Editorial Comments

Abstract

Hopf’s argument can be considerably simplified. After “blowing up” the equation (1.1) to (2.3), one wishes to show that for each sufficiently small ε there is a µ(ε), a period T(ε) and initial conditions y⁰(ε) (suitably normalized), so that (2.16) holds; the family of solutions to (1.1) asserted in the theorem is then x(t,ε) = εy(t,µ(ε),ε,v⁰). Now (2.16) is satisfied if µ = ε = 0, y⁰ = z⁰. Hence, the existence of the functions µ(ε), T(ε), y⁰(ε) follows from an implicit function theorem argument, provided that the n × n matrix

$$ \left( {\frac{{\partial \mathop y\limits_{-} }} {{\partial \mathop t\limits_{-} }},\;\frac{{\partial \mathop y\limits_{-} }} {{\partial \mu }},\;\frac{{\partial \mathop y\limits_{-} }} {{\partial \mathop {{y^0}}\limits_{-} }}} \right)\left| {t = {T_0}} \right.,\;\mu = 0,\;\varepsilon . = 0,\;{\mathop y\limits_{-}^0} = {\mathop z\limits_{-}^0} $$

has maximal rank. (Here $\frac{{\partial \mathop y\limits_{-} }} {{\partial \mathop {{y^0}}\limits_{-} }}$ is an n × (n-2) matrix representing the derivative of y with respect to (n-2) initial directions; there are two restrictions on the initial conditions from the normalization.) We show below how the rank of this matrix may be computed more easily.