Bounds for the period of periodic orbits of dynamical systems
Let U be an open subset of a Banach space E and let f:U\(\to E\) be a Lipschitz map with constant \(L:\| f(x)-f(y)\| \leq L\| x-y\|.\) We consider the autonomous first order differential equation (1) \(x'=f(x)\) with the corresponding Euler difference equation (2) \(x_{i+1}=x_ i+hf(x_ i)\) and derive sharp lower bounds for the period of any periodic orbit of (2). As an application of these bounds we obtain results relating the periods of solutions of (1) to the Lipschitz constant L, as well as results on the fixed points of iterates of maps on Banach and Hilbert spaces. In one of our theorems we prove that, if T is the period of a solution of (1), then \(T\geq 4.5/L\). This improves the bound \(T\geq 4/L\) obtained by Lasota and Yorke. For (2) in Hilbert space we obtain the sharp bound hL\(\geq 2 \sin (\pi /n)\). The bound for (1) in Banach spaces has been improved to \(T\geq 6/L\) which is the best possible by the authors and \textit{D. Fisher} in Proc. Am. Math. Soc. 98, 376-378 (1986).
autonomous first order differential equation
scientific article; zbMATH DE number 3998273