Rotating periodic solutions of second order dissipative dynamical systems (Q495776)

The authors investigate the dissipative dynamical system \[ u''+cu'+\nabla g(u)+h(u)=e(t), \] where \(c>0\) is a constant, \(g(u)=g(|u|)\), \(h\in {\mathcal C}(\mathbb R^n, \mathbb R^n)\), \(n\geq 2,\) \(h(u)=Qh(Q^{-1}u),\) \(e\in {\mathcal C}(\mathbb R, \mathbb R^n)\), \(e(t+T)=Qe(t)\) for some \(Q\in O(n)\), \(O(n)\) is the orthogonal group on \(\mathbb R^n\). They prove the following results. When \(g\in {\mathcal C}^1(\mathbb R^n, \mathbb R)\), \(\lim_{|u|\to\infty}|\nabla g(u)|=\infty\), \(|h(u)|\leq M\) for all \(u\in\mathbb R^n\) and for some positive constant \(M\), then the considered dynamical system admits a \(Q\)-rotating periodic solution. If \(\mathrm{Ker}(I_n-Q)\neq \emptyset\), \(g\in {\mathcal C}^1(\mathbb R^n\backslash \{0\}, \mathbb R),\) \(\lim_{|u|\to +\infty}|\nabla g(u)|=\infty,\) \(|h(u)|\leq M\) for all \(u\in\mathbb R^n\) and for some positive constant \(M\), \(\lim_{u\to 0}|\nabla g(u)\cdot u|=\infty\), and there exists a function \(\phi\in {\mathcal C}^1(\mathbb R^n\backslash \{0\}, \mathbb R)\) such that \(\lim_{|u|\to 0}\phi(u)=+\infty\) and \(|\nabla \phi(u)|^2\leq \eta_1|\nabla g(u)\cdot u|+\eta_2\) for some \(\eta_1\), \(\eta_2>0,\) when \(0<|u|<\delta_0,\) \(\delta_0>0\) is small enough, then the considered dynamical system admits a non-collision \(Q\)-rotating periodic solution.