Mathematics of chaos

Photo Claire Chavaudret / Lecturer and Researcher, Laboratoire J. A. Dieudonné, Nice Sophia-Antipolis University / September 11th, 2015

In many areas, decision-making is affected by the difficulty in producing reliable forecasts. The behavior of financial markets, consumers or weather phenomena, the evolution of an ecosystem or the movement of certain celestial bodies provide some examples of unpredictable events that have an impact on human activity. Some developments of mathematics can help reduce this unpredictability or, at the very least, analyze it from a strategic point of view. The theory of probability plays such a role but so do fluid mechanics in the study of turbulence, or dynamic systems in the study of so-called chaotic phenomena, which belong to a specific class of unpredictable phenomena.

Historically, the theory of dynamic systems did not immediately provide forecasting tools. On the contrary, the seminal work of Henri Poincaré, at the end of the 19th Century, allowed to account for the excessive complexity of certain systems that were considered as simple and deterministic (i.e. where the concept of chance would not intervene).

Tools that allowed to circumvent unpredictability appeared only later, during a second phase, for instance the Sinai-Ruelle-Bowen measure, a mathematical tool developed in the 1970s. This measure makes the connection between deterministic systems and the probability theory by quantifying the probability by which a chaotic system will eventually reach a given state. But the contribution of the mathematics of chaos is not limited to forecast, which remains necessarily limited.

The mathematical simulation of real phenomena may be of practical interest: for example, the film industry uses fractals to generate more realistic scenery.

Simulation can also help better understand these phenomena, by revealing similarities and developing intuition and understanding. Studies on chaos have, to a certain degree, an explanatory value. One of the challenges of science is to understand the emergence of complexity from simple rules. Dynamical systems are contributing to this challenge.

Ultimately, mathematics can provide tools for a controlled feedback on reality. Artificial satellites provide a good example as they are controlled most of the time following conventional methods, with a sufficient amount of fuel. The knowledge of chaos could allow substantial savings: that’s within the scope of the theory of control of chaos.

The study of chaotic phenomena is therefore of practical interest. But what is chaos, really?

Chaos, an experimental concept
The concept of chaos is firstly experimental, primarily used by physicists. It expanded later to other experimental sciences such as ecology, meteorology or medicine. By experiment, we refer both to the observation of real phenomena, sometimes recreated in lab work, as well as to the analysis of numerical simulations, made possible by the explosive growth of computing power since the 1970s.

This is how astronomers have observed chaos in the orbit of a natural satellite of Saturn, Hyperion. Although on a generally regular orbit, Hyperion isn’t spherical and displays sudden changes in its rotation axes, because of simple gravitational interactions with the planet and its other satellites.

In a completely different field such as ecology, chaotic phenomena are observed even when lab ecosystems are reduced to their simplest expression: flies in a jar or fluctuations of the numbers of species are not only variable, they are also aperiodic – this is even more so in the case of real ecosystems.

We see that chaos is experienced first as a disruption of periodicity, where one was expecting to observe regular cycles. This also happens in the human body: studies on electrical signals in the brain have shown the presence of chaos i.e. the expression of an aperiodicity which is not due to external or experimental noise. Various forms of chaos can be observed according to cerebral status (wake, sleep or even epilepsy): this could pave the way to designing new tools for diagnosing Parkinson’s and Alzheimer’s disease.

Another feature of chaos, perhaps the best known, is its sensitivity to initial conditions. Movement is even more unpredictable that it depends on its state at a particular point: any infinitesimal modification generates a completely different movement. Thus, reproducing twice the same lab conditions is virtually impossible. At best, one can find broad patterns which will repeat themselves in every experiment.

Both aspects are present in commonly used mathematical models. This led mathematicians to formulate their own definition of chaos.

A first mathematical definition of chaos
While mathematicians often define chaos differently, the following definition, precise and restrictive, is more or less supported by most members of the scientific community. Its main advantage is that it allow to “express things” i.e. to simulate, predict and control.

To be considered as chaotic, a system must:

-be sensitive to initial conditions i.e. small causes may have great effects. The sensitivity to initial conditions is the other name of instability, a concept explored by the Russian school of Lyapunov.

- show recurrence i.e. a movement from a given point will revert an infinity of times as close as we wish to its starting point. More specifically, many periodic motions, after a given time, will come back exactly to their starting point.

The first condition is not enough per se, because the so-called hyperbolic systems, which aren’t chaotic, also have a certain sensitivity to initial conditions. Let’s take the example of a mountain ridge: if a ball is let fall at a determined point, it will end up in a determined valley; but if released a few meters away, it could end up in a completely different valley. This is absolutely not chaotic, it’s even perfectly predictable.

The second property is not enough to obtain unpredictably since it happens in systems where all orbits are periodic (such as idealized solar systems), the archetype of predictable movements.

Therefore, for a system to be unpredictable, it needs at least these two features. Mathematically, this can happen in very simple systems, on two conditions.

A first condition is the presence of a non-linear relationship. The effect will not be proportional to the cause, as in a linear model. On the contrary, it will be disproportionate. Even the simplest nonlinear equation, the Ricatti equation, has solutions that “explode” over a finite time. In this precise case, it still isn’t chaos, because the solutions are simple to calculate.

The second condition is the number of interacting variables.

Chaos may emerge in small-scale systems, but never in systems with one or two variables. This is a consequence of the Poincaré-Bendixson theorem, which is based on the fact that in a plane, a closed curve has an inner and an outer part. The Poincaré-Bendixson theorem shows that in spaces of dimension 2, we can observe either stationary states or periodic movements.

