Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic

A "quasiclassical" approximation to the quantum spectrum of the Schrödinger equation is obtained from the trace of a quasiclassical evolution operator for the "hydrodynamical" version of the theory, in which the dynamical evolution takes place in the extended phase space [q(t), p(t), M (t)] = [qi, ∂iS, ∂i∂jS]. The quasiclassical evolution operator is multiplicative along the classical flow, the corresponding quasiclassical zeta function is entire for nice hyperbolic flows, and its eigenvalue spectrum contains the spectrum of the semiclassical zeta function. The advantage of the quasiclassical zeta function is that it has a larger analyticity domain than the original semiclassical zeta function; the disadvantage is that it contains eigenvalues extraneous to the quantum problem. Numerical investigations indicate that the presence of these extraneous eigenvalues renders the original Gutzwiller-Voros semiclassical zeta function preferable in practice to the quasiclassical zeta function presented here. The cumulant expansion of the exact quantum mechanical scattering kernel and the cycle expansion of the corresponding semiclassical zeta function part ways at a threshold given by the topological entropy; beyond this threshold quantum mechanics cannot resolve fine details of the classical chaotic dynamics.

We demonstrate the utility of the periodic-orbit description of chaotic motion by computing from a few periodic orbits highly accurate estimates of a large number of quantum resonances for the classically chaotic three-disk scattering problem. The symmetry decompositions of the eigenspectra are the same for the classical and the quantum problem, and good agreement between the periodic-orbit estimates and the exact quantum poles is observed.PACS numbers: 03.65.Sq, 05.45.+b It is a characteristic feature of dynamical systems of a few degrees of freedom that the motion is often organized around a few fundamental cycles. These short cycles capture the skeletal topology of the motion in the sense that any long orbit can approximately be pieced together from the fundamental cycles. Moreover, many quantities of interest can be computed as averaged over periodic orbits. In Ref. 1 a highly convergent expansion around short cycles has been introduced and applied to evaluation of classical chaotic averages. The goal of this Letter is to demonstrate that the curvature expansions 1 of periodic-orbit sums 2 " 4 are an equally powerful tool for evaluation of quantum resonances of classically chaotic systems.In this approach, the averages over chaotic dynamical systems are determined from the zeros of dynamical £ functions, 5 defined as expansions of infinite products of the form P f p with weight t p associated to every primitive (nonrepeating) periodic orbit (or cycle) p. The key observation is that the expanded product allows a regrouping of terms into dominant fundamental contributions and decreasing curvature corrections. Computations with f functions are rather straightforward; typically one determines lengths and stabilities of a finite number of shortest periodic orbits, substitutes them into (1), and estimates the zeros of (1) from such polynomial approximations. We shall apply here the expansion (1) to evaluation of repeller escape rates. The classical repeller escape rate y is determined 1,6 ' 7 by the largest zero of 1/fCy) (s real) with each prime cycle weighted by t p (s)-Af l e~s T '.Here T p is the period of the prime cycle p and JA P -"ln(Ap) is its stability exponent, where A p is the leading eigenvalue of the cycle Jacobian. The associated quantum amplitude is essentially the square root of the classical weight. This follows from the stationary phase formula 2 " 4 for determining the poles of the scattering matrix in terms of cycles, rewritten 8 ' 9 as the logarithmic derivative of the infinite product of f functions Z(k) m *Yir am o£j~l(k)> where the weights of the prime cycles for the different f/s are t^^exp[-fi p ({+j) + U/h)S p (k) + inv p /2] , (3)where S p (k) is the action and v p is the Maslov index (in the three-disk example considered below, k is the quantum wave number). The zeros of Z(k) in the complex k plane determine the eigenvalues or resonances of the quantum system; here we shall compute only those closest to the real energy axis, which are given by the zeros of l/£o(k).As it stands, the Eul...

The strange sets which arise in deterministic low-dimensional dynamical systems are analysed in terms of (unstable) cycles and their eigenvalues. The general formalism of cycle expansions is introduced and its convergence discussed.

We present ten new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new traveling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space onto suitably chosen 2-or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. arXiv:0808.3375v2 [physics.flu-dyn]