Table of contents

The physics of non-linear phenomena has developed in a remarkable way over the last couple of decades and has accelerated over the last few years, in particular because of the recent progress in the study of chaotic behaviour. In particular the discovery of the universal properties of the transition into chaos for certain classes of systems has stimulated much recent work in different directions both theoretically and experimentally. Chaos theory has become a real challenge to physicists in many different fields and also in many other disciplines such as astronomy, chemistry, medicine, meteorology and economics and social theory.

The study of chaos-related phenomena has a truly interdisciplinary character and makes use of important concepts and methods from other disciplines. For the description of chaotic structures one needs a new, recently developed geometry called fractal geometry. For the discussion of the enormous richness of ordered structures which appear, one uses the theory of pattern recognition. In order to study even the simplest theoretical models describing chaos, a computer is essential. It should finally be mentioned that important aspects of computer science are related to the theory of order and chaos.

A Nobel Symposium provides an excellent opportunity to bring together a group of prominent scientists for a stimulating exchange of new ideas and results. The Nobel Symposia are very small meetings by invitation only and the number of key participants is typically in the range 20-40. These symposia are organized through a special Nobel Symposium Committee after proposals from individuals. This symposium was sponsored by the Nobel Foundation through its Nobel Symposium Fund with grants from The Tercentenary Fund of the Bank of Sweden and The Knut $ Alice Wallenberg Foundation. Additional support was obtained from the Royal Academy of Sciences, The Nordic Institute for Theoretical Atomic Physics (NORDITA), Chalmers University of Technology and Gothenburg University.

The idea to arrange a Nobel symposium on the physics of chaos and related problems came up more than three years ago. The rapid progress in the field since then seemed a bit frightening, to say the least, in view of the small format of the meeting. Nevertheless, we found the idea attractive - provided that we could restrict the programme to a few selected topics of current interest in order to generate a strong interaction between the participants and produce an intensive discussion.

I feel that I need to express my apologies to all prominent scientists who could not be invited as a result of our planning. In the first place we did not attempt to review areas which seemed to be well established and have reached a certain level of maturity or saturation, irrespective of how great the individual contributions might have been. We decided firmly to concentrate on just a few of the recent developments which seemed to be in the focus of interest, deliberately leaving out important areas equally exciting.

These proceedings contain practically all the material presented in the papers given at the Symposium. We felt that some participants might have found it inconvenient to prepare a full-length paper, which in some cases would have been merely modified versions of material due to appear in regular journals. We therefore took a liberal attitude and accepted everything from a brief abstract with some key references, up to a full-length paper.

We would like to place on record our sincere thanks to all the participants who have contributed substantially in the planning of the Symposium by making valuable comments and suggestions about participants and topics. In particular, Jerry Gollub and Pierre Hohenberg helped me in organizing the programme and they also did a beautiful job with the concluding session and the conference summary. My co-organizers played a crucial role in the planning and during the Symposium week and always seemed to show an outstanding patience with my often rather chaotic actions. Our secretary, Yvonne Steen, deserves very special thanks for her outstanding work for the symposium on top of all her regular duties.

I would finally like to say something about Gräftåvallen and our hosts, Annica and Tommy Hagström. We decided to take this symposium out of the cities and away from the academic environment and found this charming tiny mountain resort on a mountain slope in the northern Swedish mountains about 20 miles from the nearest village. Annica and Tommy Hagström welcomed us with such a warm hospitality and offered us throughout the week the best of the local mountain specialities such as reindeer, bear and beaver and a variety of mountain fishes. Also the local community greeted us as some very special guests and arranged an afternoon programme at a nearby shieling with goats, sheep, dairy maids, folk music and folk dancing. They also arranged a wonderful concert in their beautiful church from the 12th century. Altogether it was a very special week also outside the lecture room.

We, the organizers, experienced this symposium as an unforgettable scientific event thanks to the outstanding contributions of our participants. We hope that these proceedings will convey to the reader something of the excitement felt by the participants during the symposium week.

