Dynamical systems:

A barrier is broken between determinism and randomness.

The exact sciences are based on determinism; it is the postulate that: “the state of a system at the present moment determines its future evolution” (of course in the absence of external intervention).  The mathematical study of the long-term evolution of deterministic systems is called “Dynamical Systems”. Systems whose law of evolution is probabilistic (random evolution) are the subject of "Ergodic Theory".

For the simplest systems, the evolution consists usually to tend to an equilibrium state:  the system stabilizes little by little,   or to tend to a periodic evolution (like the moon around the earth).  Until the 50s/60s, it was thought that systems with random, chaotic, unpredictable behavior were systems which depended on a very large (infinite) number of parameters, like meteorology.

At the beginning of the 1960s, scientists from various disciplines (May: biology, Henon and Lorenz: meteorology, Smale: mathematics, etc.) each in their field find examples of very simple systems, depending on very few parameters, but whose evolution seemed random, unpredictable. This is the phenomenon known in the media as "chaos theory", with its fauna of "strange attractors" and "fractals".

In the 1970s, mathematicians (Sinai, Ruelle, Bowen)  showed that the “simplest” chaotic systems had a very good description using probability. This means that, if we cannot predict how the system will have evolved at a given date, we can predict its statistical behavior, error margins, etc. Deterministic systems are then described as random systems (as “heads or tails” or “dice draw”)!

One of the current challenges of research in Dynamical Systems is to determine whether this good description of Chaotic Systems using Probabilities is a general phenomenon, or if, on the contrary, this approach only sees the simplest Dynamical Systems.

Among the works of amny other people, my work with Lorenzo Diaz and Marcelo Viana, researchers in Rio de Janeiro, attempts to geometrically characterize Dynamical Systems which have an evolution well described by probabilities, even if the system is only known approximately: in other words, it is necessary that this good probabilistic behavior resists small perturbations of the system.

--- We have shown that such a system must exhibit a weakened form of hyperbolicity (called “partial hyperbolicity” or “dominated decomposition”).

--- We are currently seeking to determine whether (reciprocally) partial hyperbolicity is sufficient to ensure good probabilistic behavior: our (partial) results point in this direction.

So, you solve equations?

In many fields of science, scientis seek to predict long-term evolution. To do this, it is first necessary to give “a complete description” of the system, that is to say, to measure quantities (temperature, pressure, position, speed) which are supposed to be sufficient to determine the future evolution. The state of the system is the data of these values: it is therefore a set of numbers.

The scientist must then find the law of evolution of the system: that is to say that, given the state of the system at time 0, he must be able to determine what its state will be at time 1 (one secont after). If he succeeds in determining the law of evolution we say to ourselves: "it's won! We can predict evolution!" ...Well yes, what: to have the state of the system after two days, just apply the law of evolution twice! And then, so on!

If you allow me a little formula...

Let's use the letter x to represent the state of the system at time t, and F to represent the law of evolution, that is to say that the state at time t+1 second is F(x).

At time t+2seconds it will be F(F(x)).

At time t+3 it will be F(F(F(x)))

At time t+25second it will therefore be F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(x)))))))))))))))))))))))))...So try asking your favorite computer to calculate this for me, when x is a real number, and F is the nice function :

F(x)= -x² +1.

But it's not t+25seconds that interests us, it's the state of the system next year, i.e. t+31536000 seconds!

In general, it is not possible to give the law of long-term evolution (next year, for the example above), and in any case, it would probably be of no use: the smallest error on the initial state x or on the law F, and the prediction would be completely wrong!

So: no I don't solve differential equations, because for chaotic systems, it is useless.

The mathematician who does dynamical systems does not seek to predict the evolution of a particular system.  He's not looking to solve a particular equation: he's looking for methods that other scientists can use so that everyone knows what to do, or what to expect, when they have to describe the long-term evolution of a particular system. The answer is often “geometric” and that’s why Dynamic Systems appealed to me:

In mathematics, a set of numbers represents a point in a geometric space: one number is a point on a line, two numbers is a point on a plane, three numbers is a point in space , 250 numbers is a point in a space of dimension 250.

Let us take the case of a space of states which would be of dimension 2. A law of evolution is then a transformation of the plane. In this plane, certain curves will be important: those which are invariant by the transformation. Drawing these curves helps structure the study of dynamics.

At the end of the 19th century, Poincaré realized that certain drawings with invariant intersecting curves implied a very complex dynamic: for example Birkhoff in the 1930s had shown that this simple drawing implied the existence of an infinity of periodic states whose period tended towards infinity!

Poincaré's drawing found its explanation in another drawing, "Smale Horseshoe", in the 50s/60s, which is the starting point of the theory of hyperbolic chaotic systems.