Pattern Formation in Continuous and Coupled Systems

This paper reviews past results from and future prospects for experimental studies of Rayleigh-Bénard convection with rotation about a vertical axis. At dimensionless rotation rates 0 ≤ Ω ≤ 20 and for Prandtl numbers σ ≃ 1, Küppers-Lortz-unstable patterns offered a unique opportunity to study spatio-temporal chaos immediately above a supercritical bifurcation where weakly-nonlinear theories in the form of Ginzburg-Landau (GL) or Swift-Hohenberg (SH) equations can be expected to be valid. However, the dependence of the time and length scales of the chaotic state on ε ≡ ΔT/ΔT _C - 1 was found to be different from the expected dependence based on the structure of GL equations. For Ω ≳ 70 and 0.7 ≲ σ ≲ 5 patterns were found to be cellular near onset with local four-fold coordination. They differ from the theoretically expected Küppers-Lortz-unstable state. Stable as well as intermittent defect-free rotating square lattices exist in this parameter range.

Smaller Prandtl numbers ( 0.16 ≲ σ ≲ 0.7) can only be reached in mixtures of gases. These fluids are expected to offer rich future opportunities for the study of a line of tricritical bifurcations, of supercritical Hopf bifurcations to standing waves, of a line of codimension-two points, and of a codimension-three point.

We examine some properties of attractors for symmetric dynamical systems that show what we refer to as ‘chaotic intermittency’. These are attractors that contain points with several different symmetry types, with the consequence that attracted trajectories come arbitrarily close to possessing a variety of different symmetries. Such attractors include heteroclinic attractors, on-off and in-out intermittency and cycling chaos. We indicate how they can be created at bifurcation, some open problems and further reading.

A new heteroclinic cycle is demonstrated in the case of thermal convection in a layer heated from below and rotating about a horizontal axis. This system can be realized experimentally through the use of the centrifugal force as effective gravity in the system of the rotating cylindrical annulus.

Oscillation patterns predicted by the Hopf bifurcation with the symmetries O(2) × O(2), D _m × O(2) and D _m × D _n are reviewed and discussed in the context of spatially continuous and discrete systems.

We describe some basic concepts and techniques from symmetric bifurcation theory in the context of coupled systems of cells (‘oscillator networks’). These include criteria for the existence of symmetry-breaking branches of steady and periodic states. We emphasize the role of symmetry as a general framework for such analyses. As well as overt symmetries of the network we discuss internal symmetries of the cells, ‘hidden’ symmetries related to Neumann boundary conditions, and spatio-temporal symmetries of periodic states. The methods are applied to a model central pattern generator for legged animal locomotion.

A flat, premixed flame stabilized on a circular porous plug burner at low pressure appears as a luminous disk. As the control parameters are adjusted, methaneair and methane-oxygen mixtures produce pulsating flames that exhibit both traveling-wave and standing-wave dynamic states, and isobutane-air and propane-air flames form ordered patterns of concentric luminous rings of cells that bifurcate into dynamics states in which the cells move in collective motion. In this paper the measurement and presentation of stability boundaries are discussed for representative examples of pulsating and cellular flames.

A characterization of textured patterns, referred to as the disorder function δ(β), is used to study the dynamics of patterns generated in the Swift-Hohenberg equation (SHE). The evolution of random initial states under the SHE exhibits two stages of relaxation. The initial phase, where local striped domains emerge from a noisy background, is quantified by a power law decay δ(β) ~t ^-1/2 Beyond a sharp transition a slower power law decay of δ(β), which corresponds to the coarsening of striped domains, is observed. The transition between the phases advances as the system is driven further from the onset of patterns, and suitable scaling of time and δ(β) leads to the collapse of distinct curves.

In this paper we want to present some ideas concerning the behavior of equivariant systems under small perturbations of their symmetry. We will touch these questions, discuss some applications and provide references to the literature. These references include details of the mathematical issues as well as further potential applications.

Some spatiotemporal patterns that have been observed in metal electrodissolution and in electrocatalytic reactions are discussed. The coupling through the electric field plays a major role in the formation of the patterns. This coupling is long range but not global since its strength depends on position. The coupling can be either positive or negative depending on geometry and this type of coupling produces some phenomena not seen in other types of chemically reacting systems.

Different mechanisms believed to be responsible for the generation of bursts in hydrodynamical systems are reviewed and a new mechanism capable of generating regular or irregular bursts of large dynamic range near threshold is described. The new mechanism is present in the interaction between oscillatory modes of odd and even parity in systems of large but finite aspect ratio, and provides an explanation for the bursting behavior observed in binary fluid convection by Sullivan and Ahlers.

