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On the attracting set for Duffing’s equation. II: A geometrical model for moderate force and damping. (English) Zbl 0574.58024
Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica 7D, 111-123 (1983).
[For the entire collection see Zbl 0536.00007; Part I, cf. Res. Notes Math. 101, 211-240 (1984; Zbl 0554.58046)]
After a brief review of some earlier work on Duffing’s equation in the small force and damping regions, we use the results of numerical integrations to construct a geometrically defined Poincaré map which captures the qualitative features of the attracting set of larger force and damping levels. This map has a (small) constant Jacobian determinant and can be regarded as a perturbation of a non-invertible one-dimensional map. We give a partial analysis of the map and pose some important open questions regarding perturbations of one-dimensional maps and the creation of ”strange attractors” during bifurcation to horseshoes.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
58J90 Applications of PDEs on manifolds
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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