Schematic illustration of approach I (a) and approach II (b) in the extended phase space.
The errors in the computation of a repelling LCS grow exponentially as the LCS is advected forwards in time. The same statement holds for the backward-time advection of an attracting LCS.
(a) A forward strain-surface evolves into a backward stretch-surface. (b) A forward stretch-surface evolves into a backward strain-surface.
Trajectories of system (9) . The homoclinic orbits are shown in red.
(a) Forward stretchline through the origin for three integration times T = 0.5 ( ), T = 1 ( ) and T = 2 ( ). (b) Forward strainline for the same integration times, as in panel (a). (c) The asymptotic position of the strainline (−°−) and the stretchline (−°−) compared to the classic stable and unstable manifolds (black).
(a) Classical stable and unstable manifolds (black) are shown together withthe stretchline through the origin (magenta). Three blobs of tracers with radii (blue), (yellow), and (red) are centered at the origin. The tracers and the manifolds are then advected to time t = 0.1 (b), t = 0.2 (c), and t = 0.4 (d). Over the time interval [0,2], the stretchline is the de facto unstable manifold for spreading tracers. For larger advection times, this de facto unstable manifold practically coincides the classic unstable manifold of the origin.
Stretchline (blue) and the advected image of an attracting LCS (red) at t = 0. The exponential growth of errors in backward-time advection of the LCS results in a jagged curve that deviates from the true attracting LCS.
(a) The concentric tracers with radii 0.05 (blue), 0.1 (yellow), and 0.2 (red). The stretchline (black) passing through the center is computed from the time interval [0, 50] (i.e., a = 0 and b = 50). The tracers and the stretchline are then advected forward in time to t = 10 (b), t = 15 (c), t = 25 (d).
(a) Forward stretchlines at t = 0. The attracting LCSs (i.e., locally most-stretching stretchlines) are highlighted in red. The green closed curves show the boundaries of elliptic regions. Tracers (blue circles) are used to visualize the overall mixing patterns. (b) Advected image of the attracting LCSs, tracers, and elliptic barriers at time t = 50.
(a) A spherical tracer surface (blue) at time t = 0 and the corresponding approximate stretch-surface (red) passing through its origin. (b) The advected positions of these surfaces at the final time t = 4.
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