Nonlinear Processes in Oceanic and Atmospheric Flows


Generalizing fixed points to aperiodic dynamical systems

Author: José Antonio Jiménez Madrid, Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3M).

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Generalizing fixed points to aperiodic dynamical systems

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In the context of stationary flows the idea of {\em fixed point} is a keystone to describe geometrically the solutions of the dynamical system. It is extended to time periodic flows by means of the Poincar\'e map, as periodic orbits become fixed points on the Poincare map.
Recent articles provide an important step-forwards to extend the concept of hyperbolic fixed point to aperiodic dynamical systems. In this presentation, we propose a new formal definition of {\em Distinguished trajectory} (DT) in aperiodic flows. We numerically test this definition in forced Duffing type flows with known exact distinguished trajectories. The definition accurately locates these trajectories.
We also check the definition for examples of aperiodic flows in oceanographic contexts and we find that it overcomes some technical difficulties of other approaches.
Also, this definition of distinguished trajectory is valid both for hyperbolic and non-hyperbolic type of stabilities and does not depend on the dimension $n$ of the vector field.

*Satellite images from NASA and ESA

Nonlinear Processes in Oceanic and Atmospheric Flows. July 2-4, 2008. Castro Urdiales, Cantabria, Spain.