The potted fractal and the history.

A very easy experiment that you may comfortably do in your kitchen. Let some water rest in a pot or other container for a sufficiently long time at a low temperature. I can't say exactly how much time is necessary, but leaving the pond for a whole day in the refrigerator would suffice. I happened to look at the results after casually leaving some water in a pot for a winter's night in the kitchen. The salt dissolved in water is going to aggregate in an irregular structure that floats on the surface, consisting of an incredible sequence of clusters, which resembles the image that I quoted in the title. That structure has the properties of a fractal, and arises from a process of gradual growth due to the fortuitous meeting of the salt particles dissolved in water and with the floating growing object. As new particles are drifting into contact with a particle already established, a chemical bond is formed and the new molecular grain of salt aggregates to the solid and increases it.

This process is described mathematically as the Diffusion Limited Aggregation (DLA in English - further details and animations are available on these pages of the Hong Kong Polytechnic University), which may well represent the physical process of crystallization far from the equilibrium point. Yes because under conditions of chemical and physical equilibrium there are no interactions between particles: all possible binds would be established, and the water molecules in Brownian motion oppose to the salt aggregation process. The process take-off consists of an accidental event: a pair of particles, which met first, begins to set other grains around them. This event accidentality makes it impossible to generate the same geometry if we were to repeat the experiment a second time: whatever the fortuitous circumstance that led to the formation of the first granular aggregate, this one will characterize the subsequent development and the form of the final structure. It is impossible, in retrospect, to understand why the solid has taken a specific form and not another equally plausible, because the structure will be fully determined by its history.

Short navigation among chaos, complexity and chance.

The mathematics of chaos is far from mature and sufficiently developed to be applicable to large systems. Nevertheless it has successfully found its own way into the doctrines that form the evolving body of the so-called complexity theory. Historically, you could say that it is the true progenitor.

Apparently the two disciplines have little in common. The chaotic systems with known differential equations sets are only few, and all can be considered simple in the sense that they have few degrees of freedom - the famous Lorenz system has only three, for example. This is probably due to the areas of study in which the systems known in the literature have been developed, because the principles of chaos seem easily applicable for large systems with many degrees of freedom, or with several state variables. The analysis of the measurements authorize the conclusion that also those systems that are likely huge in terms of dimensions may show an intrinsic chaotic nature, at least under some conditions.

The theory of complexity originated from reflections around systems with many dimensions, or even infinite. The latter, however, is a specific case because of the mathematics and theoretical aspects that it brings with it: a system with infinite dimensions, or an infinite number of state variables, would need infinite information to be described. In fact, every variable of state is independent of all others, and a system with an endless quantity of variables could carry infinite information in it, and could exhibit an infinitely extended dynamic. Perhaps such a system would be a totally random one.

If a system of small dimensions (such as Lorenz's one, or another chaotic system known in the scientific literature) may have such complex, varied and uneven dynamics that it can even be mistaken for a random system, then imagine what can be surprisingly motley, in theory, the dynamics of a non-linear system with a large number of state variables, or with a state space with many dimensions. This fact can not surprise any longer, once we have already observed amazing behaviours in much smaller systems.

The surprise is, however, in the case of these systems that we can clearly call complex, when we observe an entirely ordered and regular behaviour, where we would have rather expected a turbulent disorder reigning indomitably. I should make examples here, I know: I will make it good in a future post. Russian Nobel prize for physics Lev Landau, who has done major and seminal studies on turbulence, attributed this phenomenology to the infinte sum of frequencies or modes in the fluid dynamics, thereby reflecting the idea that complex behaviors were to be generated by complex systems characterized by a large number, tending to infinity, of degrees of freedom. Today we know that idea was wrong, and that a complex dynamic may be generated by a simple system like the Lorenz's one, and that a complex system can show a surprisingly smooth dynamic.

If the theory of chaos is the study of disordered behavior (or complex) of small systems, the theory of complexity - assuming you can find defined boundaries - has focused mainly on the study of organized and ordered behavior of large systems (hence complex). No surprise then that a complex system can also have a complex dynamic. In my view, even a "simple" system showing chaotic behavior can be said complex, because a system and its dynamic have a very inherent link, and you can not keep the two things apart.

Strange attractions.

