The twist coefficient of periodic solutions of a time-dependent Newton’s equation. (English) Zbl 0761.34036

A \(T\)-periodic solution \(\varphi\) of a \(T\)-periodic equation \(x''+f(t,x)=0\) is considered. It is supposed that \(\varphi\) is generic non zero twist type up to the 4th order terms. If \(f(t,x)=a(t)x+b(t)x^ 2+c(t)x^ 3+\dots\) and \(\varphi(t)\equiv 0\) then conditions for the coefficients (boundedness and sign type) are formulated, which guarantee that the trivial solution is of twist type and as a consequence Lyapunov stable. The results are illustrated by examples including the pendulum of variable length.


34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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