Flows of fluids of low viscosity may become unstable when large gradients of kinetic and/or potential energy are present. The flow field set up by the instability generally tends to smooth out the velocity and temperature differences causing it. The available kinetic or potential energy released by the instability may be so large that transition to a fully developed turbulent flow occurs.

Transition is influenced by many parameters. An important one is the level of preexisting disturbances in the fluid; a high level would generally cause early transition. Another cause for early transition in the case of wall-bounded shear flows is surface roughness. The manner in which transition occurs may also be very sensitive to the detailed flow properties.

For shear flows the basic nondimensional flow parameter measuring the tendency toward instability and transition is the Reynolds number; for high Re values, kinetic energy differences can be released faster into fluctuating motion than viscous diffusion will have time to smooth them out. For a heated fluid subject to gravity the Rayleigh number is the main stability parameter.

Of crucial importance for the tendency of a flow to become unstable and go through transition is the detailed distribution of mean velocity and/or temperature in the field. The analysis that follows is intended to illustrate this.

Although the flow processes involved in instability and transition might at a first glance appear to have only a slight resemblance to those observed in fully developed turbulence, they are nevertheless related to it in important ways. In a gross sense turbulence may be regarded as a manifestation of flow instability occurring randomly in space and time. The linear instability problem is the simplest flow model incorporating the interaction between unsteady fluctuations and a background shear or density distribution. With the aid of nonlinear instability theory one may also possibly be able to clarify some of the mechanisms whereby turbulence is maintained.

Instability to small disturbances

Because of the mathematical difficulties in the analysis of flow instability, only idealized cases for which the basic fluid flow properties vary with one spatial coordinate can be analyzed in a reasonably simple manner.

In this chapter we discuss in detail two examples of dynamical systems. The first example shows the use of one-dimensional maps for the interpretation of chaotic and periodic regimes observed in experiments with a periodically forced chemical oscillator. The second example illustrates the use of numerical techniques for the investigation of chaotic phenomena in a model of coupled reaction diffusion cells.

Periodically forced continuous stirred tank reactor

There are several reasons for the interest in periodically forced chemical reactors. First, advances in electronics and computer technology make the controlled operation of chemical systems under periodically varying conditions feasible. Periodically operated processes might possess advantages over steady state processes as discussed by Bailey. Second, forced chemical systems serve as simple experimental analogues of various dynamical systems arising at different levels of organization of living systems, for example synthesis of proteins, cell cycles, tissue growth, excitatory tissues, physiological control systems, etc. Third, chemical systems, particularly oscillatory reactions, serve as prototype experimental systems for studies of transition to chaos in dissipative systems.

A particularly interesting and important case arises when undamped oscillations of the limit cycle type occur in the flow-through reaction cell.

This type of dynamical systems is called an oscillator. Another important case of chemical dynamical systems is an excitable system or excitator. Excitators are characterized by a non oscillatory behaviour in the absence of external forcing, but they can be excited and become oscillatory under superthreshold periodic perturbations.

Studies of nonlinear phenomena which occur in mathematical models and which are observed in experiments profit both from a general knowledge of the theory of dynamical systems and bifurcations, and from the experience accumulated in an interpretation of specific examples. The most interesting and important nonlinear phenomenon that has come to prominence recently is the chaotic behaviour of deterministic dissipative systems. The investigation of chaotic dynamics has undergone an explosive development over the past ten years but the results are still mostly scattered throughout the journal literature.

The number of interested students and research workers from diverse fields, ranging from mathematics and physics to engineering sciences and biology, increases continuously and many of them will find it useful to have an introductory text, that surveys both theoretical and experimental aspects of chaotic behaviour. We have attempted to provide this in the present book.

The introductory chapter discusses the significance of chaos as a model of many seemingly random processes in nature and a definition of the class of dissipative systems that we will study.

The second chapter considers basic notions of the theory of dynamical systems. The difference between linear and nonlinear systems is illustrated and asymptotic behaviour is discussed in more detail. Definitions of chaos and of strange attractors and a description of chaotic behaviour in the frame of ergodic theory are then surveyed.

The third chapter deals with qualitative changes of asymptotic behaviour as a chosen parameter is varied. These changes (‘bifurcations’) may lead to chaos in several well-defined routes. The role of bifurcation theory in understanding the onset of chaos is illustrated by a number of characteristic examples.

Time evolution and dynamical systems

Physical, chemical, biological or social phenomena can be seen as systems characterized by a time evolution of their properties. Such evolution systems are ubiquitous in nature. Often we are able to express the rate of change of the properties of a considered evolution system in the form of equations, applying and combining the relevant laws of nature. Solutions of the constructed mathematical model then mimic the time evolution of the real system.

Our aim is to predict this evolution using a proper mathematical model. An instantaneous state of the model system can be given by a finite set of numbers or by a finite set of functions. A set of all states of the system will be called a state space (in physics literature it is also called a phase space). A system will be considered as deterministic if its future and past are fully determined by its current state. In a semideterministic system only the future is uniquely determined, while in a stochastic system neither the past nor the future is unique (this type of system will not be treated here).

A system of bodies moving according to laws of classical mechanics, electronic circuits or interacting populations in a closed ecological system may be considered as deterministic systems. An isothermal chemical reaction in an ideally stirred (homogeneous) environment is another example of a deterministic system, while the consideration of molecular diffusion makes the system semideterministic.

A substantial difference between the last two examples of evolution systems is in the dimension of the corresponding state space. Let X denote the state space and x its elements (states of the system).

