A Brief History and Introduction to Chaos Theory

A Brief History and Introduction to Chaos Theory

In physics and mathematics, chaos theory is the study of aperiodic behaviour in deterministic systems that are extremely sensitive to initial conditions. A deterministic system is a system whose present state is, in principle, fully determined by its initial conditions. Despite the fact that the term "chaos" was first associated with the subject by James Yorke and Tien-Yien Li in 1975, the theory and supporting mathematics can be traced significantly further back.

In 1887, keen amateur mathematician King Oscar II, who was simultaneously both the King of Sweden and the King of Norway, set up a prize in his own name, to be awarded on the day of his 60th birthday, for "an important discovery in the realm of higher mathematical analysis". Of particular interest to King Oscar II was the so-called three body problem, a problem first established in 1687 by Isaac Newton, in his Principia Mathematica, which concerns solving the motion of three orbiting bodies from their initial positions and momenta. This contest motivated the great French polymath Henri Poincarè to look into the subject. In his treatment of the problem, Poincarè found that there could be orbits of bodies that are non-periodic, and yet do not permanently increase, nor approach a fixed point. Although Poincarè failed to obtain the power-series solution required to solve the problem comprehensively, so impressive was his treatise that he was awarded the King Oscar Prize nonetheless. Building on the mathematics of Poincarè, the three body problem was eventually solved in 1912 by Finnish mathematician Karl Sundman, and generalised to the n>3 case in 1991 by Chinese mathematician Qiudong Wang.  Poincarè’s work introduced key concepts that would later lead to the development of chaos theory. Indeed it was Poincarè who, in 1885, coined the term "bifurcation" and was the first to study bifurcation theory by considering a mass of uniform, incompressible fluid, undergoing constant uniform rotation, subject to its own gravitational attraction, and theorising what shape the body of fluid would form if it were in dynamical equilibrium.

A dynamical, or nonlinear system is a set of nonlinear equations that may depend on given parameters. A dynamical system  in which time is discrete is called a map. A bifurcation occurs where the solutions of a nonlinear system change their qualitative character as a parameter changes. Bifurcation theory is the study of how the number of steady solutions of a system depends on parameters, and therefore concerns all nonlinear systems. Bifurcation theory has enjoyed a multitude of applications in physics, namely in traversing the bridge between the behaviour of quantum systems and their classical dynamical counterparts, such as in atomic systems, molecular systems, and semiconductor lasers. A period doubling bifurcation is a bifurcation in which a dynamical system  changes to have a period twice that of the original system. The opposite of a period doubling bifurcation, a period halving bifurcation, is a bifurcation in which the system switches to a new behavior with half the period of the original system. In 1975, American physicist Mitchell Feigenbaum discovered that the ratio of the difference between the values at which consecutive period doubling bifurcations occur tends to a constant, now known as the first Feigenbaum constant (the value of the ratio resembles the first Feigenbaum constant better with each subsequent bifurcation, and agrees with an accuracy of 4 significant figures after just 8 period doublings). The second Feigenbaum constant is defined as the ratio of the width of one branch (or tine), of the bifurcation diagram and one of its two subbranches (or subtines). Feigenbaum also proved that this behaviour would arise in a substantial amount of different mathematical functions, thus displaying the universality of chaos, establishing its versatility in describing a wide variety of different physical processes. The period doubling route to chaos was first experimentally observed in a Rayleigh–Bènard convection system (a system in which a thin plane of fluid is heated from below, resulting in the beautiful development of a regular pattern of Bènard cells) in 1979 by French physicist Albert Libchaber, thus confirming the theoretical predictions of Feigenbaum. The universality of the Feigenbaum constants themselves was proved in 1982 by American mathematician Oscar Landford III, as such, both Feigenbaum constants are seen as being fundamental mathematical constants, much the same as π or e.

