This book is an introduction to the patterns and dynamics of sustained nonequilibrium systems at a level appropriate for graduate students in biology, chemistry, engineering, mathematics, physics, and other fields. Our intent is for the book to serve as a second course that continues from a first introductory course in nonlinear dynamics. While a first exposure to nonlinear dynamics traditionally emphasizes how systems evolve in time, this book addresses new questions about the spatiotemporal structure of nonequilibrium systems. Students and researchers who succeed in understanding most of the material presented here will have a good understanding of many recent achievements and will be prepared to carry out original research on related topics.

We can suggest three reasons why nonequilibrium systems are worthy of study. First, observation tells us that most of the Universe consists of nonequilibrium systems and that these systems possess an extraordinarily rich and visually fascinating variety of spatiotemporal structure. So one answer is sheer basic curiosity: where does this rich structure come from and can we understand it? Experiments and simulations further tell us that many of these systems – whether they be fluids, granular media, reacting chemicals, lasers, plasmas, or biological tissues – often have similar dynamical properties. This then is the central scientific puzzle and challenge: to identify and to explain the similarities of different nonequilibrium systems, to discover unifying themes, and, if possible, to develop a quantitative understanding of experiments and simulations.

A second reason for studying nonequilibrium phenomena is their importance to technology.

In this chapter, we will use the amplitude equations introduced in the previous two chapters to gain an understanding of defects and fronts in stripe patterns near onset. Our discussion also serves to illustrate more advanced applications of the amplitude equations, in particular the ability to treat patterns that vary spatially.

The importance of defects in patterns was introduced in Section 4.4.Adefect can be thought of as a local imperfection in an otherwise perfect pattern, and experimental and natural patterns often contain many defects, either due to the effect of boundaries or due to a spatially inhomogeneous initial condition. To approach the complicated realistic case of many defects, it is useful first to study single isolated defects and then attempt to build an understanding of natural patterns from these elementary ingredients. In Section 4.4.2, we introduced the notion of a topological defect, namely one that can be identified from properties of the pattern far from the location of the defect (such as the winding number of the phase, Eq. (4.58)), where the pattern approaches the ideal state. These defects are particularly important because they can persist for long times and so are robust. We will focus on two types of topological defects in stripe states, dislocations, and grain boundaries.

We first discuss the structure of stationary defects. Since defects represent deviations from the ideal pattern, the intensity of the pattern is reduced in their vicinity. The region over which the intensity is reduced, and where the pattern strongly deviates from the ideal one, is called the core of the defect.

In the previous chapter, we introduced and used a one-dimensional amplitude equation to study slow spatiotemporal modulations of a stripe pattern but with the restriction that the spatial variation could only be longitudinal (in the direction normal to the stripes). In the present chapter, we extend the previous chapter in two ways to study patterns that depend on two extended coordinates. The first generalization is the obvious one, which is to write down a two-dimensional amplitude equation that can treat a stripe pattern with modulations that vary along, as well as normal to, the stripes. The second generalization is to study superpositions of stripe states with different orientations, which will allow us to study quantitatively the periodic lattice states that we discussed in Section 4.3. It turns out that many useful insights about the stability of, and competition between, lattice states can be obtained by using just the zero-dimensional (no spatial dependence) amplitude equation similar to Eq. (4.6) to describe each stripe participating in the superposition. At the end of the chapter, we will discuss briefly the more general but also more difficult case of using a two-dimensional amplitude equation to describe general slow distortions of each stripe associated with lattice states and other stripe superpositions.

The two-dimensional amplitude equation that describes modulations along as well as normal to stripes turns out to have two different forms depending on whether the system is rotationally invariant (for example, Rayleigh–Bénard convection) or anisotropic (for example, a liquid crystal or a conducting fluid in the presence of a magnetic field). We first discuss the two-dimensional amplitude equation for rotationally invariant systems.

In Section 2.2 of the previous chapter we used a simplified mathematical model Eq. (2.3) to identify the assumptions, mathematical issues, and insights associated with the linear stability analysis of a stationary uniform nonequilibrium state. In this chapter, we would like to discuss a realistic set of evolution equations and the linear stability analysis of their uniform states, using the example of chemicals that react and diffuse in solutions (see Fig. 1.18). Historically, such a linear stability analysis of a uniform state was first carried out in 1952 by Alan Turing [106]. He suggested the radical and highly stimulating idea that reaction and diffusion of chemicals in an initially uniform state could explain morphogenesis, how biological patterns arise during growth. Although reaction–diffusion systems are perhaps the easiest to study mathematically of the many experimental systems considered in this book, they have the drawback that quantitative comparisons with experiment remain difficult. The reason is that many chemical reactions involve short-lived intermediates in small concentrations that go undetected, so that the corresponding evolution equations are incomplete. Still, reaction–diffusion systems are such a broad and important class of nonequilibrium systems, prevalent in biology, chemistry, ecology, and engineering, that a detailed discussion is worthwhile.

