Title : On Dynamical Systems in n-Dimensional Torus
AuthorInfo : By Tosiya SAITO
AuthorInfo : (Tokyo Metropolitan University)
section : �� 1. In studying global behavior of dynamical systems,
section : �� 2. We denote by $(x_{1^{ }},\cdots, x_{n})$ the coordinates of a point in an w-dimension- al Euclidian space $R_{n}$ . Identifying all the points
section : �� 3. In this section, we introduce a notion of almost periodicity of the solution of (1) which will play an important role in further discussions.
section : �� 4. In what follows, we shall establish following two theorems.
section : �� 5. Here we first prove
section : �� 6. Since stability of the system implies almost periodicity of $\varphi j(t;x_{0})$
section : �� 7. In this section, we shall prove that $f$ is a continuous mapping of $\Omega_{n}$ onto $\Omega_{n}^{\prime}$ .
section : �� 8. Now it remains for us to show that $f$ defines one-to-one correspon- dence between $\Omega_{n}$ and $\Omega_{n^{J}}$ .
section : �� 9. Thus we have seen that $\Omega_{n}(=\overline{G})$ can be regarded as a compact Lie group which is homomorphically mapped onto a compact Lie group $\Omega_{n^{\prime}}$ by the mapping /.
section : �� 10. In this section, the proof of Theorem 4 will be given.
section : Bibliography