Nonlinear dynamics, chaos and thermal effects

Beyond Newtonian physics, two scientific revolutions marked the last century, each associated with the fact that a certain quantity, previously considered infinite, turned out to be finite.

The first revolution is the consequence of the finite nature of the speed of light, a constant independent of the observer, giving rise to the Einsteinian world of relativity. The universe of relativity, or at least of special relativity, is deterministic and predictable.

The second revolution results from the fact that there is a minimal quantum of action h/2π (the inverse of this action is therefore not infinite), leading to quantum mechanics. The evolution of a quantum state is governed by a deterministic and predictable equation (the Schrödinger equation) between two measurements, but the measurement process itself (wave packet reduction) is non-deterministic and non-predictable, except in a statistical sense.

In this article, we focus on the dynamics of nonlinear systems, and in particular on the chaotic behaviours that reflect what some authors call a third revolution. In this context, it appears that irregular behavior does not necessarily result from the interaction between a large number of degrees of freedom ( ). In particular, in the case of one-dimensional nonlinear applications, a single degree of freedom (a single variable) may be sufficient to generate "deterministic chaos" behavior combining determinism and unpredictability. The unpredictability (beyond a certain time horizon) of chaotic behavior is the consequence of its sensitivity to initial conditions, which implies that, in a phase space, two chaotic trajectories, initially close to each other, move exponentially apart over time. There is therefore a predictability horizon that is not infinite.