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

Abstract

This paper demonstrates that the geometry and topology of material lines in 2D time-periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality. This property implies the existence of local asymptotic orientations at each point within the chaotic region, determined by the unstable eigendirections of the Jacobian matrix of the nth iterative of the Poincaré map associated with the flow. Asymptotic directionality also determines the geometry of the invariant unstable manifolds, which are everywhere tangent to the field of asymptotic eigendirections. This fact is used to derive simple non-perturbative methods for reconstructing the global invariant manifolds to any desired level of detail. The geometric approach associated with the existence of a field of invariant unstable subspaces permits us to introduce the concept of a geometric global unstable manifold as an intrinsic property of a Poincaré map of the flow, defined as a class of equivalence of integral manifolds belonging to the invariant unstable foliation. The connection between the geometric global unstable manifold and the global unstable manifold of hyperbolic periodic points is also addressed. Since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the Poincaré map, in a sense that will be made clear in the article, the reconstruction of unstable integral manifolds can be used to obtain a quantitative characterization of the topological and statistical properties of partially mixed structures. Two physically realizable systems are analyzed: closed cavity flow and flow between eccentric cylinders. Asymptotic directionality provides evidence of a global self-organizing structure characterizing chaotic flow which is analogous to that of Anosov diffeomorphisms, which turns out to be the basic prototype of mixing systems. In this framework, we show how partially mixed structures can be quantitatively characterized by a nonuniform stationary measure (different from the ergodic measure) associated with the dynamical system generated by the field of asymptotic unstable eigenvectors.