## 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.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
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### References:

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