Symmetry reduction in high dimensions, illustrated in a turbulent pipe.

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

Periodic orbit analysis of a system with continuous symmetry--A tutorial.

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. The system's relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the "method of slices," which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the "slice" can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.

Reduction of SO(2) symmetry for spatially extended dynamical systems.

Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the "first Fourier mode slice," a very simple, easy to implement reduction of SO(2) symmetry. While the method exhibits rapid variations in phase velocity whenever the magnitude of the first Fourier mode is nearly vanishing, these near singularities can be regularized by a time-scaling transformation. We show that after application of the method, hitherto unseen global structures, for example, Kuramoto-Sivashinsky relative periodic orbits and unstable manifolds of traveling waves, are uncovered.

Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system.

The finest state-space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation, the stochastic neighborhoods of deterministic periodic orbits can be computed from distributions stationary under the action of a local Fokker-Planck operator and its adjoint. We derive explicit formulas for widths of these distributions in the case of chaotic dynamics, when the periodic orbits are hyperbolic. The resulting neighborhoods form a basis for functions on the attractor. The global stationary distribution, needed for calculation of long-time expectation values of observables, can be expressed in this basis.

Cartography of high-dimensional flows: a visual guide to sections and slices.

Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a "template." Together these charts provide an atlas of the symmetry-reduced "slice" of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits. Visualizations of these solutions and their unstable manifolds reveal their interrelations and the role they play in organizing turbulence/chaos.

How well can one resolve the state space of a chaotic map?

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching or contraction and the smearing due to noise. We propose to determine the "finest attainable" partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. Numerical tests of such "optimal partition" of a one-dimensional repeller support our hypothesis.

Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics.

We undertake an exploration of recurrent patterns in the antisymmetric subspace of the one-dimensional Kuramoto-Sivashinsky system. For a small but already rather "turbulent" system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. The Newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin-its backbone consists of several Smale horseshoe repellers, well approximated by intrinsic local one-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.

A perturbative analysis of modulated amplitude waves in Bose-Einstein condensates.

We apply Lindstedt's method and multiple scale perturbation theory to analyze spatio-temporal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasi-one-dimensional Bose-Einstein condensates with mean-field interactions. We determine the dependence of the amplitude of modulated amplitude waves on their wave number. We also explore the band structure of Bose-Einstein condensates in detail using Hamiltonian perturbation theory and supporting numerical simulations.

Modulated amplitude waves in Bose-Einstein condensates.

We analyze spatiotemporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt's method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits ("modulated amplitude waves") on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.

Variational method for finding periodic orbits in a general flow.

A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems. An initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop. The "Newton descent" partial differential equation that governs this evolution is an infinitesimal step version of the damped Newton-Raphson iteration. The feasibility of the method is demonstrated by its application to the Hénon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.

Manipulating epileptiform bursting in the rat hippocampus using chaos control and adaptive techniques.

Epilepsy is a relatively common disease, afflicting 1%-2% of the population, yet many epileptic patients are not sufficiently helped by current pharmacological therapies. Recent reports have suggested that chaos control techniques may be useful for electrically manipulating epileptiform bursting behavior in vitro and could possibly lead to an alternative method for preventing seizures. We implemented chaos control of spontaneous bursting in the rat hippocampal slice using robust control techniques: stable manifold placement (SMP) and an adaptive tracking (AT) algorithm designed to overcome nonstationarity. We examined the effect of several factors, including control radius size and synaptic plasticity, on control efficacy. AT improved control efficacy over basic SMP control, but relatively frequent stimulation was still necessary and very tight control was only achieved for brief stretches. A novel technique was developed for validating period-1 orbit detection in noisy systems by forcing the system directly onto the period-1 orbit. This forcing analysis suggested that period-1 orbits were indeed present but that control would be difficult because of high noise levels and nonstationarity. Noise might actually be lower in vivo, where regulatory inputs to the hippocampus are still intact. Thus, it may still be feasible to use chaos control algorithms for preventing epileptic seizures.

Identification of determinism in noisy neuronal systems.

Most neuronal ensembles are nonlinear excitable systems. Thus it is becoming common to apply principles derived from nonlinear dynamics to characterize neuronal systems. One important characterization is whether such systems contain deterministic behavior or are purely stochastic. Unfortunately, many methods used to make this distinction do not perform well when both determinism and high-amplitude noise are present which is often the case in physiological systems. Therefore, we propose two novel techniques for identifying determinism in experimental systems. The first, called short-time expansion analysis, examines the expansion rate of small groups of points in state space. The second, called state point forcing, derives from the technique of chaos control. The system state is forced onto a fixed point, and the subsequent behavior is analyzed. This technique can be used to verify the presence of fixed points (or unstable periodic orbits) and to assess stationarity. If these are present, it implies that the system contains determinism. We demonstrate the use and possible limitations of these two techniques in two systems: the Hénon map, a classic example of a chaotic system, and spontaneous epileptiform bursting in the rat hippocampal slice. Identifying the presence of determinism in a physiological system assists in the understanding of the system's dynamics and provides a mechanism for manipulating this behavior.

A Fredholm determinant for semiclassical quantization.

We investigate a new type of approximation to quantum determinants, the "quantum Fredholm determinant," and test numerically the conjecture that for Axiom A hyperbolic flows such determinants have a larger domain of analyticity and better convergence than the Gutzwiller-Voros zeta functions derived from the Gutzwiller trace formula. The conjecture is supported by numerical investigations of the 3-disk repeller, a normal-form model of a flow, and a model 2-D map.

Periodic orbit quantization of the anisotropic Kepler problem.

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.

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.