The simplest example of the onset of chaos in a Hamiltonian system is provided by the "standard" or Chirikov-Taylor model. As a nonlinearity parameter, k, is increased the long term behavior of the momentum, p, is examined. At k = 0, p is conserved. For k < k_c, for all starting points, p is of bounded variation. For some starting points its behavior is periodic, for others quasi-periodic, for others chaotic. At some critical value of k, unbounded chaotic variation first appears. A scaling analysis to describe this onset is described.

A review is given of studies of chaotic dynamics in several solid state systems. In each case the physical system is described, relevant equations of motion are given, experimental results are presented and interpreted, more or less, from the relevant equations, including numerical solutions. The systems are: (1) An electron-hole helical plasma density wave in a Ge crystal in parallel electric and magnetic fields; this shows period doubling and quasiperiodic routes to chaos. (2) Standing mode spin wave packets in ferrite spheres, excited by driving ferromagnetic resonance of the uniform mode; this system shows period doubling to chaos and periodic windows. (3) Resonantly driven p-n junctions in Si show extremely nonlinear behavior due to charge stored during injection; one junction shows period doubling to chaos and period adding (frequency locking); coupled junctions show, in addition, quasiperiodicity, entrainment, and behavior generic to coupled nonlinear oscillators. The fractal dimension is measured for these systems.

A time series of intermittent chaos is changed by bursts suddenly, and its amplitude and phase jump randomly. Between the bursts, the time series oscillates with slow amplitude modulations. Thus the time series is a random sequence of periodic states with different lifetimes, amplitude modulations and phase shifts. Such time series are observed in various systems extending from the pendulum driven by a periodic force to the Bénard convection of fluids. This universality is quantitatively represented by an inverse-power law of the envelope of equally-spaced Lorentzian lines in the power spectrum of the time series near the onset point. The exponents of the separation and widths of the Lorentzian lines are determined within a mean-field framework.

We have observed a complex sequence of periodic states that occurs as a function of flow rate in a well-stirred continuous flow reactor. Each state is an admixture of small and large amplitude oscillations; the number of oscillations per period ranges up to 26. The oscillation pattern in even the most complex states is a combination of the simplest oscillation patterns. With the flow rate as control parameter, the ratio of the number of small to the total number of oscillations per period is a function similar to a devil's staircase. Chaos is not observed in the regime of the complex transition sequence, but chaos is observed in other regions of the parameter space-the largest Lyapunov exponent is then positive and the information dimension of the attractors is between 2 and 3.

A short review of the theory and experimental measurement of the static and dynamical invariants that characterize chaotic motions is presented. There is an infinity of such quantities which remain invariant under a smooth change of coordinates, but only the low order ones are relevant for experimental applications. From the theoretical point of view we stress the many interesting relations which exist between the invariants.

After a review of the period-doubling universality, we point to its possible deeper roots and give a few examples of scaling induced by hyperbolicity (= linear instability). We present a conjecture for the case of linearly indifferent systems.

Dissipative systems with two competing frequencies exhibit transitions to chaos. We have investigated the transition through a study of discrete maps of the circle onto itself, and by constructing and analyzing return maps of differential equations representing some physical systems. The transition is caused by interaction and overlap of mode-locked resonances and takes place at a critical line where the map looses invertibility. At this line the mode-locked intervals trace up a complete Devil's Staircase whose complementary set is a Cantor set with universal fractal dimension D ∼ 0.87. Below criticality there is room for quasiperiodic orbits, whose measure is given by an exponent β ∼ 0.34 which can be related to D through a scaling relation, just as for second order phase transitions. The Lebesgue measure serves as an order parameter for the transition to chaos.

The resistively shunted Josephson junction, and charge density waves (CDWs) in r.f. electric fields are usually described by the differential equation of the damped driven pendulum. The 2d return map for this equation collapses to 1d circle map at and below the transition to chaos. The theoretical results on universal behavior, derived here and elsewhere, can thus readily be checked experimentally by studying real physical systems. Recent experiments on Josephson junctions and CDWs indicating the predicted fractal scaling of mode-locking at criticality are reviewed.

