Massera's theorem for quasi-periodic differential equations

Rafael Ortega, Massimo Tarallo

DOI: http://dx.doi.org/10.12775/TMNA.2002.003

Abstract


For a scalar, first order ordinary differential equation which
depends periodically on time, Massera's Theorem says that the
existence of a bounded solution implies the existence of a periodic solution.
Though the statement is false when periodicity
is replaced by quasi-periodicity, solutions with some kind of
recurrence are anyway expected when the equation is quasi-periodic
in time. Indeed we first prove that the existence of a bounded
solution implies the existence of a solution which is
quasi-periodic in a weak sense. The partial differential equation,
having our original equation as its equation of characteristics,
plays a key role in the introduction of this notion of weak
quasi-periodicity. Then we compare our approach with others
already known in the literature. Finally, we give an explicit
example of the weak case, and an extension to higher dimension for
a special class of equations.

Keywords


Bounded; recurrent and quasi-periodic solutions; partial differential equations on the torus

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