The most used way to graphically show the evolution of a dynamic system is to draw its trajectory in the phase space: a space with as many dimensions as state variables (or even degrees of freedom) in the system. Each point of this space represents a very specific state in which the system can come to find itself. The phase space of the pendulum - for instance - consists of the Cartesian coordinates system identified by two variables: the angle from the vertical and the angular speed of the pendulum.
Not all points of the infinite phase space (also known as state space) are possible points for the system: indeed, most of them are impossible points and the system will never reach them. Mathematically this means that those points (combination of status variables) are not a solution of the differential equations governing the dynamics.
If we consider all possible points of the system in a chronological order, we will find that it draws a trajectory line in the phase space, which looks like the flight of a fly. The study of these trajectories tells much about systems.
Generally speaking the development of a trajectory depends on the initial state and thus by the initial conditions.
Left free to evolve a system displays normally a short-term dynamic, temporary, known as transient stage, and a long-term dynamic conditions. The latter can be a null state (think of a spring that is discharged, or the swings of a pendulum at the end of its oscillations, when it doesn't receive additional forces) or a steady state. The current in a transformer is periodic indefinitely, for example.
The state we are interested to is the chaotic one. In this case, different initial conditions cause different trajectories, which never overlap nor cross both a given point. The chaotic trajectories are an infinite development of a curve in the phase space that do not ever repeat twice the same even for a tiny tract.
Should we draw it, we would see that this curve never intersects itself. If you think about it, this fact is closely related to the impredicibility of chaos: should an overlap ever occur periodicity would settle up, because from that moment on the curve will repeat itself, giving rise to a periodic state that allows easy predictions of the system dynamics.
The bizarre geometric figure drawn by this trajectory is called strange attractor: the name betrays the surprise of those who saw it for the very first time and did not know how else that rich dynamic could be called. The set of all states that give raise to trajectories evolving with this curious geometry is called the attractor's basin of attraction.

A chaotic system can produce more than one strange attractor, with basins of attraction hardly distinguishable from each other or even overlapped - such as riddled basins, which are fractal basins. In these cases, a dynamic system might jump from one to another attractor due to external causes and perturbations. My words fail to depict the complexity that can surface from a similar case.

But the question is not purely theoretical, rather it is very real and closely concerns the world we live in. An example might be the climate, which so often deserves the honor of the first columns in these years.

What is Chaos?

We have been hearing that deterministic chaos was found in this or that natural phenomenon for over 20 years now, since the moment we opened our eyes almost suddenly to a reality that we had decided not to observe: that of real non-linear phenomena (ie almost all real phenomena tout court).
Personally one of the findings that impressed me most is chaos in the human heartbeat: in a healthy individual dynamics of the heartbeat as you can measure by means of an electrocardiogram is chaotic, while a regular dynamic is often symptom of a disease. Even the electroencephalogram of a healthy person has shown the typical and unambiguous marks of chaos.
What does it mean that a natural phenomenon shows chaotic behaviour? How is that a natural system is chaotic?
If we know the equations of a system then we can rely on some tools such as mathematics analysis, computer simulation that allows us to study how the behaviour of the system changes with its parameters.
The onset of chaos is always preceded by well detectable mathematical phenomena, points of transition from regular to complex behaviour: I mean, for example, period doubling that happens at given intervals and that ultimately flows into true chaos. These paths can be viewed in the so called bifurcations diagrams.
An universal characteristic of chaos is that the convergence ratio in the sequence of bifurcations (the so-called route to chaos), which is observed as the control parameter varies, is almost constant and is constant as the parameter approaches the infinte. This universal ratio is called delta of Feigenbaum, and allows us to know exactly when the next bifurcation will happen in a system that evolves towards chaos. So if we see a constant ratio that is close to the Feigenbaum's delta in the intervals between one bifurcation and the next one, we have an important clue that system has the characteristics of chaos in itself and is evolving towards a chaotic dynamic as consequence of the change of one of its structural parameters (eg tension of a diode).
The calculation of the so-called Lyapunov's exponents yields the conclusion that two close trajectories of the system differ on average: the mathematical consequence of this fact is the system's chaoticism for some given parameters (for other parameters the same system may not be chaotic). This is of course possible only if you know the equations of the system, in which case you can draw the exponents of Lyapunov mathematically.
If we do not know the system's equations, then we must carry out a measurement of its dynamic based on a series of samples long enough. On this series of measures we can quantitatively estimate some parameters: these are the coefficients of Lyapunov, or the Kolmogorov's entropy, or fractal dimension, or the size of correlation. These parameters can tell us whether we have chaos.
Finally we can say chaos has well detectable traces, and its traces have been discovered in many natural systems, whether living or not.

This should unanmbiguously answer the question: what is chaos. Yet it tells as little of what chaos means.

Some philosophical aspects about chaos.

Pierre Simon Laplace was one of the mathematicians who mostly influenced on Western thought. He was probably the supreme representative of scientific determinism.
I imagine that by now many of us have come to realize how deeply the deterministic thought - born within the calssic post-newtonian science - influenced our civilization, our way of thinking and feeling.
He wrote in 1812:

An intellect which at a certain moment would know all forces that set
nature in motion, and all positions of all items of which nature is composed, if
this intellect were also vast enough to submit these data to analysis, it would
embrace in a single formula the movements of the greatest bodies of the universe
and those of the tiniest atom; for such an intellect nothing would be uncertain
and the future just like the past would be present before its eyes.

