Note 8

Dynamical systems can be divided in :

Stable systems (a small difference in [assessing the] starting condition corresponds to a small difference in the evolution of the system [also the longer-term evolution] ).
If the initial condition is statistically depicted as a blob in phase space, then upon evolution of the system the shape of the blob (and also its volume) remains (virtually) unchanged.
Unstable systems (chaotic systems) (a small difference in [assessing the] starting condition makes a very large difference with respect to the system's long-term evolution).
If the initial condition is statistically depicted as a blob in phase space, then upon evolution of the system the shape of the blob (but not its volume) changes dramatically. With fine branches it speads out over all regions of phase space. There are several degrees of instability : we have so-called mixing-flows, and we have K-flows, which latter have a higher degree of instability than mixing-flows.

Another classification is that in : ergodic systems and non-ergodic systems.
While in the non-ergodic case the system explores only a limited part of its phase space (as it is the case in a pendulum), an ergodic system eventually explores all areas of its phase space. Some stable systems (i.e. systems not sensitive to small differences in initial conditions) are ergodic (while others are not), while all unstable systems are ergodic.

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