A Translation of Hopf’s Original Paper

Let $$ \mathop x\limits^. = {F_1}({x_1}, \ldots, {x_n},\mu )\quad (i = 1, \ldots, n) $$ or, in vector notation, (1.1) $$ \mathop x\limits_{-}^. = \mathop F\limits_{-} (\mathop x\limits_{-}, \mu ) $$ be a real system of differential equations with real parameter µ, where F is analytic in x and µ for x in a domain G and |µ| < c. For |µ| < c let (1.1) possess an analytic family of stationary solutions $$ \mathop x\limits_{-} = \mathop x\limits_{-}^\sim (\mu ) $$ lying in G: $$ \mathop F\limits_{-} (\mathop x\limits_{-}^\sim (\mu )\,,\mu ) = 0 $$ . As is well known, the characteristic exponents of the stationary solution are the eigenvalues of the eigenvalue problem $$ \lambda \mathop{{{\text{ }}a}}\limits_{ - } = {{\mathop{{{\text{ }}L}}\limits_{ - } }_{\mu }}\mathop{{{\text{ }}a}}\limits_{ - } $$ where Lµ stands for the linear operator, depending only on µ, which arises after neglect of the nonlinear terms in the series expansion of F about $$ \mathop x\limits_{-} = \mathop x\limits_{-}^\sim $$ . The exponents are either real or pairwise complex conjugate and depend on µ.