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Estimate of the number of periodic solutions via the twist number. (English) Zbl 0881.34063
The authors consider the dynamical system of the form $$\ddot x+ V'(x)=0,$$ where $x\in\bbfR^N$, $V\in C^2(\bbfR^N,\bbfR)$ and the gradient $V'(x)$ is asymptotically linear for $|x|\to\infty$. It is assumed also that the potential $V$ has a finite number of non-degenerate critical points $z_1,\dots,z_n$. Starting from the positive eigenvalues of the Hessian matrices $V''(z_i)$, the authors define the global twist number of the system and using this characteristic they give a lower estimate for the number $n(T)$ of non-constant $T$-periodic solutions of the system for $T$ sufficiently large.

##### MSC:
 34C25 Periodic solutions of ODE 37-99 Dynamic systems and ergodic theory (MSC2000)
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