Order and chaos

We review integrable and chaotic systems and the transition from Order to Chaos. We give the various types of orbits in Stiickel potentials of 2 and 3 degrees of freedom. Then we discuss integrable systems in general. In particular we consider Li ouville systems and the Painleve test of integrability. Formal integrals (like the “third” integral) are an important tool in galactic dynamics. We discuss their construction, and the limits of their applicability by means of Nehoroshev's theory. Then we enumerate the various routes to chaos in dissipative and conservative systems. The creation of large degree of chaos follows the destruction of the last KAM (Kolmogorov, Arnold, Moser) torus and the formation of cantori. We discuss -the structure of the chaotic regions, that are filled with the asymptotic curves of the unstable periodic orbits. In systems of 3 degrees of freedom we have 3 main new phenomena: Complex instability, Collisions of bifurcations, and Arnold diffusion. Finally we discuss applications of the theory of fractals on the forms of the orbits.

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