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

TL;DR

This paper establishes a universal lower bound for the minimal period of solutions to Lipschitz differential equations, showing it is at most 2π, and characterizes when this bound is attained across different spaces and norms.

Contribution

It proves a universal inequality for the minimal period of solutions to Lipschitz differential equations, independent of the space and norm, and identifies conditions for equality.

Findings

Minimal period normalized by Lipschitz constant is at most 2π.

Equality in the minimal period bound is achieved in certain spaces and norms.

The results are independent of the specific space, applying to both $C^n$ and $R^n$.

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.