Predrag Cvitanović

Recycling of strange sets: I. Cycle expansions

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.

Visualizing the geometry of state space in plane Couette flow

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 Reynolds number 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 Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate- Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

Recycling of strange sets: II. Applications

For pt.I see ibid., vol.3, no.2, p.325-59 (1990). Cycle expansions are applied to a series of low-dimensional dynamically generated strange sets: the skew Ulam map, the period-doubling repeller, the Henon-type strange sets and the irrational winding set for circle maps. These illustrate various aspects of the cycle expansion technique; convergence of the curvature expansions, approximations of generic strange sets by self-similar Cantor sets, effects of admixture of non-hyperbolicity, and infinite resummations required in presence of orbits of marginal stability. A new exact and highly convergent series for the Feigenbaum delta is obtained.

Periodic orbits as the skeleton classical and quantum chaos

A description of a low-dimensional deterministic chaotic system in terms of unstable periodic orbits (cycles) is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos, comparable to the familiar perturbation expansions for nearly integrable systems. The infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a Selberg product form, dominated by the short cycles, and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits. Methods for implementing this computation for finite subshift dynamics are introduced.

Spatiotemporal chaos in terms of unstable recurrent patterns

Spatiotemporally chaotic dynamics of a Kuramoto - Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as an accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

Equilibrium and travelling-wave solutions of plane Couette flow

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-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 three-dimensional 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 on to suitably chosen two-or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

Periodic orbit expansions for classical smooth flows

The authors derive a generalized Selberg-type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow. This relation is a classical analogue of the quantum trace formulae and Selberg-type zeta functions.

Scaling Laws for Mode Lockings in Circle Maps

The self-similar structure of mode lockings for circle maps is studied by means of the associated Farey trees. We investigate numerically several classes of scaling relations implicit in the Farey organization of mode lockings and discuss the extent to which they lead to universal scaling laws.

Gauge invariance structure of quantum chromodynamics

Abstract Perturbative QCD may be subdivided into separately gauge-invariant sectors according to the projection of non-abelian color weights onto linearly independent basis elements. We exploit the general Lie group structure of the theory to give an algorithm for finding these gauge-invariant sets and present several examples of its use. The planar sector and the systematics of the non-planar corrections are defined for any gauge theory. Our gauge set classification has implications for QCD bound states, finite order perturbative QCD calculations, the study of QCD infrared singularities and for the question of convergence of the perturbation series.

Investigation of the Lorentz gas in terms of periodic orbits

The diffusion constant and the Lyapunov exponent for the spatially periodic Lorentz gas are evaluated numerically in terms of periodic orbits. A symbolic description of the dynamics reduced to a fundamental domain is used to generate the shortest periodic orbits. Applied to a dilute Lorentz gas with finite horizon, the theory works well, but for the dense Lorentz gas the convergence is hampered by the strong pruning of the admissible orbits.