One striking aspect of the orderliness of chaos is that many of its geometric aspects are governed by fractals, and that many of its physical aspects are governed by its fractal geometry. The present work reports several observations concerning the dynamics of a continuous interpolate, forward and backward, of the quadratic map of the complex plane. In the difficult limit case |λ| = 1, the dynamics is known to have rich structures that depend on whether Arg λ/2π is rational or is a Siegel number. This paper establishes that these rich structures have counterparts for |λ| < 1. The observations concern an intrinsic tiling that covers the interior of a J-set and rules the Schröder interpolation of the forward dynamics, its intrinsic inverse, and the periodic or chaotic limit properties of the intrinsic inverse.

Spatial chaos may arise as a consequence of competing spatial periodicities just as temporal chaos may be caused by competing temporal periodicities or frequencies. In particular, spatial chaos occurs in condensed matter systems where one of the periods is that of the crystal lattice, and the other period that of a modulated ordered structure. As an example we review recent results on an axial Ising model where the chaotic states appear as trajectories of a four-dimensional map.

We discuss the new chaotic phenomena arising from the feedback coupling of the response of a nonlinear oscillator to the driving force. The dynamics is described by two coupled autonomous second order differential equations. The fourth dimension in the state space is found to be important. For example, the simple tangent bifurcation of forced oscillations is replaced by a Feigenbaum sequence followed by a crisis, which can be understood in terms of two-dimensional Henon return map. We also argue that cross over from period doubling to intermittency scaling could be found in these systems. We concentrate in this paper to the global phase space structure and suggest the detailed study on universality and scaling of the routes to chaos as an object for further investigations.

Using real-time neutron scattering we have studied convection in a nematic liquid crystal subjected to a vertical magnetic field (H). As a function of the vertical temperature gradient (ΔT) we find two flow regimes. The one at low ΔT's is multistable with interesting transient and hysteretic behaviour. The width of this regime depends on H and vanishes at a multicritical point (H_m, ΔT_m). Each of the competing intensity levels is an attractor. The fractal dimensions of the attractors are derived from the intensity times series, and we find two coexisting fractal dimensions. This observation is interpreted as an indication of spatially dependent dissipation. In part of the parameter space the attractors are coupled by a larger attractor, i.e., there is a crisis.

Universal scaling properties have been predicted for the transition from quasiperiodicity to chaos. We have tested the applicability of these predictions, which are based on the behavior of circle maps, to a hydrodynamic system (Rayleigh-Bénard convection) in which a limit cycle was perturbed at a second frequency to generate a flow on a torus in phase space. The ratio of the two frequencies was adjusted to within about 0.1% of the inverse of the golden mean. Some evidence for spectral self-similarity was found. Phase locking at a nearby rational ratio changes the spectrum qualitatively.

A challenging problem in non-equilibrium statistical mechanics is that regarding the insurgence of ordered structures starting from a chaotic (maximum entropy) condition, in a system strongly perturbed at its boundary as a quantum optical system. For still higher perturbations, the ordered structures become more and more complex, until reaching deterministic chaos. Three experimental situations for CO₂ lasers (a laser with modulated losses, a ring laser with competition between forward and backward waves, and a laser with injected signal) are analyzed as examples of the onset of chaos in systems with a homogeneous gain line and with a particular time scale imposed by the values of the relaxation constants. I stress the coexistence of several basins of attraction (generalized multistability) and their coupling by external noise. This coupling induces a low frequency branch in the power spectrum. Comparison is made between the spectra of noise-induced jumps over independent attractors and that of deterministic diffusion within subregions of the same attractor. At the borderline between the two classes of phenomena a scaling law holds, relating the control parameter and the external noise in their effect on the mean escape time from a given stability region.

This talk is intended to provide an introduction to the study of pattern selection and propagation in nonequilibrium systems. For simplicity we shall confine the discussion to systems under steady external conditions. Examples from hydrodynamics are Rayleigh-Bénard convection and Taylor-Couette flow in large aspect-ratio containers, or Faraday crispations; related phenomena appear in flame-front propagation, chemical reactions, plate buckling, and various problems in crystal growth. Questions of interest include: (i) the prediction of steady vs time dependent behavior and of the relevant time scales for reaching this behavior; (ii) an analysis of spatial patterns, both ordered and disordered; (iii) elucidation of the mechanisms for the onset of temporal chaos in systems with many active degrees of freedom (e.g., are the attractors low-dimensional?).

