Minimal period of solutions to Lipschitz differential equations with arbitrary vector norm

Abstract

The Lipschitz differential equation, $\dot x=f(x)$, in spaces $X \in C^n$ and $X \in R^n$ is considered. The minimal period problem is to find the exact lower bound for peri-ods of non-constant solutions, expressed in the Lipschitz constant $L$. In this paper, some inequality for the components, $x_k(t)$, which is independent on the space $X$ is found. As a result, it is proved that for any $X$ and $n$, the normalized minimal period, $k=TL \leq 2\pi$. In the space $C^n$, the equality $k= 2\pi$ is reached for any $X$. For $R^n$, this equality is attained for univercally adopted norms.