Looking back behind us it is not difficult to note that, starting from the end of the eighteenth century and throughout the twentieth century, the common perception of the future collective and individual has often been marked by an aura of inevitability, predetermination (scriptum est): the future seen as something that moves from starting state perfectly defined towards a destination just as perfectly determined a priori. The scientific causality has clashed with religious finality, overwhelming it.
Many scientific discoveries seemed to corroborate that conclusion (at least until the advent of quantum mechanics with its portfolio of probability and uncertainty), with the result that - if not consciously, at least subconsciously -- few have shown any skepticism.
It would be interesting to develop the theme of political, social and cultural relapses of this trend of the century just past, which still influences our way of thinking and acting.
The deterministic vision that leaves no escape to free will is broken up at the end of last century due to some simple and (moreover) deterministic systems - the chaotic ones - rushing into the scientific landscape: despite their determinism these systems show a behaviour entirely impredictable even for an Intelligence, such as the one imagined by Laplace, which had knowledge of all the physical data of universe at any given moment. This Intelligence, even if aware of these data with infinite precision, yet it could not have present all the future and all the past before its eyes, because it could not find the closed-form solution of the differential equations governing a chaotic system.
So there are namely deterministic systems that exhibit a behavior that is impredicibile or indeterminabile: what sounds an oxymoron is actually fully supported by mathematics.

"Therefore, even God must let [the dynamics of] these chaotic systems evolve to
see what will happen in the future. There is no shortcut to the prediction for
chaotic systems."

Robert C. Hilborn - Chaos and Nonlinear Dynamics.

Lorenz's Butterfly.

It was 1960 when Edward Lorenz developed a model for forecasting weather conditions at first sight rather simple. He must have considered it simple too.
The model could be written as a system of nonlinear differential equations, quite simple, which however could not find a solution in the closed form. In other words it was not possible to integrate its differential equations, so as to make evident time dependence of the three variables.
This explanation would have helped to write each of the three variables as a function of time (eg y = f (t)) - which is tantamount to find a closed-form solution for the system of equations, as the mathematicians love to say, and that would have helped get a punctual forecast of the weather, or at least a prediction consistent with the assumptions Lorenz made in the development of its model.
This approach was perfectly compatible with the physical thought that dominated almost the entire last century, that all phenomenologies might be considered within the mathematic determinism, becaus the observable reality is an expression of some stability form of the observed system. And I apologise for the extreme synthesis of these concepts.
Although closed-form solution cannot be found, the system developed by Lorenz can be solved numerically, with the help of computer. The geometric representation - in the space of phases - of the trajectories of state that emerges from these simulations is surprising, and is today the most famous example of a chaotic system. The trajectories never repeat, neither two of them will ever cross or overlap.
The system is unpredictable, because you can not write the integral functions, nonetheless it is stable and shows some regularity. A hidden order, impossible to imagine a priori.
The system is sensitive to initial conditions: the smallest variation of these can give rise to a dynamic evolution completely different from what you would have if the change had not been there.
Then the famous reflection of Edward Lorenz:

can the beat of wings of a butterfly in Brazil generate a hurricane in Texas?

The "simplicity" of the simple pendulum.

Anyone who has done a course in physics at secondary schools met the so-called "simple pendulum", one of the most commonly used examples in teaching.
Known since antiquity, it has inspired the fundamental insights of Galileo Galilei that later brought the great man in the formulation of the properties of isocronism, and opened the way for concepts of motion and momentum that are the foundation of classic mechanics.
Apart from these historical notes, since our first meeting with the pendulum what we notice most is its "simplicity": the motion of the pendulum can be described by means of a simple equation. Depending on certain initial conditions - angle, mass, length - you can predict all the future behaviour of the pendulum without possibility of errors. The behaviour is periodic in time, provided you can neglect some forces that have no direct relation with the mechanism itself, but are inevitable in a real pendulum: friction and resistance.
However, you can incorporate these components into the motion of the pendulum, which from ideal and linear becomes real (at least a little' more, because the whole mass is considered as condensed in a single point and concentrated in the lower summit of the shaft) and non-linear. The equation is significantly modified, but at first glance one could not say that the effects may disrupt our first understanding of the pendulum: it should remain a simple and absolutely predictable dynamic system.
I invite you to make direct experience of such pendulum playing with the simulator available on the website MyPhysicLab.com.
Depending on the choice of system parameters the behaviour of the pendulum becomes very similar to a totally random behavior, although it remains absolutely deterministic. This is a chaotic dynamics, where you can not predict the kinematic conditions of the system (e.g. speed, position, acceleration) even if initial conditions are well-known.
This is apparently in disagreement with the deterministic nature of the pendulum.
If one looks at the trajectories of state variables of the system they never repeat: this can be demonstrated mathematically. There is no periodicity for the oscillations of the pendulum. The trajectories will never overlap, infinitely. The state variables are no longer constrained to repeat the same routes, but free to experience an infinite variety of possible trajectories within the state space. The pendulum, from simple, has become something much more complex.