Using Invariant Manifolds to Construct Symbolic Dynamics for Three-Dimensional Volume-Preserving Maps

Abstract

Topological techniques are powerful tools for characterizing the complexity of many dynamical systems, including the commonly studied area-preserving maps of the plane. However, the extension of many topological techniques to higher dimensions is filled with roadblocks preventing their application. This article shows how to extend the homotopic lobe dynamics (HLD) technique, previously developed for 2D maps, to volume-preserving maps of a three-dimensional phase space. Such maps are physically relevant to particle transport by incompressible fluid flows or by magnetic field lines. Specifically, this manuscript shows how to utilize two-dimensional stable and unstable invariant manifolds, intersecting in a heteroclinic tangle, to construct a symbolic representation of the topological dynamics of the map. This symbolic representation can be used to classify system trajectories and to compute topological entropy. We illustrate the salient ideas through a series of examples with increasing complexity. These examples highlight new features of the HLD technique in 3D. Ultimately, in the final example, our technique detects a difference between the 2D stretching rate of surfaces and the 1D stretching rate of curves, illustrating the truly 3D nature of our approach.

Computational method for phase space transport with applications to lobe dynamics and rate of escape

“…Furthermore, the methods presented in this article can also be used in conjunction with homotopic lobe dynamics technique that is an extension of transport in twodimensional maps to higher dimensional systems in Ref. [21]. The output file (<rslt>)contains one line with 4 numbers: the area of the lobes inside, the area of the lobes outside and the relative error on these two values.…”

Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve

“…The purpose of this example is to introduce the techniques used in Refs. [54,55]. Figure 4a shows a cross section of the trellis, while Fig.…”

“…Following Ref. [54] we identify pairs of pseudoneighbor intersection curves from the ETPs. Two heteroclinic curves α n and β n form a pair of pseudoneighbors if α n and β n , or some iterate α m and β m , are adjacent on both the forward and backward ETPs, more precisely, if a line can be drawn between the two curves on both the forward and backward ETPs without intersecting any other heteroclinic curve.…”

“…Based on the preceding studies of 2D maps, we have been motivated to study 3D volume-preserving maps by extending HLD to analyze the global structure of 2D stable and unstable manifolds [54,55]. The input to the 3D HLD technique is (finite-area) pieces of intersecting 2D stable and unstable manifolds attached to hyperbolic fixed points (or as we discuss in Sec.…”

“…The prior work on 3D HLD [54,55] was restricted to a particular class of 3D maps that have a socalled equatorial heteroclinic intersection curve; the union of the stable and unstable manifolds of two fixed points up to this intersection curve divides phase space into two distinct regions. This is entirely analogous to how a primary intersection point is used to define a resonance zone for 2D maps.…”