A Translation of Hopf’s Original Paper

Abstract

Let

$$ \mathop x\limits^. = {F_1}({x_1}, \ldots, {x_n},\mu )\quad (i = 1, \ldots, n) $$

or, in vector notation,

$$ \mathop x\limits_{-}^. = \mathop F\limits_{-} (\mathop x\limits_{-}, \mu ) $$

((1.1))

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.

“Abzweigung einer periodischen Lösung von einer stationären Losung eines Differentialsystems” Berichten der Mathematisch-Physischen Klasse der Sächsischen Akademie der Wissenschaften zu Leipzig. XCIV. Band Sitzung vom 19. Januar 1942.

Bifurcation of a Periodic Solution from a Stationary Solution of a System of Differential Equations by Eberhard Hopf

Dedicated to Paul Koebe on his 60th birthday