Chaotic behavior in a deterministic dynamical system results from the interplay in state space of two geometric mechanisms, stretching and squeezing, which conspire to separate arbitrarily close trajectories while confining the dynamics to a bounded subset of state space, a strange attractor. A topological method has been designed to classify the various ways in which stretching and squeezing can organize chaotic attractors. It characterizes knots and links formed by unstable periodic orbits in the attractor and describes their topological organization with branched manifolds. Its robustness has allowed it to be successfully applied to a number of experimental systems, ranging from vibrating strings to lasers. Knotted periodic orbits can also be used as powerful indicators of chaos when their knot type is associated with positive topological entropy and thus implies mixing in state space. However, knot theory can only be applied to three-dimensional systems. Extension of this approach to higher-dimensional systems will thus require alternate formulations of the principles upon which it is builT, determinism and continuity.
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