The geometry of mixing in time-periodic chaotic flows. I. Asymptotic directionality in physically realizable flows and global invariant properties

This paper demonstrates that the geometry and topology of material lines in 2D time-periodic chaotic flows is controlled by a global geometric propert y referred to as asymptotic directionality. This property implies the exist ence of local asymptotic orientations at each point within the chaotic regi on, determined by the unstable eigendirections of the Jacobian matrix of th e nih iterative of the Poincare map associated with the flow. Asymptotic di rectionality also determines the geometry of the invariant unstable manifol ds, which are everywhere tangent to the field of asymptotic eigendirections . This fact is used to derive simple non-perturbative methods for reconstru cting the global invariant manifolds to any desired level of detail. The ge ometric approach associated with the existence of a field of invariant unst able subspaces permits us to introduce the concept of a geometric global un stable manifold as an intrinsic property of a Poincare map of the flow, def ined as a class of equivalence of integral manifolds belonging to the invar iant unstable foliation, The connection between the geometric global unstab le manifold and the global unstable manifold of hyperbolic periodic points is also addressed. Since material lines evolved by a chaotic flow are asymp totically attracted to the geometric global unstable manifold of the Poinca re map, in a sense that will be made clear in the article, the reconstructi on of unstable integral manifolds can be used to obtain a quantitative char acterization of the topological and statistical properties of partially mix ed structures. Two physically realizable systems are analyzed: closed cavit y flow and flow between eccentric cylinders. Asymptotic directionality prov ides evidence of a global self-organizing structure characterizing chaotic how which is analogous to that of Anosov diffeomorphisms, which turns out t o be the basic prototype of mixing systems. In this framework, we show how partially mixed structures can be quantitatively characterized by a nonunif orm stationary measure (different from the ergodic measure) associated with the dynamical system generated by the field of asymptotic unstable eigenve ctors. (C) 1999 Elsevier Science B.V. All rights reserved.