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New Jersey Institute of Technology
New models for chaotic dynamics
2005, vol. 10, no. 3, pp. 307-321
New type of strange chaotic "attractor" models for discrete dynamical systems of dimension greater than one are constructed geometrically. These model, unlike most of the standard examples of chaotic attractors, have very complicated dynamics that are not generated by transverse (homoclinic) intersections of the stable and unstable manifolds of fixed points, and may include transverse heteroclinic orbits. Moreover, the dynamics of these model are not generally structurally stable (nor $\Omega$-stable) for dimensions greater than two, although the topology and geometry of the nonwandering set $\Omega$ are invariant under small continuously differentiable perturbations. It is shown how these strange chaotic models can be analyzed using symbolic dynamics, and examples of analytically defined diffeomorphisms are adduced that generate the models locally. Possible applications of the exotic dynamical regimes exhibited by these models are also briefly discussed.