Harry L. Swinney

DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES

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.

Flow regimes in a circular Couette system with independently rotating cylinders

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.

Dynamical instabilities and the transition to chaotic Taylor vortex flow

We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/R c = 10·1, where R c is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/R c ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/R c = 10·1 disappears at R/R c = 19·3, and the remaining sharp component disappears at R/R c = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landaus picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.

Pattern formation by interacting chemical fronts

Experiments on a bistable chemical reaction in a continuously fed thin gel layer reveal a new type of spatiotemporal pattern, one in which fronts propagate at a constant speed until they reach a critical separation (typically 0.4 millimeter) and stop. The resulting asymptotic state is a highly irregular stationary pattern that contrasts with the regular patterns such as hexagons, squares, and stripes that have been observed in many nonequilibrium systems. The observed patterns are initiated by a finite amplitude perturbation rather than through spontaneous symmetry breaking.

Collective motion and density fluctuations in bacterial colonies

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.

Observation of a strange attractor

Abstract Phase space portraits have been constructed and analyzed for noisy (nonperiodic) data obtained in an experiment on a nonequilibrium homogeneous chemical reaction. The phase space trajectories define a limit set that is an “attractor” - following a perturbation, the trajectory quickly returns to the attracting set. This attracting set is shown to be “strange” - nearby trajectories separate exponentially on the average. Moreover, the Poincare sections exhibit the stretching and folding that is characteristic of strange attractors.

Resonant pattern formation in achemical system

A periodic force applied to a nonlinear pendulum can cause the pendulum to become entrained at a frequency that is rationally related to the applied frequency, a phenomenon known as frequency-locking. A recent theoretical analysis showed that anarray of coupled nonlinear oscillators can exhibit spatial reorganization when subjected to external periodic forcing. We present here experimental evidence that reaction–diffusion processes, which govern pattern evolution and selection in many chemical and biological systems, can also exhibit frequency-locking phenomena. For example, periodic optical forcing of the light-sensitive Belousov–Zhabotinsky (BZ) reaction transforms a rotating spiral wave to a labyrinthine standing-wave pattern (Fig. 1). As the forcing frequency is varied, we observe a sequence of frequency-locked regimes, analogous to the frequency-locked ‘tongues’ of a driven nonlinear pendulum, except that in the reactor different frequencies correspond to different spatial patterns. Resonant interactions leading to standing-wave patternshave not been observed previously in chemical or biological media, but periodic forcing (such as circadian rhythm) is abundant in nature and may lead to similar pattern-forming phenomena.

Dynamic Fracture in Single Crystal Silicon

We have measured the velocity of a running crack in brittle single crystal silicon as a function of energy flow to the crack tip. The experiments are designed to permit direct comparison with molecular dynamics simulations; therefore the experiments provide an indirect but sensitive test of interatomic potentials. Performing molecular dynamics simulations of brittle crack motion at the atomic scale we find that experiments and simulations disagree showing that interatomic potentials are not yet well understood.

Long-wavelength surface-tension-driven Bénard convection: experiment and theory

Surface-tension-driven Benard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Benard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm 2 s −1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.

Observations of order and chaos in nonlinear systems

Experiments on nonlinear electrical oscillators, the Belousov-Zhabotinskii reaction, Rayleigh-Benard convection, and Couette-Taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom. Experimental results are presented illustrating the following transition sequences; period doubling and the U-sequence, intermittency, the periodic-quasiperiodic-chaotic sequence, frequency locking, and an alternating periodic-chaotic sequence.