The phase portrait of aperiodic non-autonomous dynamical systems

Details The phase portrait of aperiodic non-autonomous dynamical systems The phase portrait of aperiodic non-autonomous dynamical systems

2011-06-07

arxiv.org/abs/1106.1306v1

Nonlinear Sciences

Chaotic Dynamics

35 pages, 21 figures

Scientific paper

Geometrical representations on the phase space are at the basis of Poincar\'e ideas of seeking structures that divide it into regions corresponding to trajectories with different dynamical fates. These ideas have been a very useful approach for studying dynamical systems. However while these representations are well achieved for autonomous and time dependent periodic dynamical systems, there is not a well established theory for describing those systems with general aperiodic time dependence. In this article we propose a methodology to build Lagrangian descriptors for arbitrary time dependent flows based on intrinsic bounded positive geometrical and physical properties of trajectories. We analyze the convenience of different descriptors from several points of view: regularity conditions requested on the vector field, rate at which the Lagrangian information is achieved and computational performance. Comparisons with other traditional methods such as FTLE are also reported.

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