The torus illustrates further essentials at the transition to chaos\:
1. The motion may no longer be dissolved into independent periods in each of the system's dimensions, it loses symmetry - the system becomes non-integrable (for the three body problem Earth-Sun-Jupiter discovered by Poincar\u00E9 around 1900).
2. It moves along orbits that become unstable each and remains predictable from observation only until it changes to another one - whatsoever the duration and precision of studying its behaviour.
3. The resonant, but unstable periodic orbits form a dynamic skeleton
, however, which describes each finitely long aperiodic motion approximately - insofar as it is known.
Since each attractor is an entity of reduced dimensionality, some internal synchronization has to appear between quantities of the system which are otherwise independent (coordinates of its phase space). The strongly correlated motion along a sequence of unstable periodic orbits, for every short period of time, is a deeper essential of chaotic systems which makes them distinguishable from others that are simply subjected to random fluctuations. The sensible dependence of a specific motion on its initial conditions, trademark of the deterministic chaos
, follows from there.
Classical Chaos