Estimate of the number of periodic solutions via the twist number (Q678037)

The authors consider the dynamical system of the form \[ \ddot x+ V'(x)=0, \] where \(x\in\mathbb{R}^N\), \(V\in C^2(\mathbb{R}^N,\mathbb{R})\) 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.