Generalized fixed points and the Lagrangian structure of time dependent geophysical flows

Aperiodic geophysical flows are poorly understood as theory which is well established in autonomous or periodic flows is not directly applicable to them. In stationary flows the idea of {\it fixed point} is a keystone to describe geometrically the solutions of the dynamical system. The concept of fixed point is extended to time periodic flows by means of the Poincar\'e map, as periodic orbits with $T$ period become fixed points on the Poincar\'e map. To gain insight on the geometrical structure of aperiodic flows typically are used concepts such as Lyapunov exponents and its finite time versions (FSLE and FTLE). In this presentation we propose to this end a generalisation of the concept of fixed point to aperiodic dynamical systems: the distinguished trajectory. In the context of highly aperiodic realistic flows our definition characterizes trajectories and states that they hold the property of being distinguished in a finite time interval. Previous works by Ide {\em et al.} and Ju {\em et al.} have addressed the existence of {\it distinguished hyperbolic trajectories} but our new definition shows that non-hyperbolic orbits may also fall within this category. This type of trajectories might be of special interest for their applications in oceanography as they are related to eddies or vortices.