Existence and Computation of Hyperbolic Trajectories of Aperiodically Time Dependent Vector Fields and Their Approximations

Abstract

In this paper we give sufficient conditions for the existence of hyperbolic trajectories in aperiodically time dependent vector fields. These conditions do not require the a priori introduction of hyperbolicity into the dynamics of the vector field or assumptions of "time scale separation". The hyperbolic trajectory is obtained as a solution of an integral equation over an infinite time interval. We give an expression for the error obtained when the solution is approximated over a finite time interval. Finally, we show how the method can be numerically implemented in a specific example.

Controlling the Unsteady Analogue of Saddle Stagnation Points

“…The reason is that a spatially uniform time-dependent forcing makes all trajectories uniformly hyperbolic. Hyperbolicity of the controlled trajectories in this linear instance can be easily verified analytically (see, e.g., section 2.1 in Ju, Small, and Wiggins [24]). Thus we just need to check thatthese trajectories are the only bounded trajectories of the forced system, staying in an ε-neighborhood of the saddle stagnation point.…”

Computation of hyperbolic trajectories and their stable and unstable manifolds for oceanographic flows represented as data sets

“…We develop an improvement (to be described below) of algorithms given in Ide et al (2002) and Ju et al (2003) for computing hyperbolic trajectories. This algorithm is then combined with an algorithm given in Mancho et al (2003) for computing the stable and unstable manifolds of hyperbolic trajectories.…”

“…ISPs are related to the Eulerian flow structure, and themselves are not particle trajectories. Nevertheless, the work in Ide et al (2002) and Ju et al (2003) shows how hyperbolic ISPs can be used to initialize an iterative process that may converge to a hyperbolic trajectory of the flow. This iterative method is based on certain integral equations and is therefore global in time for the entire length of the time interval of interest.…”

“…In this situation Fourier and convolution type methods can be used to construct a hyperbolic trajectory over the entire length of the finite time interval of interest. The algorithm developed in Ju et al (2003) is fully nonlinear and, as mentioned above, utilizes an iterative technique to solve an integral equation, whose solution is a hyperbolic trajectory. The first step in this iterative procedure yields the answer one would obtain by the algorithm developed in Ide et al (2002).…”

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

“…Among others, techniques developed are finite-size Lyapunov exponents (FSLEs) (Aurell et al, 1997) and finitetime Lyapunov exponents (FTLEs) (Haller, 2000;Haller and Yuan, 2000;Haller, 2001;Shadden et al, 2005). 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).…”