ABSTRACT. We consider the periodic boundary value problem for the non-autonomous scalar second order equation x''+F(x,x')=e(t); with e(t) a continuous and T-periodic forcing term. Using a continuation theorem adapted from Capietto et al. (Trans. Amer. Math. Soc. 329 (1992) 41–72), we propose some new conditions for the existence of T-periodic solutions to the forced equation in terms of the dynamical properties of the trajectories of the associated autonomous equation x''+F(x,x')=0: Special emphasis will be addressed to the study of the case in which the presence of an unbounded separatrix for the autonomous system in the phase-plane allows to obtain a priori bounds for the T-periodic solutions of the homotopic equation x''+F(x,x')=l*e(t).
A geometrical approach to periodically forced dynamical systems in presence of a separatrix / G. VILLARI; F. ZANOLIN. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - STAMPA. - 208:(2005), pp. 292-311. [10.1016/j.jde.2003.09.005]
A geometrical approach to periodically forced dynamical systems in presence of a separatrix
VILLARI, GABRIELE;
2005
Abstract
ABSTRACT. We consider the periodic boundary value problem for the non-autonomous scalar second order equation x''+F(x,x')=e(t); with e(t) a continuous and T-periodic forcing term. Using a continuation theorem adapted from Capietto et al. (Trans. Amer. Math. Soc. 329 (1992) 41–72), we propose some new conditions for the existence of T-periodic solutions to the forced equation in terms of the dynamical properties of the trajectories of the associated autonomous equation x''+F(x,x')=0: Special emphasis will be addressed to the study of the case in which the presence of an unbounded separatrix for the autonomous system in the phase-plane allows to obtain a priori bounds for the T-periodic solutions of the homotopic equation x''+F(x,x')=l*e(t).File | Dimensione | Formato | |
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