Distinguished trajectories in time dependent vector fields

TL;DR

This paper introduces a new numerical method to identify distinguished trajectories in time-dependent vector fields, extending fixed point and periodic orbit concepts to aperiodic systems, with applications to realistic flows.

Contribution

The paper proposes a generalized definition of distinguished trajectories applicable to aperiodic systems and provides a numerical procedure for their identification, including in finite time intervals.

Findings

Successfully applied to known examples of distinguished trajectories.

Characterizes distinguished trajectories in highly aperiodic realistic flows.

Trajectories outside finite intervals are not distinguished.

Abstract

We introduce a new definition of distinguished trajectory that generalises the concepts of fixed point and periodic orbit to aperiodic dynamical systems. This new definition is valid for identifying distinguished trajectories with hyperbolic and non-hyperbolic types of stability. The definition is implemented numerically and the procedure consist in determining a path of limit coordinates. It has been successfully applied to known examples of distinguished trajectories. In the context of highly aperiodic realistic flows our definition characterises distinguished trajectories in finite time intervals, and states that outside these intervals trajectories are no longer distinguished.