On the Angular Variation of Solutions of Second Order Linear Systems
Steven D. Taliaferro · SIAM Journal on Mathematical Analysis · 1980
Upper bounds for the angular variation of extremal solutions of the second order linear system \[x'' + P(t)x = 0,\] where $P(t)$ is a symmetric $n \times n$ matrix, are obtained. The angular variation of a solution of the above equation is defined to be the length of its radial projection on the $n - 1$ dimensional sphere. Also, if $n = 2$ and $x(t)$ is an extremal solution, then an upper bound, depending on the angular variation of $x(t)$, is obtained for the number of zeros of each component of $x(t)$. The proofs are based on variational arguments.