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There is no universally agreed definition of chaos. However, most people would accept the
following working definition:
Chaos is aperiodic time-asymptotic behaviour in a deterministic system which
exhibits sensitive dependence on initial conditions.
This definition contains three main elements:
- Aperiodic time-asymptotic behaviour--this implies the existence of
phase-space trajectories which do not settle down to fixed points or
periodic orbits. For practical reasons, we insist that these trajectories
are not too rare. We also require the trajectories to be bounded:
i.e., they should not go off to infinity.
- Deterministic--this implies that the equations of motion of the system possess no
random inputs. In other words, the irregular behaviour of the system arises
from non-linear dynamics, and not from noisy driving forces.
- Sensitive dependence on initial conditions--this implies that nearby
trajectories in phase-space separate exponentially fast in time: i.e., that the
system has a positive Liapunov exponent.