Classical chaos

Chaos in conservative systems

Integrable vs. Chaotic dynamics in a nutshell

In classical mechanics, a canonical dynamical system is said to be integrable if it has as many independent conserved quantities (energy, momentum, angular momentum) as degrees of freedom (dimension d of the configuration space). The motion of a particle is then restricted to a d-dimensional invariant torus.
On the opposite, if the only conserved quantity is the energy, the motion can a priori take place on the whole (2d-1-dimensional) energy surface (which we suppose to be situated in a bounded region of phase space). Such a system
is said to be ergodic if a "typical" initial point will visit all regions of the energy surface, therefore being at the "antipodes" of an integrable system. The word "typical" refers here to the invariant (Liouville) measure on the energy surface: "typical" means "for all initial point on the energy surface except a set of zero measure". If the trajectories are very sensitive to initial conditions, the dynamics is said to be unstable, or hyperbolic, and for large times the motion becomes genuinely chaotic. This generally encompasses several properties:

  • the motion around each trajectory is characterized by nonzero Lyapunov exponents, which measure the unstability of the motion.
  • Mixing: let us consider the evolution of a distribution density on the energy surface, rather than of single points. The motion is mixing if any initial density becomes, for large times,  scattered throughout the full energy surface. It converges to the uniform invariant density, which is the "equilibrium density" for the system.
  • Complexity: the  periodic trajectories form a discrete set on the energy surface. However, their number grows exponentially fast when one includes longer and longer orbits. The (positive) factor in the exponential is called the topological entropy: it thus characterizes the complexity of the set of periodic orbits, which form the "backbone" of the motion.
  • Through a Markov partition of the energy surface, the (deterministic) dynamics is in one-to-one correspondence with a stochastic Markov process
Maps as models of chaotic dynamics
Since any 1-dimensional autonomous Hamiltonian system preserves energy, the motion is necessarily integrable. To obtain a chaotic Hamiltonian dynamics, one needs to consider time-varying Hamiltonians, or consider autonomous Hamiltonians of at least 2 degrees of freedom (both are formally equivalent, the time dependence can be understood as adding one degree of freedom).
In the latter case, the Hamiltonian flow takes place
on energy surfaces of dimension 3, which is not very easy to visualize. It is very convenient to consider the dynamics induced on a Poincaré section, which is a 2-dimensional surface in the energy shell, transverse to the flow. Each trajectory is described through its successive intersections with that section: one obtains a Poincaré map on the section, which will contain all the "nontrivial features" of the flow (ergodicity, hyperbolicity, mixing) as long as the section has been correctly chosen.

Probably the simplest such Hamiltonian system is the free evolution of a particle inside a plane billiards, with specular reflection at the boundary. The natural Poincaré map for this system is the bounce map, which describes the successive bouces on the boundary. The 2 coordinates of this section are the position on the boundary and the sinus of the angle of bouncing (with respect to the normal direction). One obtains a canonical map on that reduced phase space (the plot below represents the bounce map for the limaçon billiard).
We represent one trajectory of the (chaotic) stadium billiard

Bounce map for the stadium billiard: the points belong to a single long orbit

One step further, one can also consider a canonical map on a 2-dimensional phase space, independently of any underlying Hamiltonian flow. The simple examples of hyperbolic systems are indeed maps on the 2-dimensional torus, like Arnold's Cat map, or the baker's map.

This picture (courtesy of Leon Poon)
represents one iteration of "Arnold's cat" map on the 2-torus

Mixed phase space
Integrable systems, as well as purely chaotic systems, are "rare" among all conservative dynamical systems. A typical system will rather be "inbetween", which means that the energy surface roughly splits between two subsets: stable islands where the motion is regular and organized into invariant tori, and a chaotic sea. The junction between the two regions is organized into complicated hierarchical resonance structures.

Long trajectories of the bounce map of the limaçon billiard, for various initial positions. If the initial point is inside an elliptic island, its trajectory is restricted on a 1-d curve (deformed circle around a stable fixed point). On the opposite, an initial point in the "chaotic sea" will explore the whole sea.
 This picture was produced by Arnd Bäcker.


Retour en haut