When an oscillatory nonlinear system is driven by a periodic external stimulus, the system can lock at rational multiples p : q of the driving frequency. The frequency range of this resonant locking at a given p : q depends on the amplitude of the stimulus; the frequency width of locking increases from zero as the stimulus amplitude increases from zero, generating an “Arnol’d tongue” in a graph of stimulus amplitude vs stimulus frequency. Physical systems that exhibit frequency locking include electronic circuits [1, 2], Josephson junctions [3], chemical reactions [4], fields of fireflies [5, 6], and forced cardiac systems [7, 8]. Most studies of frequency locking have concerned either maps or systems of a few coupled ODEs. The Arnol’d tongue structure of the sine circle map has been extensively studied, and the theory of periodically driven ODE systems has been well developed [9], but there has been very little analysis of frequency locking phenomena in PDEs, except for a few studies of the parametrically excited Mathieu equation with diffusion and damping [10, 11, 12] and the parametrically excited complex Ginzburg-Landau equation [13, 14]. Our interest here is in the effect of periodic forcing on pattern forming systems such as convecting fluids, liquid crystals, granular media, and reaction-diffusion systems. Such systems are often subject to periodic forcing (e.g., circadian forcing of biological systems), but the effect of forcing on the bifurcations to patterns has not been examined in experiments or analyzed in PDE models of these systems.

This article is a review of double-layer convection in which the pattern formation arises due to a competition between the bulk motions in each fluid (see Figure 1). An instability takes place when the temperature difference between the upper and lower walls reaches a threshold value, and the response of the two-layer system depends on the properties of the constituent fluids. Our motivation is the search for patterns formed in non-equilibrium fluid dynamical systems which exhibit time-dependence at or near the onset of a pattern. Such a time-dependent state is predicted for the two-layer Rayleigh-Benard system and is accessible experimentally as well as theoretically. We expect to see oscillatory and spatio-temporal chaotic behavior. The presence of the interface and the coupling between the fluids in this model problem may provide an understanding of generic behaviors in related applications, such as the modeling of the earth’s mantle as a two-layer convecting system [3, 11], and liquid encapsulated crystal growth [19].

Surface reactions exhibit unique features as model systems for nonlinear effects in chemical reactions. In addition they have an immense importance in heterogeneous catalysis in the chemical industry. Dynamic processes on surfaces, like the Pt — catalyzed CO-oxidation, can be described by a set of reaction-diffusion equations. For a certain range of reactants partial pressures and temperature of the sample, pattern formation like spiral waves, target patterns or solitary waves can be observed. When global coupling via the gas phase is introduced strong temporal oscillations may occur, sometimes exhibiting spatio-temporal patterns like standing waves, period doubling and chaotic behavior. The patterns mentioned were found under isothermal conditions. Of course, when increasing the reaction pressure, due to the exothermic nature of the CO-oxidation, temperature variations can be explored, observable with a sensitive InfraRed (IR) camera.

Recent results on the dynamical behavior of patterns in two and three spatial dimensions are reviewed. Based upon spatio-temporal symmetries of patterns, it is shown that transitions to other patterns can be explained by analyzing low-dimensional model equations. Examples include the dynamics of periodically forced twisted scroll waves and transitions from rigidly-rotating spiral waves to meandering or drifting spirals.

We review recent results on pattern selection in the one- or two-dimensional reaction-diffusion system xt - Δx = f(x, y, λ), y _t = εg(x, y), subject to global (〈x〉 = x ₀) or long-range interaction; the source functions may be realistic kinetic functions or simple cubic or quintic f(x) functions for which the system admits inversion symmetry. This review discusses: (i) physical sources of such interactions and experimental observations in catalytic and electrochemical systems; (ii) the main emerging patterns and their classification according to their symmetry; (iii) the bifurcation between patterns; (iv) patterns when f(x) = 0 is tristable and can sustain several fronts.

The rich class of patterns simulated in a ribbon can be classified as stationary-front solutions (including oscillating fronts and antiphase oscillations) and moving pulse solutions (unidirectional, back-and-forth and source-points). Patterns on a disk may be classified as circular (including oscillatory or moving target patterns), rotating (stationary or moving spiral wave) and other patterns.

We present an experimental as well as theoretical study of kink motion in one-dimensional arrays of Josephson junctions connected in parallel by superconducting wires. The boundaries are closed to form a ring, and the waveform and stability of an isolated circulating kink is discussed. Two one-dimensional rings can be coupled which provides an interesting and clean platform to study interactions between kinks. These studies form foundations for investigating the more difficult two-dimensional arrays in which vortices move along rows but with some inter-row coupling. We introduce recent progress in the analysis of vortex dynamics in 2D arrays.

Among the best studied many oscillator systems in recent years are Josephson junction arrays. This paper summarizes some theoretical progress likely to be especially useful in the quest to design practical electromagnetic sources at very high frequencies. Outstanding puzzles and future theoretical directions are also described. The central problem is one which is pervasive in the general study of nonlinear oscillators: under what conditions will the various elements oscillate with the same frequency and phase?