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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Floquet theory ： ウィキペディア英語版 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:\dot = A(t) x,\,with \displaystyle A(t) a piecewise continuous periodic function with period T and defines the state of the stability of solutions. 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 \displaystyle y=Q^(t)x with \displaystyle Q(t+2T)=Q(t) 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 \phi\,(t) is called a ''fundamental matrix solution'' if all columns are linearly independent solutions. A matrix \Phi(t) is called a ''principal fundamental matrix solution'' if all columns are linearly independent solutions and there exists t_0 such that \Phi(t_0) is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using \Phi(t)=\phi\,(t)^(t_0). The solution of the linear differential equation with the initial condition x(0)=x_0 is x(t)=\phi\,(t)^(0)x_0 where \phi \,(t) is any fundamental matrix solution.== Floquet's theorem == Floquet theorem redirects to this section -->Let \dot= A(t) x be a linear first order differential equation,where x(t) is a column vector of length n and A(t) an n \times n periodic matrix with period T (that is A(t + T) = A(t) for all real values of t). Let \phi\, (t) be a fundamental matrix solution of this differential equation. Then, for all t \in \mathbb,: \phi(t+T)=\phi(t) \phi^(0) \phi (T).\ Here:\phi^(0) \phi (T)\ is known as the monodromy matrix.In addition, for each matrix B (possibly complex) such that:e^=\phi^(0) \phi (T),\ there is a periodic (period T) matrix function t \mapsto P(t) such that:\phi (t) = P(t)e^\textt \in \mathbb.\ Also, there is a ''real'' matrix R and a ''real'' periodic (period-2T) matrix function t \mapsto Q(t) such that:\phi (t) = Q(t)e^\textt \in \mathbb.\ In the above B, P, Q and R are n \times n matrices.== Consequences and applications ==This mapping \phi \,(t) = Q(t)e^ gives rise to a time-dependent change of coordinates (y = Q^(t) x), under which our original system becomes a linear system with real constant coefficients \dot = 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 \phi \, (t) = P(t)e^ is called a ''Floquet normal form'' for the fundamental matrix \phi \, (t).The eigenvalues of e^ 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 \mu such that e^ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since e^)T}=e^, where k 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. 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 :$\backslash dot\; =\; A(t)\; x,\backslash ,$ with $\backslash displaystyle\; A(t)$ a piecewise continuous periodic function with period $T$ and defines the state of the stability of solutions. 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 $\backslash displaystyle\; y=Q^(t)x$ with $\backslash displaystyle\; Q(t+2T)=Q(t)$ 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 $\backslash phi\backslash ,(t)$ is called a ''fundamental matrix solution'' if all columns are linearly independent solutions. A matrix $\backslash Phi(t)$ is called a ''principal fundamental matrix solution'' if all columns are linearly independent solutions and there exists $t\_0$ such that $\backslash Phi(t\_0)$ is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using $\backslash Phi(t)=\backslash phi\backslash ,(t)^(t\_0)$. The solution of the linear differential equation with the initial condition $x(0)=x\_0$ is $x(t)=\backslash phi\backslash ,(t)^(0)x\_0$ where $\backslash phi\; \backslash ,(t)$ is any fundamental matrix solution. == Floquet's theorem == Let $\backslash dot=\; A(t)\; x$ be a linear first order differential equation, where $x(t)$ is a column vector of length $n$ and $A(t)$ an $n\; \backslash times\; n$ periodic matrix with period $T$ (that is $A(t\; +\; T)\; =\; A(t)$ for all real values of $t$). Let $\backslash phi\backslash ,\; (t)$ be a fundamental matrix solution of this differential equation. Then, for all $t\; \backslash in\; \backslash mathbb$, :$\backslash phi(t+T)=\backslash phi(t)\; \backslash phi^(0)\; \backslash phi\; (T).\backslash$ Here :$\backslash phi^(0)\; \backslash phi\; (T)\backslash$ is known as the monodromy matrix. In addition, for each matrix $B$ (possibly complex) such that :$e^=\backslash phi^(0)\; \backslash phi\; (T),\backslash$ there is a periodic (period $T$) matrix function $t\; \backslash mapsto\; P(t)$ such that :$\backslash phi\; (t)\; =\; P(t)e^\backslash textt\; \backslash in\; \backslash mathbb.\backslash$ Also, there is a ''real'' matrix $R$ and a ''real'' periodic (period-$2T$) matrix function $t\; \backslash mapsto\; Q(t)$ such that :$\backslash phi\; (t)\; =\; Q(t)e^\backslash textt\; \backslash in\; \backslash mathbb.\backslash$ In the above $B$, $P$, $Q$ and $R$ are $n\; \backslash times\; n$ matrices. == Consequences and applications == This mapping $\backslash phi\; \backslash ,(t)\; =\; Q(t)e^$ gives rise to a time-dependent change of coordinates ($y\; =\; Q^(t)\; x$), under which our original system becomes a linear system with real constant coefficients $\backslash dot\; =\; R\; y$. 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 $\backslash phi\; \backslash ,\; (t)\; =\; P(t)e^$ is called a ''Floquet normal form'' for the fundamental matrix $\backslash phi\; \backslash ,\; (t)$. The eigenvalues of $e^$ are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps $x(t)\; \backslash to\; x(t+T)$. A ''Floquet exponent'' (sometimes called a characteristic exponent), is a complex $\backslash mu$ such that $e^$ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since $e^)T\}=e^$, where $k$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「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:\dot = A(t) x,\,with \displaystyle A(t) a piecewise continuous periodic function with period T and defines the state of the stability of solutions. 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 \displaystyle y=Q^(t)x with \displaystyle Q(t+2T)=Q(t) 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 \phi\,(t) is called a ''fundamental matrix solution'' if all columns are linearly independent solutions. A matrix \Phi(t) is called a ''principal fundamental matrix solution'' if all columns are linearly independent solutions and there exists t_0 such that \Phi(t_0) is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using \Phi(t)=\phi\,(t)^(t_0). The solution of the linear differential equation with the initial condition x(0)=x_0 is x(t)=\phi\,(t)^(0)x_0 where \phi \,(t) is any fundamental matrix solution.== Floquet's theorem == Floquet theorem redirects to this section -->Let \dot= A(t) x be a linear first order differential equation,where x(t) is a column vector of length n and A(t) an n \times n periodic matrix with period T (that is A(t + T) = A(t) for all real values of t). Let \phi\, (t) be a fundamental matrix solution of this differential equation. Then, for all t \in \mathbb,: \phi(t+T)=\phi(t) \phi^(0) \phi (T).\ Here:\phi^(0) \phi (T)\ is known as the monodromy matrix.In addition, for each matrix B (possibly complex) such that:e^=\phi^(0) \phi (T),\ there is a periodic (period T) matrix function t \mapsto P(t) such that:\phi (t) = P(t)e^\textt \in \mathbb.\ Also, there is a ''real'' matrix R and a ''real'' periodic (period-2T) matrix function t \mapsto Q(t) such that:\phi (t) = Q(t)e^\textt \in \mathbb.\ In the above B, P, Q and R are n \times n matrices.== Consequences and applications ==This mapping \phi \,(t) = Q(t)e^ gives rise to a time-dependent change of coordinates (y = Q^(t) x), under which our original system becomes a linear system with real constant coefficients \dot = 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 \phi \, (t) = P(t)e^ is called a ''Floquet normal form'' for the fundamental matrix \phi \, (t).The eigenvalues of e^ 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 \mu such that e^ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since e^)T}=e^, where k 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.」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.