A geometric approach to periodically forced dynamical systems in presence of a separatrix |
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Authors: | Gabriele Villari Fabio Zanolin |
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Institution: | a Dipartimento di Matematica “U.Dini,” Università, viale Morgagni 67/A, 50137 Firenze, Italy b Dipartimento di Matematica e Informatica, Università, via delle Scienze 206, 33100 Udine, Italy |
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Abstract: | We consider the periodic boundary value problem for the non-autonomous scalar second-order equation , with e(·) 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 . 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 . |
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Keywords: | Continuation theorem Non-autonomous equations Periodic solutions |
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