Distinguished trajectories in time dependent vector fields

Abstract

We introduce a new definition of distinguished trajectory that generalizes the concepts of fixed point and periodic orbit to aperiodic dynamical systems. This new definition is valid for identifying distinguished trajectories with hyperbolic and nonhyperbolic types of stability. The definition is implemented numerically and the procedure consists of 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 characterizes distinguished trajectories in finite time intervals, and states that outside these intervals trajectories are no longer distinguished. This paper attempts to generalize the concepts of fixed point and periodic orbit to time dependent aperiodic dynamical systems. Fixed points and periodic orbits are keystones for describing solutions of autonomous and time periodic dynamical systems, as the stable and unstable manifolds of these hyperbolic objects form the basis of the geometrical template organizing the description of the dynamical system. The mathematical theory of aperiodic dynamical systems is far from complete. In this context, this work deals with a general definition that encompasses the concepts of fixed point and periodic orbit and which when applied to finite time and aperiodic dynamical systems identifies special trajectories that play an organizing role in the geometry of the flow.

A dynamical systems perspective for a real-time response to a marine oil spill

“…In the unfortunate event of a marine oil spill, it is crucial to take advantage 40 of different scientific tools and mathematical models that will help identify the spill and provide a description of the oceanic transport processes driving the currents in the region where this environmental catastrophe occurs. Therefore, the synergy between different approaches undoubtedly plays a key role in the development of contingency plans and strategies to assess effectively these dis- 45 asters, in order to reduce their ecological impact as much as possible in future events.…”

“…The initial green circle approaches the coast line following the stable manifold, and elongates itself along the unstable manifold. Figure 2D) provides further details of the underlying skeleton, by plotting two unstable A skeleton of the stable and unstable manifolds of hyperbolic trajectories in time dependent flows can be constructed with the technique that is referred to as Lagrangian descriptors, based on the function M (see [40,26,27]), defined as follows: Another tool for revealing invariant manifolds in time dependent flows, which has been extensively applied in geophysical contexts [12,25,9] In order to deal with Eq. (1), we have assumed that particle motion is mainly horizontal on a sphere of radius R:…”

A simple kinematic model for the Lagrangian description of relevant nonlinear processes in the stratospheric polar vortex

“…Other techniques include DHTs (Ide et al, 2002;Ju et al, 2003) and the direct calculation of manifolds as material surfaces (Mancho et al, 2003(Mancho et al, , 2004(Mancho et al, , 2006b, the geodesic theory of Lagrangian coherent structures (LCS) (Haller and Beron-Vera, 2012) and the variational theory of LCS (Farazmand and Haller, 2012), etc. Our choice in this work will be the use of the Lagrangian descriptor (LD) function M introduced by Madrid and Mancho (2009) and Mendoza and Mancho (2010). The function M has been applied in a variety of geophysical contexts.…”

Coherent structures and the saturation of a nonlinear dynamo

“…In this paper, the emphasis is on the detection of coherent structures and the transport of passive scalars and magnetic field lines in the transition from the kinematic to the saturated phase. Eulerian structures are detected using the Q-criterion and, for the detection of LCSs, the traditional technique of finite-time Lyapunov exponents (FTLEs) is compared with the recently proposed function M (Madrid & Mancho 2009).…”

“…From a Lagrangian point of view, coherent structures are seen as material surfaces that form the boundaries between regions of the flow with different behavior, such as vortex surfaces. They are found by following trajectories of fluid particles, while computing quantities such as the maximum rate of divergence of neighboring trajectories (Haller 2001;Shadden et al 2005) or the arc-length of the trajectory (Madrid & Mancho 2009). Lagrangian tools are naturally suited for unsteady flows, since they take into account the temporal variations of the vector field, not just instantaneous snapshots.…”