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Existence of the global attractor is proved for the strong solutions to the 3D viscous Primitive Equations (PEs) modeling large scale ocean and atmosphere dynamics. This result is obtained under the natural assumption that the external heat source Q is square integrable. Furthermore, it is shown in [20] that the fractal and Hausdroff dimensions of the global attractor for 3D viscous PEs are both finite.

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In this paper we give sufficient conditions for the existence of hyperbolic trajectories in aperiodically time dependent vector fields. These conditions do not require the a priori introduction of hyperbolicity into the dynamics of the vector field or assumptions of "time scale separation". The hyperbolic trajectory is obtained as a solution of an integral equation over an infinite time interval. We give an expression for the error obtained when the solution is approximated over a finite time interval. Finally, we show how the method can be numerically implemented in a specific example.

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