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 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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