A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

We study coupled map lattices with a scaling form of connectivity and show that the dynamics of these systems exhibit a transition from spatial disorder to spatially uniform, temporal chaos as the scaling is varied. We numerically investigate the eigenvalue spectrum of the random matrix characterizing fluctuations from spatial uniformity, and find that the spectrum is real, bounded, and has a gap between the largest eigenvalue (corresponding to the uniform solution) and the remaining $N\ensuremath{-}1$ eigenvalues (nonuniform solutions). The width of this gap depends on the scaling exponent. We associate the transition to the coherent state with the appearance of this gap.

Spatially Coherent States in Fractally Coupled Map Lattices

The purpose of the paper is to address the synchronous chaos in coupled map lattices with general connectivity topology. Our main results contain the following. First, the master stability functions also hold for general connectivity topology with coupling through a nonlinear function that needs to be exactly the individual chaotic map. Second, the synchronization curve, composed of pieces of transverse Lyapunov exponent curves, is constructed. Third, necessary and sufficient conditions on coupling strength for yielding the synchronous chaos of the system are given. Moreover, the coupling strength $d_c$ giving the fastest convergence rate of the initial values toward the synchronous state is explicitly obtained. It is also proved that such $d_c$ is independent of the choice of the individual map. Finally, our results here can be applied to address questions of wavelength bifurcations and size instability.

https://ir.nctu.edu.tw:443/bitstream/11536/9802/1/000260847600003.pdf

Synchronous Chaos in Coupled Map Lattices with General Connectivity Topology

Pattern formation and evolution in the desynchronizing process of scale-free complex networks are investigated. Depending on how far the system is away from the synchronizable regime, two types of synchronous patterns are identified, namely, the giant-cluster state (GCS) and the scattered-cluster state (SCS). GCS is observed when a system is immediately outside of the synchronizable regime, where the dynamics undergoes a process of on-off intermittency and the patterns are signatured by the existence of a giant synchronous cluster. As the system leaves away from the synchronizable regime, GCS gradually transforms into SCS, accompanied by the continuous dissolving of the giant cluster. Both the two types of patterns are non-stationary, reflected as the timely changed size and content of the clusters. By introducing a new form of synchronization, the temporal phase synchronization, we investigate the dynamical and statistical properties of these non-stationary patterns. An interesting finding is that the unstable nodes of GCS, i.e. nodes that escape from the giant cluster more frequently, are independent of the coupling strength but are sensitive to the bifurcation types. The intermittent behavior of GCS is analyzed by a theory of snapshot attractors, and the theoretical predications fit the numerical observations qualitatively well.

Pattern evolution in non-synchronizable scale-free networks

Roughness effects on lubricant flows are investigated via 2D lattice Boltzmann simulations. At a Reynolds numbers of order 1000 a transition from laminar to unsteady flow is observed by an increase of the roughness height from about 10% to about 25% of the channel width. At lower Reynolds numbers (where the flow is laminar in both channels), the transition is observed when increasing the wall roughness further. In other words, the critical Reynolds number for the transition from laminar toward unsteady flow decreases at higher wall roughness. Wall roughness may, therefore, qualitatively change the flow properties in confined geometry. Due to the ubiquitous presence of the wall roughness, the phenomenon is relevant in all cases where relatively high Reynolds number flow occur in strongly confined channels such as lubricant flow during the deformation of solid surfaces. For a fixed Reynolds number and channel geometry, time and spatial dependence of the velocity field and fluctuating quantities obey the sca...

/pdf/chaotic-flows-in-microchannels-a-lattice-boltzmann-study-4muktaqt4e.pdf

Chaotic flows in microchannels :A lattice Boltzmann study

We show how increasing spatial interaction leads to the merging of coherent structures from chaos in some systems of coupled map lattices. This phenomenon reflects the arising of new ground states in the corresponding model of statistical mechanics. If we further increase the coupling then, new ground states appear showing the coexistence of a large-scale coherent structure with a small-scale chaotic motion. This allows us to propose a generalization of the notion of spatial intermittency.

The Emergence of Coherent Structures in Coupled Map Lattices

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The Emergence of Coherent Structures in Coupled Map Lattices