弗洛凱理論

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弗洛凱理論是常微分方程理論的一種，討論有關下列微分方程類型的解答類別，

$\dot{x} = A(t) x$ ,

其中，A(t)是一週期為T的連續週期函數。

弗洛凱理論的主要定理－弗洛凱定理給出了一般線性系統的每個基本解的正規形式。它給定了一座標轉變 $y = Q - 1 (t) x$ ，其中 $Q (t + 2 T) = Q (t)$ ，用以來轉變週期系統至有常數及實係數的傳統線性系統。

在固態物理中，其類比的結果(推廣至三維)為布拉赫定理。

Note that the solutions of the linear differential equation form a vector space. A Matrix $φ(t)$ is called fundamental matrix solution if all columns are linearly independent solutions. It is called a principal fundamental matrix at $t 0$ if $φ(t 0)$ is the identity. Because of existence and uniqueness of the solutions there is a principal fundamental matrix $Φ(t 0) = φ(t)φ - 1 (t 0)$ for each $t 0$ . The solution of the linear differential equation with the initial condition $x (0) = x 0$ is $x (t) = φ(t)φ - 1 (0) x 0$ where $φ(t)$ is any fundamental matrix solution.

[编辑] 弗洛凱定理

If $φ(t)$ is a fundamental matrix solution of the periodic system $\dot{x}= A(t) x$ , with $A (t)$ a periodic function with period $T$ then, for all $t \in \mathbb{R}$ ,

φ(t + T) = φ(t)φ - 1 (0)φ(T).

In addition, for each matrix $B$ (possibly complex) such that:

e T B = φ - 1 (0)φ(T)

there is a periodic (period $T$ ) matrix function $t \to P(t)$ such that

φ(t) = P (t) e t B

for all $t \in \mathbb{R}$ .

Also, there is a real matrix $R$ and a real periodic (period $2 T$ ) matrix function $t \to Q(t)$ such that

φ(t) = Q (t) e t R

for all $t \in \mathbb{R}$ .

[编辑] 結論與應用

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

The representation $φ(t) = P (t) e t B$ is called a Floquet normal form for the fundamental matrix $φ(t)$ .

The eigenvalues of $e T B$ are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps $x(t) \to x(t+T)$ . A Floquet exponent (sometimes called a characteristic exponent), is a complex $μ$ such that $e μ T$ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since $e^{\mu + \frac{2 \pi i k}{T}}=e^{\mu T}$ . 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 to the study of dynamical systems
Floquet theory shows stability in Hill's equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field.

[编辑] 參考

Chicone, Carmen. Ordinary Differential Equations with Applications. Springer-Verlag, New York 1999
Gaston Floquet, "Sur les équations différentielles linéaires à coefficients périodiques," Ann. École Norm. Sup. 12, 47-88 (1883).