Instability of periodic minimals
Antonio J. Ureña
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 345-357 doi: 10.3934/dcds.2013.33.345
We consider second-order Euler-Lagrange systems which are periodic in time. Their periodic solutions may be characterized as the stationary points of an associated action functional, and we study the dynamical implications of minimizing the action. Examples are well-known of stable periodic minimizers, but instability always holds for periodic solutions which are minimal in the sense of Aubry-Mather.
keywords: Periodic minimizers quasi-asymptotic solutions minimals instability.
Some new qualittative properties on the solvability set of pendulum-type equations
Antonio Cañada Antonio J. Ureña
Conference Publications 2001, 2001(Special): 66-73 doi: 10.3934/proc.2001.2001.66
Please refer to Full Text.
keywords: Pendulum-type equations asymptotic results linear damping Baire category. Dirichlet boundary conditions solvability set Riemann-Lebesgue lemma
Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force
Alessandro Fonda Antonio J. Ureña
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 169-192 doi: 10.3934/dcds.2011.29.169
We consider planar systems driven by a central force which depends periodically on time. If the force is sublinear and attractive, then there is a connected set of subharmonic and quasi-periodic solutions rotating around the origin at different speeds; moreover, this connected set stretches from zero to infinity. The result still holds allowing the force to be attractive only in average provided that an uniformity condition is satisfied and there are no periodic oscillations with zero angular momentum. We provide examples showing that these assumptions cannot be skipped.
keywords: periodic and quasi-periodic solutions nonlinear dynamics. central force

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