We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor.The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

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Our flow-visualization and spectral studies of flow between concentric independently rotating cylinders have revealed a surprisingly large variety of different flow states. (The system studied has radius ratio 0.883, aspect ratios ranging from 20 to 48, and the end boundaries were attached to the outer cylinder.) Different states were distinguished by their symmetry under rotation and reflection, by their azimuthal and axial wavenumbers, and by the rotation frequencies of the azimuthal travelling waves. Transitions between states were determined as functions of the inner- and outer-cylinder Reynolds numbers, Ri and Ro, respectively. The transitions were located by fixing Ro and slowly increasing Ri. Observed states include Taylor vortices, wavy vortices, modulated wavy vortices, vortices with wavy outflow boundaries, vortices with wavy inflow boundaries, vortices with flat boundaries and internal waves (twists), laminar spirals, interpenetrating spirals, waves on interpenetrating spirals, spiral turbulence, a flow with intermittent turbulent spots, turbulent Taylor vortices, a turbulent flow with no large-scale features, and various combinations of these flows. Some of these flow states have not been previously described, and even for those states that were previously described the present work provides the first coherent characterization of the states and the transitions between them. These flow states are all stable to small perturbations, and the transition boundaries between the states are reproducible. These observations can serve as a challenge and test for future analytic and numerical studies, and the map of the transitions provides several possible codimension-2 bifurcations that warrant further study.

Flocking birds, fish schools, and insect swarms are familiar examples of collective motion that plays a role in a range of problems, such as spreading of diseases. Models have provided a qualitative understanding of the collective motion, but progress has been hindered by the lack of detailed experimental data. Here we report simultaneous measurements of the positions, velocities, and orientations as a function of time for up to a thousand wild-type Bacillus subtilis bacteria in a colony. The bacteria spontaneously form closely packed dynamic clusters within which they move cooperatively. The number of bacteria in a cluster exhibits a power-law distribution truncated by an exponential tail. The probability of finding clusters with large numbers of bacteria grows markedly as the bacterial density increases. The number of bacteria per unit area exhibits fluctuations far larger than those for populations in thermal equilibrium. Such "giant number fluctuations" have been found in models and in experiments on inert systems but not observed previously in a biological system. Our results demonstrate that bacteria are an excellent system to study the general phenomenon of collective motion.swarming | self-organization | active suspensions D espite differences in the length scales and the cognitive abilities of constituent individuals, collective motion in systems as diverse as bird flocks, mammal herds, swarming bacteria, and vibrating granular particles (1-8) produces similar patterns of extended spatiotemporal coherence, suggesting general principles of collective motion. One approach to unveil these principles has been to model individuals as interacting self-propelled particles, which align their motions with neighbors (8-13). Some models also include repulsive and attractive interactions between particles in addition to the local alignment of velocities. With empirically chosen parameters such as the range over which the local alignment occurs, self-propelled particle models produce motions qualitatively similar to the observations. For example, within certain parameter regimes, models (8, 13) predict that collectively moving individuals form dynamic clusters, as often seen in fish schools or mammal herds (1); these clusters lead to large fluctuations in population density. Quantitatively, analytic theories based on liquid crystal physics (14-16) have predicted that these density fluctuations should scale with system size differently from fluctuations in thermal equilibrium systems.In contrast to numerical models and analytical theories, quantitative experiments have been limited (2-8), though decisive experiments are urgently needed to test the theoretical assumptions, determine sensitive modeling parameters, and verify theoretical predictions. The lack of experimental data is mainly due to technical difficulties. In conventional macroscopic systems, such as bird flocks, it is exceedingly challenging to track individual motions of a large population over long periods of time, and studies with systematic parameter ...

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