The chapter is divided into two halves. In the first part, we introduce the simple model put forward by Turing, and give a careful analysis of the instability of the uniform states. In the second part, we apply these ideas to realistic models of experimental systems.

The waves that are most familiar in daily life are sound waves, light waves, electrical waves, water waves, and mechanical waves (say a standing wave on a piano string). These familiar waves have the property that their magnitude decreases as they propagate away from their source or, for standing waves, once their source is turned off. The decrease in magnitude is a result of dissipative effects in the medium such as fluid viscosity, electrical resistance, or friction that drain energy from the wave and that restore the medium to thermal equilibrium.

These familiar waves have the additional property of often being accurately described by a linear evolution equation such as the wave equation. Because the evolution equation is linear, one can superimpose sinusoidal waves to get localized pulses of arbitrary shape, and these pulses can also propagate. (For example, clapping your hands once loudly creates a localized sound pulse that propagates away.) Because each Fourier component in the superposition is itself damped in typical media, the propagating pulses also damp out and disappear over time. Even in the absence of damping, dispersive effects can cause the different Fourier components to travel at different speeds so, again, waves and pulses change their shape and decrease in magnitude during propagation.

We have seen in earlier chapters that sustained nonequilibrium systems allow many dynamical states that can propagate or exist in a local spatial region but are such that these states do not damp out over time or they preserve their shape and speed as they propagate away from a source.

The linear stability analysis of Chapter 2 predicts that a small perturbation about a uniform state will grow exponentially in magnitude when the uniform state becomes unstable. Over time, the magnitude of a perturbation will grow so large that the nonlinear terms that were neglected when deriving the linearized evolution equation can no longer be ignored. These nonlinear terms play a fundamental role in the resulting pattern formation: they saturate the exponential growth, and they select among different spatial states. It is the essential role of nonlinearity in a spatially extended system that makes the study of pattern formation novel and hard.

We can gain a great deal of insight about the nonlinear regime of pattern formation by considering spatially periodic patterns. This is natural when considering the fate of a single exponentially growing Fourier mode of the linearized evolution equations associated with a linear stability analysis. Nonlinearities in the evolution equations for the system generate spatial harmonics (Fourier modes with wave vectors nq with n an integer) of this growing mode so that the finite-amplitude solution maintains the periodicity over the length 2π/q. A key role of the nonlinearity is to quench the exponential growth of the solution, leading to steady spatially periodic solutions for a stationary instability, and nonlinear oscillations or waves for an oscillatory instability. If this steady or periodic solution is to be physically relevant, we must also require that it be stable with respect to small perturbations. Thus we will study the existence and stability of steady or oscillatory spatially periodic (for qc ≠ 0) solutions.

This appendix provides some of the background for Chapter 4, especially for Eq. (4.14), by reviewing some of the elementary bifurcation theory that is often discussed in an introductory undergraduate course on nonlinear dynamics. Bifurcation theory is concerned with the change in the nature of solutions as parameters are varied. The changes can involve changes in the numbers or types of attractors, in the structure of the basins of attraction, or in even more subtle details of the phase space that are not easily detected by experiment. Sufficiently close to the onset of a bifurcation of a fixed point, a combination of a perturbation expansion and of nonlinear changes of variables can reduce the evolution equations to a much simpler dynamical system (usually a few odes) called a normal form. The normal form captures the essential behavior of the evolution equations sufficiently close to the bifurcation point and can be used to classify the possible bifurcations. For our purposes, the classification and associated language (e.g. pitchfork, Hopf, and other kinds) are the more important topics so we do not show how to reduce a set of equations describing a physical system to normal form, which can involve lengthy calculations, even with a computer mathematics program.

We begin our discussion by analyzing the bifurcations of some simple one variable dynamical systems and then discuss how these systems are related to the normal forms of more complicated evolution equations.

This appendix describes and gives some examples of multiple scales perturbation theory. This is a widely used technique in applied mathematics, physics, engineering, and other fields that systematically yields approximate solutions to ordinary and partial differential equations for which there is a small parameter e such that the mathematical problem can be solved without too much effort when the small parameter is set to zero. In the context of pattern formation, the formalism provides a systematic way to analyze the spatiotemporal behavior of fields near the supercritical instability of a spatially uniform state. Near such a bifurcation, for reasons clarified by the multiple scales theory, the physical system can be accurately analyzed as a slowly varying spatiotemporal modulation of a fast oscillatory behavior in space or in time.

The perturbation theory is based on two key features. First is the idea of multiple scales, which is to introduce scaled space and time coordinates that capture the slow modulation of the pattern. These new scaled variables will be treated as mathematically independent of the original variables that are used to describe the pattern state itself. The second key feature is the use of what are known as solvability conditions. In the formalism, these conditions arise as mathematical statements that prevent a resonant driving of a higher-order term by a lower-order term that would cause the perturbation method to fail after a short time. The lowest-order nontrivial solvability condition often ends up being an evolution equation for a slowly varying multiplicative factor of the unperturbed solution, what we have called an amplitude equation.