以下搬運(yùn)自維基百科：

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.

以下翻譯自谷歌翻譯（建議直接看英文）：

Floquet理論是常微分方程理論的一個(gè)分支，與形式為周期線性微分方程的解的類別有關(guān)

與組成的分段連續(xù)周期函數(shù)，周期為，它定義了解的穩(wěn)定性狀態(tài)。

Floquet理論的主要定理，即由Gaston Floquet（1883）提出的Floquet定理，為該常見線性系統(tǒng)的每個(gè)基本矩陣解提供了標(biāo)準(zhǔn)形式。給出坐標(biāo)更改與將周期系統(tǒng)轉(zhuǎn)換為具有恒定實(shí)系數(shù)的傳統(tǒng)線性系統(tǒng)。

在固態(tài)物理學(xué)中，類似的結(jié)果稱為布洛赫定理。

注意，線性微分方程的解形成向量空間。如果所有列都是線性獨(dú)立的解，則矩陣被稱為基本矩陣解決方案。如果所有列都是線性獨(dú)立的解，并且存在使得是身份，則矩陣被稱為主基本矩陣解。可以使用。初始條件為的線性微分方程的解為，其中是任何基本矩陣解。

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?,

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 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 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.

• 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.

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