In 1993, while studying the relationship between chaos and models of animal immigration and population dynamics, Israeli biomathematician and zoologist Lewi Stone discovered that the 'universal' Feigenbaumian behaviour could be inhibited. Stone analysed a modification of the logistic population growth model, put forward in 1992 by Australian ecologist Hamish McCallum. This new model varied from the original logisitc growth model, first developed in 1838 by French mathematician Pierre-François Verhulst, by considering that numerous sub-populations rely either on refuges for protection, or on an influx of individuals by immigration, and thus must havesome non-zero base value below which the population level never falls. Whilst simulating McCallum’s model, Stone observed the expected period doubling bifurcation process curtail and reverse, giving rise to period halving bifurcations which formed a distinctive "bubbling" effect known as period bubbling. The period bubbling phenomena appears in many different systems across nature, such as in models of insect populations, psychotic human behaviour, and magnetoconvection.

In 1898, French mathematician Jacques Hadamard was the first to examine a chaotic dynamical system. Hadamard’s dynamical system considered the motion of a free particle on a frictionless, two-dimensional surface of genus two with constant negative curvature, known as the Bolza surface (introduced in 1887 by German mathematician Oskar Bolza). Hadamard showed that the trajectory of every particle diverges from the trajectory of every other particle, that the system has a positive Lyapunov exponent. The Lyapunov exponent of a dynamical system, named after Russian mathematician Aleksandr Lyapunov, represents a measure of the exponential separation of infinitesimally close trajectories. Hadamard’s dynamical system, also known as the Hadamard-Gutzwiller model in acknowledgment of the work of Swiss-American physicist Martin Gutzwiller who studied the system in the first half of the 1980’s, has seen application in different areas of physics. It has been used to simply model the relationship between classical chaotic dynamical systems and quantum mechanics, an area of physics known as quantum chaos. In 1963, while working on his own billiard model, Russian mathematician Yakov Sinai showed that the motion of the atoms in the  classical Boltzmann–Gibbs ensemble for an ideal gas followed the trajectories of the Hadamard-Gutzwiller model.

In 1961, whilst simulating weather patterns, American mathematician and meteorologist Edward Lorenz serendipitously discovered that small changes to the initial conditions of a  dynamical system can have a significant effect on the long term outcome. Lorenz used two-dimensional convection in a horizontal layer of fluid, heated from below and cooled from above, as a simple model for atmospheric convection. Partway through one of his simulations, in order to reexamine a specific series of data, Lorenz restarted the simulation, but in order to save time, entered data from a printout that corresponded to the conditions in the middle of the original simulation. Upon rerunning the program, Lorenz found that his computer was predicting weather that was completely different to that of the original simulation from which the starting conditions were taken. This aberration arose from the computer working to 6 significant figures, but the printout from which Lorenz took the data only printing to 3 significant figures. This large change in the weather pattern from a small change in the initial conditions proved that even extremely precise, detailed models of the atmosphere fail to make accurate long-term weather predictions. This is why meteorologists are still only able to make weather predictions for a 7 day period, and why their forecasts tend to be so frequently wrong. Lorenz’s discovery and subsequent 1963 paper are considered to be the provenance of chaos theory. The Lorenz system would later find application in many different areas, including laser physics, electrical circuits used in private communications, and biological systems.

Chaos only occurs in three or more spatial dimensions. A Poincarè section is a projection of a higher dimensional trajectory onto a lower dimension. It is the intersection of a periodic phase space orbit with a lower dimensional surface. When the period of a system doubles, so does the number of points on the associated Poincarè section. Its usefulness arises from the difficulties of determining the dynamics of particles from their trajectories in higher dimensions, and provides an easy visual representation for determining if a system is chaotic or not. An attractor is the set of points to which trajectories approach. As the system enters the chaotic regime, the points on the Poincarè section converge on a curve known as a strange attractor, which is an attractor that has a fractal structure and can exhibit sensitive dependence on initial conditions. While it may appear quite daunting or abstract to someone without a mathematical or physics background, I hope this article has gone some way to show how accessible chaos theory is.