There is a close relationship between the spatial structure of hydrodynamic flows and the onset of chaos in these systems. In this lecture, I will describe this relationship in the context of three experiments. (1) Competing patterns or modes of oscillation of the free surface of a fluid layer gives rise to chaotic time evolution that can be explained by a relatively simple model. (2) Competing spatial periodicities in the electrohydrodynamics of a layer of nematic liquid crystal gives rise to commensurate, incommensurate, and apparently spatially chaotic phases. (3) New experiments on the evolution of Rayleigh-Bénard convection patterns provide a test of models in which a functional of a two-dimensional order parameter field evolves toward a minimum.

We describe an apparatus and procedures for simultaneous heat transport measurements and computer enhanced shadowgraph flow-pattern imaging in a shallow horizontal layer of fluid heated from below. The heat transport measurements have a resolution of better than 0.1%, and the shadowgraph technique can detect the flow field for ≡ (R-R_c)/R_c as small as 10^-2 (R_c is the critical value of the Rayleigh number R for onset of convection).

The apparatus and procedures were used to study pattern and wave-number evolution in a cylindrical layer of water with radius-to-height ratio L = 7.5 and Prandtl number σ = 6.1. We found that dynamic sidewall forcing during the early thermal transients after a change in the heat current from a subcritical to a supercritical value establishes a cylindrical flow pattern. Once created, this pattern is stable in our apparatus over the wide range 0.16 ≲ ≲ 8 even after the transients have decayed.

With changing , adjustment in the wave number k takes place discontinuously by hysteretic changes at the cell center in the number of convection roll pairs. When is increased, the discontinuous changes at the cell center are towards smaller k and are preceded by a continuous loss of cylindrical symmetry (the middle roll pair moves off center). The selected wave numbers coincide neither with the zig-zag instability of the infinite system, as once suggested, nor with a linear extrapolation to = O(1) of the recent prediction to lowest order in of Manneville and Piquemal and of Cross. Comparison of the selected k with measurements by others reveals no dependence upon L and σ.

For < 0.16, the cylindrical pattern is unstable and decays on a time scale much longer than a horizontal diffusion time to patterns of rolls which tend to be perpendicular to the sidewalls and which contain defects. Once formed, these latter patterns will persist at large values of . These patterns also undergo a wave-number adjustment process with hysteretic changes mediated mostly by focus singularities near the walls. In these cases, larger values of also tend to produce smaller values of k.

We first briefly sketch a general approach how to reduce the equations of systems composed of many subsystems to equations for, in general, few order parameters at instability points. As special case generalized Ginzburg-Landau equations are obtained. Recent results based on these equations showing the pattern formation in the convection instability and flames are presented. Bifurcations from tori to other tori are treated and some general conclusions are drawn. Analogies between fluid dynamics and lasers which for the first time led to the prediction of laser light chaos are pointed out. Finally the suspension of a class of discrete one-dimensional maps is discussed and explicitly presented for a typical case.

Weiss and McConnell recently have observed the formation of chiral, solid-like domains of a phospholipid monolayer at an air-water interface. A tentative explanation for this phenomenon is that the phospholipid molecules in the two-dimensional fluid-like phase undergo chiral diffusion, that is, that there is a component of the diffusion flux in a direction perpendicular to the concentration gradient. Such an effect seems possible, and consistent with time-reversal symmetry of the underlying dynamics, if the viscous forces on the molecule couple translational and rotational motions. This possibility is investigated by studying a molecular model that has some of the properties of a flexible pinwheel. A formula is derived from the transverse diffusion constant; and a rough estimate of the magnitude of this quantity seems consistent with the experimental observations.

The class of first order non-linear partial differential equations possessing travelling wave solutions is considered. We find a dynamical selection mechanism that picks a state out of a one parameter family of marginally stable solutions with different propagation velocities. The travelling wave that is singled out possesses a minimal velocity allowed by the initial state.

