Subharmonic solutions of second-order equations arising near harmonic solutions
The author completes his study of \(x''+g(x)=Ef(t)\) where g(x) is an infinitely differentiable odd function and f(t) is a \(2\pi\)-periodic even function such that \(f(t+\pi)=-f(t)\) (that is: f is odd harmonic). Suppose for \(E=E_ 0\), solutions of the associated variation equation \(y''+g'[x(t,E)]=0\) are periodic with a period which is a multiple of \(2\pi\). That is, the characteristic multipliers are exp (\(\pm ip| q)\) where p and q are relatively prime integers such that \(0<p<q\). The period of the solutions is 2q\(\pi\). In favorable cases, there are 2q subharmonics if p and q are odd and 4q subharmonics if either is even. Of these exactly half are stable in a sense defined by the author.
scientific article; zbMATH DE number 3923173