This is the simplest self-consistent version of a continuation program developed and used in our research group. The program is able to find a starting point in a state space and to continue this point as one or two parameters are varied. The description of the method of continuation used in the program as well as references are contained in Chapter 4. Hints for running the program are found in comment lines to the main program. The subroutine PRSTR defining the equations and the Jacobi matrix must be supplied by the user. As a test model, two coupled Brusselators discussed in detail in Chapter 6 are defined in the sample subroutine PRSTR. Also included are input data for obtaining solution diagrams in Figs 6.14 and 6.17

CONT package is available through Internet upon request. For information send and e-mail message to marek@ vscht.cs or schrig@ vscht.cs.

Introduction

Several strategies have been followed in theoretical investigations of spatio-temporal coherence and chaos. Spatially distributed nonlinear dynamical systems have been studied as cellular automata coupled map lattices and as partial differential equations (PDEs) which, after a proper reduction or discretization, are studied by various numerical or analogue methods.

Spatially extended dynamical systems may be classified according to continuity or discreteness in space, time and local state. Thus PDEs are continuous in space, time and local state, iterated functional equations are discrete in time and continuous in space and local state, chains of oscillators (discussed in Chapter 5) are discrete in space and continuous in time and local state, lattice dynamical systems are discrete in space and time but continuous in local state and, finally, cellular automata are discrete in all three aspects.

Computer time requirements for the study of evolution of spatio-temporal patterns increase from cellular automata to PDEs drastically. However, seemingly different models of different physical systems have revealed the same types of phenomena (stationary space-time patterns, similar routes to chaos, space-time intermittency). Relations between the results reached through different approaches are gradually being developed. It is expected that in some cases mappings from classes of coupled-map lattices and cellular automata to classes of PDEs will be established in the future.

Various examples of experimental observations of chaotic behaviour in distributed electronic, solid state, chemical and hydrodynamic systems have been discussed in Chapter 5.

Introduction

Chaos has been observed in a large number of both natural and artificial nonlinear dissipative dynamical systems. For example, it has been shown in observations and computations of the evolution of meteorological patterns in the Earth's atmosphere that the patterns of evolution are very sensitive to initial conditions and predictions necessarily diverge after several days. These observations can be interpreted as an indication of the presence of a chaotic attractor in the corresponding state space. Also in the studies of populations dynamics it is often observed that the sizes of populations fluctuate widely from one season to the next; this can again be interpreted on the basis of the existence of chaotic trajectories in the population state space.

Recent developments in the theory of nonlinear dynamical systems coupled with the use of computers in the acquisition and analysis of long time series of experimental data have supported an exponential increase of detailed studies of various nonlinear dissipative systems displaying aperiodic behaviour and lead to both the confirmation of the presence and a detailed characterization of chaotic attractors.

Several model systems have been studied mostly with the aim of verifying theoretical predictions. Among them are classical hydrodynamical systems for the study of the development of turbulence – the Taylor–Couette flow and the Rayleigh–Bénard convection, nonlinear electronic oscillatory circuits, oscillatory mechanical systems, lasers, chemical systems (well stirred reactors) and various oscillating or excitatory structures in biology (heart cells, neurones).

Experimental observations on nonlinear circuits give results which are closest to theoretical predictions.

The equations of motion in classical physics differ considerably depending upon the subject they describe: a particle, an electromagnetic field, or a fluid. However our natural yearning for unification in the description of different phenomena has long since led to the development of universal formalisms. Among these the Lagrangian and Hamiltonian formalisms are the most advanced. This can be explained by the nature of the phenomena discussed. The popularity of each method varied at different stages in the development of physics. Throughout the whole period of advancement of relativistically invariant theories, preference was chiefly given to Lagrangian formalism (this was most conspicuous in field theory and the theory of a continuous medium). To a large extent, it was not before the generalization of the concepts of Hamiltonian formalism and introduction of Poisson's brackets that the Hamiltonian method of analysis was able to compete with the Lagrangian one.

The formation of new ideas and possibilities triggered recently by the discovery of the phenomenon of dynamic stochasticity (or simply, chaos) have brought the methods of Hamiltonian dynamics to the fore. Liouville's theorems on the conservation of phase volume and on the integrability of systems with a complete set of integrals of motion have determined both the formulation of many problems of dynamics and the methods of their study. The Hamiltonian method turned out to be of extreme importance for the theory of stability, which was advanced in this direction by Poincaré. Numerous subsequent studies have shown that Hamiltonian systems (i.e., systems which can be described by Hamiltonian equations of motion) demonstrate fundamental physical differences from other (non-Hamiltonian) systems. This chapter provides the most necessary information on Hamiltonian systems.

So far, we have been discussing various kinds of pattern with regular or almost regular symmetry. They emerged either in phase space of dynamic systems or in coordinate space of hydrodynamic flows. Common to all these cases was the method of obtaining or revealing patterns. Such patterns emerged not as the result of some artificial formal algorithm but as an expression of natural laws. In ancient times, however, people did not possess the level of knowledge available to us today. Perhaps it was the attempt to penetrate into the laws of creation of regular patterns, that gave rise to the art of ornament. Or perhaps this form of human activity had nothing to do with what was observed in nature. In either event, it would be interesting to make a number of comparisons between ancient ornaments and the pictures drawn by the trajectory of a real particle under certain conditions.

Two-dimensional tilings in art

Byzantine mosaic is one of the oldest examples of symmetrical periodic tilings of a plane (Fig. 10.1.1). Although the periodicity condition might have arisen as an independent problem, practical aims of architectural design required exactly this kind of ornament. Tiles of one shape (or of several different shapes) were to form the elementary components of a tiling. The element of an ornament was to be reproduced as many times as need, so that eventually any chosen portion of the plane could be paved.

The ornamental technique reached its peak of development in Muslim art. Elementary cells of an ornament became far more complex (Figs. 10.1.2 to 10.1.4).