In unidirectional crystal growth ordered or chaotic cellular patterns develop depending on the angle between the growth direction and the crystal main symmetry axis [⟨100⟩ direction in our experiment]. The three possible bifurcations for a cell are discussed.

"Surface kinetic dimension" was defined as a quantity to measure the logarithmic derivative of the surface with respect to the volume of growing patterns. This quantity for the growing pattern based on the "hot tip model" was found to show two kinks when the tip priority factor was increased. They were found to correspond to a microscopic roughening transition and a macroscopic morphological transition from a compact to a dendritic growth pattern, respectively. The result of the simulation based on the triangular pixel was also reported. A possible mechanism of the transition was discussed. It was pointed out that the surface kinetic dimension plays the role of a potential in the growing pattern dynamics.

For an arbitrary probability density p(x) (x ∊ Eⁿ) an "observed" probability density is defined for a given resolution. Then using this observed density, the Kullback information or "contrast" of p(x), K[p₀;p], where p₀(x) is a reference probability density, can be decomposed with respect to structure size, the differential information being the Hausdorf codimension of p(x) for the corresponding structure size. It is also possible to make a simultaneous decomposition of the Kullback information with respect to position and structure size.

All results can easily be generalized to Rényi information of arbitrary order.

It is indicated how the same concepts can be defined for a density operator in quantum statistics.

Finally a qualitative description is given of the "contrast economics" of a self-organizing chemical system.

Fully Developed Turbulence is concerned with the behaviour of turbulent flows (here assumed incompressible) as the Reynolds number R becomes very large. For such flows there is evidence of scaling (e.g. the -5/3 law for the energy spectrum). It is shown that the Kolmogorov 1941 theory is equivalent to an invariance principle: as R → ∞ all the invariance properties of the Navier-Stokes equation, possibly broken by the mechanisms producing the turbulence, are recovered at small scales in a statistical sense. Actually scale-invariance is (slightly) broken, as evidenced by the phenomenon of small-scale intermittency. We discuss the role of real and complex singularities in connection with intermittency. Computer simulations give strong evidence for the existence of singularities in the complex space-time domain. Simulations are so far inconclusive as to the existence of real singularities of the Euler equation (zero viscosity). Complex space singularities within less than one mean-free-path of the real domain could imply a breakdown for the hydrodynamical approximation. This (unlikely) occurrence is connected with the first of three possible explanations of the unpredictable character of turbulence: (1) loss of uniqueness through singularities; (2) intrinsic stochasticity ("Chaos"); (3) extrinsic stochasticity. The correct explanation is likely to be a combination of the latter two. We then turn to the problem of the minimum number of degrees of freedom needed for high Reynolds number flow: Is it O(Rⁿ) (n ≃ 9/4) as suggested by the Kolmogorov theory or, possibly, much smaller? This may depend crucially on the role played by coherent structures. Finally we briefly assess the impact on Fully Developed Turbulence of attempts to carry over techniques from Phase Transitions and Quantum Field Theory.

The ergodic theory of differentiable dynamical systems is reviewed, and applied to the Navier-Stokes equation. Using exact estimates on bound states for the Schrödinger operator one obtains upper bounds on characteristic exponents, entropy, and Hausdorff dimension of attracting sets.

A brief review of the current status of hyperscaling relations is given, and some new results for finite size systems are presented.

Upper bounds are given for the topological entropy and the Hausdorff dimension of the universal attractor of a plane Poiseuille flow.

We provide a brief introduction to questions about the fundamental physical limitations of the computational process, and to reversible computation, without any attempt to repeat more complete and self-contained discussions. We explicitly question, in connection with the existing quantum mechanical descriptions of the computational process, whether the Hamiltonians invoked in these descriptions correspond to the modest number of parts considered to be reasonable for a computer. Our main goal, however, is an analysis of classical reversible computers, in the presence of noise and friction. Such computers can be driven by an externally applied driving force, independent of time and of the computational progress. Alternatively, they can be driven by time dependent force fields, much as ordinary computers are clocked. It is shown that clocked reversible computers can minimize the ratio of diffusion, i.e. unpredictability in the progress of the computation, relative to the intended speed of motion. The analysis emphasizes the case of heavily damped systems, but indicates that lightly damped systems behave similarly.

