Chaos is a complex term. When a system is called chaotic, it is usually meant that it is sensitive to initial conditions. Such a sensitivity, in turn, means that when two, close initial conditions are considered (by close one means arbitrarily close, i.e. differing by ε>0), the resulting trajectories, when the system is evolved in time, diverge exponentially. This means that very small perturbations quickly cause very large changes. Chaotic phenomenona are present in all fields, e.g. meteorology, genetics, economy, environmental sciences (e.g., predator-prey models), nuclear physics, electrical engineering, and many more. A simple physical realisation of a chaotic system is the double pendulum. In this case, the initial conditions are the locations and velocities of the pendula. But the locations can be determined only with a finite precision due to measurement errors, so that one can never set the pendula twice in exactly the same initial position. In chaotic systems, such a small uncertainty will eventually lead to a motion that is unpredictable, contrary to a non-chaotic case when the divergence can be precisely predicted. This measurement uncertainty is the reason why forecasting weather works only for very short periods. A quantitative measure of the strength of chaos is the Lyapunov exponent: the greater it is, the smaller the predictability timescale. It is also an indicator of chaos, i.e. when it is positive, chaos is present in the system, while when it is ≤0, the behaviour is regular. A natural language to talk about chaotic systems is the theory of dynamical systems.

Figure 1. A mosaic of the most iconic depictions related to chaos and fractals. From left to right: the Lorenz attractor, the bifurcation diagram of the logistic map, Poincaré surface of section of galactic motion in the Hénon-Heiles potential, the Mandelbrot set.

In astrophysics there is a vast set of chaotic phenomena. The most well-known is the classical three-body problem occuring in celestial mechanics, but also in galactic dynamics, pulsar spin-down rates, cosmology, megnetohydrodynamics, among others. An example of an observationally verified chaotic state is the rotation of Hyperion, a satellite of Saturn. A combination of a highly aspherical shape and eccentric orbit causes Hyperion to tumble chaotically, constantly changing its axis of rotation and angular velocity. The investigation of the dynamical properties of Hyperion, and oblate satellites in general, was the topic of my PhD thesis.

Figure 2. Hyperion as seen by Cassini-Huygens spacecraft.

A non-exhaustive list of tools useful in examining chaotic dynamics are bifurcation diagrams (most famous example is of the logistic map) or the Poincaré diagrams. The latter are the intersection of a phase-space trajectory with a plane, and often result in complex pictures with a fractal structure (in case of dissipative systems). A fractal is a mathematical set that possesses the property of self-similarity, i.e. its every subset contains the whole set. A classic example of a fractal is the Mandelbrot set. Fractals can be described by a fractal dimension, often attaining fractional values. A fractal dimension of 1.5, for example, means that the set is topologically something more than a one-dimensional line, but it does not cover the whole two-dimensional space. Despite charaterising fractals per se, a fractal dimension can be used to detect and quantify clustering of celestial objects, such as the distribution of gamma-ray bursts on the sky, or galactic filaments.

Figure 3. (a) A schematic illustration of the Poincaré surface of section. When forcing is present in the system, like in the Duffing pendulum, recording the state of the system with the forcing frequency is common. (b) The attractor of the Duffing pendulum. Its fractal dimension is about 1.4. (c) Poincaré surface of section of a simplified model of an oblate satellite's chaotic rotation. The chaotic sea is enveloped from bottom and top by quasiperiodic rotational separatrices. Embedded in the chaotic zone are islands of stability, consisting of librational regular motion, with strictly periodic trajectories in their center. The system is conservative (Hamiltonian), so there is no fractal structure in the chaotic sea, and the points densely cover the whole available space.

Due to its unpredictability, chaos is usually unwanted when it comes to practical applications. The field of chaos control gives a variety of methods to decrease the strength of chaos, for example in nuclear plasma or electronic circuits. An ability to construct efficient control terms of the aforementioned rotation of small Solar System bodies, like asteroids, while still rather futuristic, might prove to be important in future asteroid capture missions and mining attempts. Construction of a control term that reduces chaos substantially, down to a strictly periodic rotation, is a recent finding of mine.

Book recommendations

An excellent popular introduction to the theory of chaos is Ian Stewart's book Does God Play Dice? The New Mathematics of Chaos/Czy Bóg gra w kości? Nowa matematyka chaosu, as well as James Gleick's Chaos. Making a New Science/Chaos. Narodziny nowej nauki.

Introductory academic textbooks are: Chaotic Dynamics. An Introduction/Wstęp do Dynamiki Układów Chaotycznych by G. L. Baker and J. P. Gollub, and Nonlinear Dynamics and Chaos by S. H. Strogatz.

A classical advanced textbook is Edward Ott's Chaos in Dynamical Systems/Chaos w Układach Dynamicznych.