Integrable vs. Chaotic
dynamics in a nutshell
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
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).
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.
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.