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We prove the existence of time-periodic solutions to non-linear massiveKlein-Gordon equations in Anti-de Sitter as well as their orbital stabilityover exponentially long times for certain values of the mass corresponding tocompletely resonant spectrum. We analyse the resonant system in the Fourierspace by relying in particular on Zeilberger's algorithm which allows for asystematic way to derive recurrence formulae for the Fourier coefficients. Wealso show that the derivation and analysis of the Fourier systems easilyextends to other semi-linear wave equations such as the co-rotational wave mapequation.

Time periodic solutions and Nekhoroshev stability to non-linear massive Klein-Gordon equations in Anti-de Sitter