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Rigorous Numerics for PDEs with Indefinite Tail : Existence of a Periodic Solution of the Boussinesq Equation with Time-dependent Forcing

Czechowski A., Zgliczyński P.

Schedae Informaticae

2015

Vol. 24

143--158

We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even though there is no smoothing effect for arbitrary smooth initial data, we are able to apply the method of self-consistent bounds to deduce the existence of smooth classical periodic solutions in the vicinity of 0. The proof is non-perturbative and relies on construction of periodic isolating segments in the Galerkin projections.

A study of chaos for processes under small perturbations II : rigorous proof of chaos

Oprocha P., Wilczyński P.

Opuscula Mathematica

2010

Vol. 30, no. 1

5-36

In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation [formula]. Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution 0 present in the system when N = 0 are also discussed.