In this chapter we study a larger class of dynamical systems that include but go beyond Hamiltonian systems. We are interested, on the one hand, in dissipative systems, i.e. systems that lose energy through frictional forces or into which energy is fed from exterior sources, and, on the other hand, in discrete, or discretized, systems such as those generated by studying flows by means of the Poincaré mapping. The occurence of dissipation implies that the system is coupled to other, external systems, in a controllable manner. The strength of such couplings appears in the set of solutions, usually in the form of parameters. If these parameters are varied it may happen that the flow undergoes an essential and qualitative change, at certain critical values of the parameters. This leads rather naturally to the question of stability of the manifold of solutions against variations of the control parameters and of the nature of such a structural change. In studying these questions, one realizes that deterministic systems do not always have the well-ordered and simple behavior that we know from the integrable examples of Chap. 1, but that they may exhibit completely unordered, chaotic behavior as well. In fact, in contradiction with traditional views, and perhaps also with one’s own intuition, chaotic behavior is not restricted to dissipative systems (turbulence of viscous fluids, dynamics of climates, etc.). Even relatively simple Hamiltonian systems with a small number of degrees of freedom exhibit domains where the solutions have strongly chaotic character. As we shall see, some of these are relevant for celestial mechanics.
KeywordsVector Field Hamiltonian System Equilibrium Position Chaotic Behavior Chaotic Motion
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- Arnol’d, V.I.: Catastrophe Theory (Springer, Berlin, Heidelberg 1986 )Google Scholar
- Berge, P., Pomeau, Y., Vidal, C.: Order within Chaos; Towards a Deterministic Approach to Turbulence (Wiley, New York 1986 ) French original ( Hermann, Paris 1984 )Google Scholar
- Collet, P., Eckmann, J.P.: Iterated Maps on the Interval as Dynamical Systems ( Birkhäuser, Boston 1990 )Google Scholar
- Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin, Heidelberg 1990 )Google Scholar
- Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra ( Academic, New York 1974 )Google Scholar
- Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems (Springer, Berlin, Heidelberg 1982 )Google Scholar
- Peitgen, H.O., Richter, P.H.: The Beauty of Fractals, Images of Complex Dynamical Systems (Springer, Berlin, Heidelberg 1986 )Google Scholar
- Ruelle, D.: Chance and Chaos ( Princeton University Press, New Jersey, 1991 )Google Scholar
- Ruelle, D.: Elements of Differential Dynamics and Bifurcation Theory ( Academic, New York 1989 )Google Scholar
- Schuster, H.G.: Deterministic Chaos, An Introduction ( Physik-Verlag, Weinheim 1987 )Google Scholar