An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

Abstract

a b s t r a c tWe present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in threedimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R 3 . We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R 3 . The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable 'material surface' in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics -a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C 2 interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady threedimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.

On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

“…In particular, the identification of vortex surfaces in vortex reconnection is important but challenging (Kida & Takaoka 1994). Recent progress on the computation of invariant manifolds (see, e.g., Krauskopf et al 2005 andBranicki &Wiggins 2009) may shed light on the construction or identification of vortex surfaces. Another extension of the present study may address the relationship between the geometry of vortex surfaces and potential finite-time singularity in Euler flows (see Hou & Li 2008).…”

Accurate control of hyperbolic trajectories in any dimension

“…A common spherical configuration is the Hadamard-Rybczynski solution for Stokes flows [31,32,35,36,66], whose kinematic structure is similar to the classical Hill's spherical vortex [75][76][77][78][79][80][81][82][83][84][85] for Euler flows with an additive solid rotation. Particle trajectories of this flow satisfẏ…”

Insights into the three-dimensional Lagrangian geometry of the Antarctic polar vortex

“…In idealized 3-D time-dependent flows Poincaré sections have been used to recognize significant Lagrangian structures (Cartwright et al, 1996;Pouransari et al, 2010;Moharana et al, 2013;Rypina et al, 2015). Invariant manifolds acting as transport barriers in 3-D flows may have the structure of convoluted 2-D surfaces embedded in a volume (Branicki and Wiggins, 2009). In oceanic contexts these surfaces have been identified by stitching together 2-D Lagrangian structures in different layers (Branicki and Kirwan, 2010) or connecting ridges computed from finite-size Lyapunov exponent fields (Bettencourt et al, 2014).…”