Holmes, Philip; Whitley, David 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. Cited in 12 Documents 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 Keywords:unstable manifolds; Duffing’s equation; Poincaré map; constant Jacobian determinant; perturbations of one-dimensional maps; strange attractors; bifurcation; horseshoes PDF BibTeX XML OpenURL References:  Andronov, A. A.; Vitt, E. A.; Khaiken, S. E.: Theory of oscillators. (1966) · Zbl 0188.56304  Bowen, R.: On axiom A diffeomorphisms. Amer. math. Soc. regional conference series in math #35 (1978) · Zbl 0383.58010  Chillingworth, D. R. J.: Differential topology with a view to applications. (1976) · Zbl 0336.58001  Collet, P.; Eckmann, J. P.: Iterated maps on the interval as dynamical systems. (1980) · Zbl 0458.58002  Collet, P.; Eckmann, J. P.; Lanford, O. 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(1980) · Zbl 0497.58007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.