1. The spirit of the presentation of chaos theory adopted in the present chapter is that of the beautiful review by P. Holmes: Poincaré, celestial mechanics, dynamical-systems theory and “chaos”, Physics Reports
193, 137–163 (1990).
2. Holmes is also one of the authors of a book that, together with the many times cited book by Wiggins, is a standard reference in research on dynamical-systems theory: J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, 1983 ).
3. Sects. 11.2–11.4: The generalization of the Smale-Birkhoff-Moser theorem and Melnikov’s method to higher-dimensional cases is treated in: S. Wiggins: Global Bifurcations and Chaos - Analytical Methods (Springer, 1988 ).
4. The theorem on the existence of horseshoe dynamics in the neighbourhood of a hyperbolic fixed point of a perturbed Hamiltonian system has been stated in this form in the paper by Holmes and Marsden quoted in Section 11.4. It relies on the fundamental work of Melnikov and on other refinements of the theory, mainly pursued by Holmes and collaborators. Some significant references are the following:
5. V. K. Melnikov: On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12, 1–57 (1963).