Mechanics pp 317-379 | Cite as

Stability and Chaos

  • Florian A. Scheck


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.


Vector Field Hamiltonian System Equilibrium Position Chaotic Behavior Chaotic Motion 
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  1. Arnold, V.I.: Catastrophe Theory (Springer, Berlin, Heidelberg 1986 )Google Scholar
  2. Berg, 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
  3. Chirikov, B.V.: A Universal Instability of Many-Dimensional Oscillator Systems, Physics Reports 52, 263 (1979)MathSciNetADSCrossRefGoogle Scholar
  4. Collet, P., Eckmann, J.P.: Iterated Maps on the Interval as Dynamical Systems ( Birkhäuser, Boston 1980 )MATHGoogle Scholar
  5. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems (Benjamin Cummings, Reading 1986) Feigenbaum, M.: J. Stat. Phys. 19, 25 (1978) and 21, 669 (1979)Google Scholar
  6. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin, Heidelberg 1983 )Google Scholar
  7. Hénon, M., Heiles, C.: Astron. Joum. 69, 73 (1964)ADSCrossRefGoogle Scholar
  8. Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra ( Academic, New York 1974 )Google Scholar
  9. Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems (Springer, Berlin, Heidelberg 1982 ) Peitgen, H.O., Richter, P.H.: The Beauty of Fractals, Images of Complex Dynamical Systems (Springer, Berlin, Heidelberg 1986 )Google Scholar
  10. Ruelle, D.: D.: Elements of Differential Dynamics and Bifurcation Theory ( Academic, New York 1989 )Google Scholar
  11. Schuster, H.G.: Deterministic Chaos, An Introduction (Physik-Verlag, Weinheim 1984) Wisdom, J.: Chaotic Behaviour in the Solar System, Nucl. Phys. B (Proc. Suppl.) 2, 391 (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Florian A. Scheck
    • 1
  1. 1.Institut für Physik, Fachbereich PhysikJohannes Gutenberg-UniversitätMainz 1Fed. Rep. of Germany

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