In three dimensional spaces, however, chaos can emerge. A good example is the simplified Lorenz system for weather. In this case, a small number of equations, containing very few variables, are enough to generate movements that are difficult to predict, especially if considering the succession of different states and not just the overall state of the system on the long term.


Observing order within chaos
Fortunately, not only do mathematics allow us to recognize the presence of a signal or of a quantity that evolves in a chaotic or unpredictable manner, but also to move beyond the slightly discouraging notion of chaos by providing us with new tools of description.

This will help describe what happens “on average” or “almost surely” (from a probabilistic point of view) or “generically” (from a topological point of view). Because for a mathematician, chaos doesn’t mean absolute disorder or total instability.

Two main ideas can be used to distinguish patterns in chaotic systems and make some predictions or descriptions: the concept of attractor, used in topology, and that of invariant measure, used in probability theory.

An attractor defines an area in which a system will most surely evolve over the long term. A chaotic system is distinguished by the fact that this area has a very complex shape, as in the Lorenz meteorological system (it is then referred to as a “strange attractor”).

An invariant measure indicates the mean time that a trajectory in the system will probably spend in a given set of states. Here, the introduction of probability theory is done only in a second phase, within the system, and not to represent the outside “noise.” This reflects a pragmatic vision of probability and randomness: the notion of randomness is introduced when it is virtually impossible to predict with certainty – nonetheless, randomness is not assumed as an intrinsic feature of observed phenomena.

The boundaries of classical chaos: randomness and unpredictability
In chaotic phenomena in the strict sense, disorder is intrinsic to the system. This excludes random systems in which disorder comes from outside. This distinction is useful to consider the question of financial markets, for example. Is unpredictability intrinsic to markets, or does it emerge because of the number of variables that come into play? The question is still fueling debate.

In any case, chaos is defined as a seemingly random behavior. It is a disorder which nevertheless fits in a deterministic system.

This calls for some clarification. Laplace, in his Philosophical essay on probabilities, gives the following definition of “determinism”: the assumption that, if our intelligence were infinite, we could calculate with infinite precision the future state of a system within an arbitrarily large time-frame, provided that we know, with infinite precision, the current state of the system and its past. In other words, past, present and future states of the system are completely interconnected.

Historically, this assumption has been interpreted as the ability to formulate the laws of nature by ordinary differential equations, in which the knowledge of the state of a system at a given time is enough to calculate both its past and its future. The use of deterministic models in science is an application of this postulate.

Some deterministic phenomena, however, are almost unpredictable and this is where chaos steps in.

In practice, to be selected and classified by the scientific community under this term, unpredictable deterministic phenomena must derive from simple rules. This is the case, for example, of the equations of classical mechanics. It is also true for iteration rules such as “pick a number, take its inverse and remove the integer part” (Gauss map).

To be labeled as “chaotic,” these phenomena must also appear in systems with few variables. Thus, fluid mechanics, which describes systems with a very large number of particles, is outside the scope of the mathematics of chaos.

Chaos, according to its most widely spread meaning among scientists, refers therefore to a form of disorder that can be grasped relatively comfortably. It derives from very simple models, while the reality around us, in order to be perfectly described by mathematics, would probably require a huge number of variables and parameters.

The complexity of nature is not exhausted by the mathematics of chaos, although chaos can explain part of it.

Most specifically, the mathematics of chaos does not cover infinite-dimensional systems.

This excludes two main types of equations. First, delayed differential equations, where the evolution of a present state depends explicitly on a past state. These equations have possible applications in aeronautics, ecology or economics.

Second, partial differential equations, which are essential in physics: these are used to predict and measure turbulence in a fluid. This turbulence is not strictly speaking covered by the mathematics of chaos. This is worth noting because, historically, the study of turbulence contributed to showing a chaotic phenomenon, that of period doubling cascades (which occur, for example, in population dynamics).

Some mathematicians, including Yakov Sinai (Abel Prize 2014), see turbulence as the physical expression of a singularity, i.e. of an “explosion” of solutions in the Navier-Stokes equations. Since these equations involve partial derivatives, the problem has, from a mathematical perspective, infinite dimensions and therefore, falls outside the scope of chaos and in the field of unpredictability, to use a distinction made by philosopher Marie Farge.

For the same reasons, natural hazards (tsunamis, hurricanes, earthquakes) are closer to unpredictability than to chaos. However, mathematicians that study both subjects, chaos on the one hand, and infinite-dimensional systems on the other, are no longer as segregated as before: some techniques are shared by both.

The boundaries of chaos, such as random systems and infinite-dimensional systems are extremely stimulating and challenging research areas, with an urgent and growing demand.


In conjunction with its emergence in many experimental sciences, the notion of chaos has gradually been defined in mathematics as some kind of seemingly random behavior displayed by the solutions of a deterministic system. Observing chaos needs a non-linear system of dimension 3, at least.

In fact, the scientific community is used the concept chaos only if this complexity, this apparent randomness, is seen in a simple system, with relatively few variables, allowing to unify the efforts of understanding and describing around a set of common techniques, that largely proceed from the theory of dynamical systems.

This restrictive meaning excludes many no less important systems that show a disordered behavior. But in return, restriction allows for the development of tools for describing and forecasting which will hopefully be exported into other areas, especially because boundaries between scientific communities are shifting quickly.

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