Minimal periods of solutions of Lipschitzian differential equations in a vector space with an arbitrary norm
Consider the autonomous systems \[ {dx\over dt}= f(x),\tag{1} \] \[ {d^{2p}x\over dt^{2p}}= f(x)\tag{2} \] in \(\mathbb{R}^n\) or \(\mathbb{C}^n\) under the assumption \[ \| f(x_1)- f(x_2)\|\leq L\| x_1- x_2\|,\tag{3} \] where \(\|.\|\) is any vector norm. The author proves the following results. Theorem 1. Consider system (1) in \(\mathbb{C}^n\) under the assumption (3). Then the minimal period of any non-trivial periodic solution of (1) is \(2\pi/L\). Theorem 2. Consider system (2) in \(\mathbb{C}^n\) or \(\mathbb{R}^n\) under the assumption (3). Then the minimal period of any non-trivial periodic solution of (2) is \(2\pi/L^{{1\over 2p}}\).
scientific article; zbMATH DE number 6106475