Complex systems, such as biological organisms and computing structures, lie between the realms of statistical mechanics and the physics of a few degrees of freedom. Moreover, they can exhibit self-organized behavior which in many cases is characterized by learning, recognition and fault tolerance. This talk will describe studies of adaptive parallel computers which are capable of reliable learning and recognition. The existence of attractors in their dynamical behavior leads to a novel self-repairing mechanism which has been tested by quantitative experiments. Moreover, we will show how these highly concurrent structures, which are capable of universal computation, can be used to study simple, fault-tolerant, perceptual tasks.

Cellular automata are simple mathematical systems that exhibit very complicated behaviour. They can be considered as discrete dynamical systems or as computational systems. Progress has recently been made in studying several aspects of them. Twenty central problems that remain unsolved are discussed.

The simplest chaotic Hamiltonian systems arise in geometry: a point particle moving freely on a 2 or 3 dimensional manifold of constant negative curvature. There is tremendous variety representing both bound state and scattering situations, quite in contrast to the non-chaotic motion on surfaces of constant positive curvature. The classical mechanics has been studied for almost a century and reduces to an enumeration and description of geodesic lines which in turn are closely connected with the topology of the manifold. Physics enters only in a trivial manner since everything can be scaled with the (kinetic) energy of the particle. In the quantum mechanics of these motions, however, the energy is the essential parameter, and one can ask how the two mechanics are related to each other. Quite surprisingly, Feynman's path integral gives the same answer as its classical approximation, just as in the harmonic oscillator. In 3 dimensions, this result holds for the propagator itself, while in 2 dimensions it is true only for the trace and yields Selberg's trace formula. A few striking examples of constant negative curvature spaces will be shown. Thus, quantum chaos can be understood in terms of classical chaos in spite of their qualitatively different features: fractal for the latter, smooth for the former.

This paper will focus on the connection between renormalization of dynamical systems with two frequencies and the discrete 1-D Schroedinger equation with an incommensurate modulation. The connection will be explored by constructing a renormalization group formulation of both problems using a binary "Farey" address to label the incommensurability. This address is closely related to the continued fraction representation which has previously been used to investigate these problems. The emphasis of this paper is to introduce a new method rather than to present a completed calculation. Future investigations will determine whether this will eventually lead to deeper insights into these problems than can be obtained using the continued fractions.

We introduce a random hierarchical model of a spin glass which allows the Parisi order function q(x) to be calculated by a simple functional recursion relation. The resulting q(x) has a devil's staircase form, and the staircase is complete if the fraction p of antiferromagnetic bonds is less than 2^-1/2. The fractal dimensionality of the Cantor ser of points where q(x) is not flat is (-2 log₂p)^-1.

We conjecture that the fractional quantized Hall effect Landau-level occupation factors are organized by the Haldane tree. This is a variant of the Farey tree, suitably refined to account for the absence of forbidden states.

Here we review some recent numerical simulations of quantum phenomena in: (1) Little's model of a one-dimensional electron-exciton system and (2) two dimensional quantum lattice gases. New algorithms and faster machines make it possible to study a variety of quantum correlations in condensed matter systems, and these examples illustrate some of the types of things which can be learned from simulation.

In preparation for the concluding session, we distributed a questionnaire to the participants in which we asked for suggestions of: interesting and potentially answerable questions; issues that were obscure and need to be elucidated; topics that are particularly stimulating; and conjectures or bets about the future of the field.

A large fraction of the participants filled out the questionnaire and provided us with enough material to form the basis for the concluding session. What follows is an edited transcript of the discussions, based on a tape recording of the session and on limited subsequent consultation with participants. We eliminated some portions which were either inaudible, unclear, or overly technical, and have added some clarifying comments where we thought those would be helpful. In general, we tried to remain faithful to the spirit, but not always to the letter of the discussion.