The Least Distance between Extrema and the Minimum Period of Solutions of Autonomous Vector Differential Equations

Abstract

Solutions x(t) of the equation \(\dot {x} = f(x)\), where \(x \in {{{\text{R}}}^{n}}\) and the function f(x) satisfies the Lipschitz condition with an arbitrary vector norm, are considered. It is proved that the lower bound for the distances between successive extrema x_k(t), k = 1, 2, …, n, is \(\frac{\pi }{L}\), where L is the Lipschitz constant. For nonconstant periodic solutions, the lower bound for the periods is \(\frac{{2\pi }}{L}\). These estimates are sharp for norms that are invariant with respect to permutations of indices.