
so that the solution is bounded on the real axis. General solution of Mathieu's equation (, non-integer) is the form
where
and are arbitrary constants.All bounded solutions --those of fractional as well as integral order-- are described by an infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency.
Another very important property of Mathieu's functions is the orthogonality [3]:
If
and are simple roots of
then:
i.e.,
where <.,.> denotes an inner product defined from 0 to π.