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Definition of Chaos

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:
  1. 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.
  2. 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.
  3. 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.

Richard Fitzpatrick 2011-03-31