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Periodic solutions of dynamical systems. (English) Zbl 0599.70010
The dynamics of a finite number of material points embedded in a conservative field of forces is studied under the assumption that the potential function is periodic in t and asymptotically linear in x. More precisely, the problem of finding T-periodic solutions of the problem \(- \ddot x-K^ 2x=\omega^ 2\nabla V(x,\omega t),\) \(K\in {\mathbb{Z}},\quad \omega =2\pi /T\) is considered in the case when V(x,t) and \(\nabla V(x,t)\) are bounded. Under these conditions the Palais-Smale condition is not in general satiesfied. Nevertheless it seems possible to find a multiplicity of T-periodic solutions for T large enough if some rather simple restrictions on V are valid.
The method used in the first part of the paper is also applied to the research of T-periodic solutions of Hamiltonian systems \[ -J\dot z=\omega H_ z(z,\omega t),\quad J=\left( \begin{matrix} 0\\ I\end{matrix} \begin{matrix} -I\\ 0\end{matrix} \right), \] under the hypothesis that there exists \(\lambda\in {\mathbb{R}}\) such that \(H_ z(z,t)=\lambda z+G_ z(z,t)\), where \(| G_ z(z,t)| /| z| \to 0\) as \(| z| \to +\infty\) uniformly in t. The assumptions on G(z,t) are given that guarantee the existence of T-periodec solutions for sufficiently large T.
Reviewer: I.Ya.Dorfman

70F10 \(n\)-body problems
70H05 Hamilton’s equations
34C25 Periodic solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
Full Text: DOI
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