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.