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Dynamical systems and attractors

A (dissipative) dynamical system can be described by a set of $n$ ordinary differential equations:
\frac{d\vec{X}(t)}{dt} = \vec{F}(\vec{X}(t)), \quad \vec{X} \in
\end{displaymath} (2.1)

and when time is not continuous but assumes only discrete values by difference equations (maps):
\vec{X}_{n+1} = \vec{F}(\vec{X}_{n}), \quad \vec{X} \in \Re^{n}
\end{displaymath} (2.2)

For dissipative systems, phase-space volumes are contracted by the time evolution. The trajectory (orbit) of such a system typically settles on a subset of $\Re^{n}$, called an attractor [Eckmann and Ruelle, 1985]. For example, the attractor of a damped pendulum is a fixed point, while the attractor of a periodically forced pendulum is a limit cycle. On a strange attractor, trajectories wander in an apparently erratic manner and they are highly sensitive to initial conditions. For a zoo of strange attractors, see Ref. [Holden and Muhamad, 1986].