### Massera's theorem for quasi-periodic differential equations

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.

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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