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Strange attractors

If the state space of a dynamical system is finite in extent, there will be a subset of the state space to which all trajectories are attracted as time passes, called the attractor of the dynamical system. For the simplest systems the attractor is a single point, which represents a state of equilibrium of the system. The physical interpretation is that for any initial state the system always approaches the state of equilibrium as time passes. For more complicated systems, the attractor can be a trajectory that is a closed curve, which represents a system performing regular oscillations. The physical interpretation is that for any initial state the system begins to perform regular oscillations as time passes. But in general nonlinear dynamical system of dimension three or more exhibit chaotic behaviour, and the attractor does not have a simple geometric description: the long-term behaviour of the system appears to be unpredictable, as with the double pendulum. For a chaotic system the attractor is called a strange attractor.

The most famous strange attractor is the Lorenz attractor, which is associated with a three-dimensional dynamical system, studied by Edward Lorenz in 1963, in connection with a problem in meteorology. See an image of the Lorenz attractor.