Theory of Orbits pp 297-354 | Cite as

# Chaos

## Abstract

In any exposition of the foundations of celestial mechanics based on analytical mechanics, the extreme subject to present is the theory of the global and long-time behaviour of dynamical systems. In this area, as in many others already discussed, the contributions of astronomers and celestial mechanicians have played a major role in the development of completely new methods of approach. Once it was realized that the Laplacian dream of calculating the complete detailed evolution of an *N*-body system is an impossible task, not just for lack of skill or patience, but for deep theoretical reasons, research was aimed at finding new tools to get more general information on the possible states of a system. In this regards, it is by no means an overestimate to again cite the leading activity of Poincaré in the construction of these tools. Linking the presentation of the subject with the stress we have laid in the rest of our treatment on the problem of integrability, we see that the elaboration of the “new science” of chaos begins just at the point at which Poincaré shows how it is that a generic dynamical system has no analytical integrals of motion other than the energy and explains in what sense we must speak of chaoticity in describing the long-time evolution of non-integrable systems. From these cues, left unheeded for a long time and taken up again in the past few decades, a well-grounded theory of “chaotic dynamics” has sprung forth with manifold applications in every field of pure and applied research.

## Keywords

Periodic Orbit Invariant Manifold Unstable Manifold Homoclinic Orbit Orbit Structure## Preview

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## References

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