The geometry of mixing in 2-d time-periodic chaotic flows

This paper demonstrates that the geometry and topology of material lines in time-periodic chaotic flows is controlled by a global geometric property r eferred to as asymptotic directionality,. This property implies the existen ce of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the Jacobian matrix of the n -period Poincare map associated with the flow. Asymptotic directionality al so determines the topology of the invariant unstable manifolds of the Poinc are map, which are everywhere tangent to the held of asymptotic eigendirect ions. This fact is used to derive simple non-perturbative methods for recon structing the invariant unstable manifolds associated with a Poincare secti on to any desired level of detail. Since material lines evolved by a chaoti c flow are asymptotically attracted to the geometric global unstable manifo ld of the flow (this concept is introduced in this article), such reconstru ctions can be used to characterize the topological and statistical properti es of partially mixed structures quantitatively. Asymptotic directionality provides evidence of a global self-organizing structure characterizing phys ically realizable chaotic mixing systems which is analogous to that of Anos ov diffeomorphisms, which turns out to represent the basic prototype of a m ixing system. In this framework we show how partially mixed structures can be quantitatively characterized by a non-uniform stationary measure (differ ent from the ergodic measure) associated with the dynamical system generate d by the field of asymptotic unstable eigenvectors. (C) 1999 Elsevier Scien ce Ltd. All rights reserved.