This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

The transition to chaos for a Hamiltonian (conservative) system is somewhat

different than that for a dissipative system (recall [2.5]). In an integrable

(nonchaotic) Hamiltonian system, the motion is "quasiperiodic", that is motion

that is oscillatory, but involves more than one independent frequency (see

also [2.12]). Geometrically the orbits move on tori, i.e. the mathematical

generalization of a donut. Examples of integrable Hamiltonian systems include

harmonic oscillators (simple mass on a spring, or systems of coupled linear

springs), the pendulum, certain special tops (for example the Euler and

Lagrange tops), and the Kepler motion of one planet around the sun.

It was expected that a typical perturbation of an integrable Hamiltonian

system would lead to "ergodic" motion, a weak version of chaos in which all of

phase space is covered, but the Lyapunov exponents [2.11] are not necessarily

positive. That this was not true was rather surprisingly discovered by one of

the first computer experiments in dynamics, that of Fermi, Pasta and Ulam.

They showed that trajectories in nonintegrable system may also be surprisingly

stable. Mathematically this was shown to be the case by the celebrated theorem

of Kolmogorov Arnold and Moser (KAM), first proposed by Kolmogorov in 1954.

The KAM theorem is rather technical, but in essence says that many of the

quasiperiodic motions are preserved under perturbations. These orbits fill out

what are called KAM tori.

An amazing extension of this result was started with the work of John Greene

in 1968. He showed that if one continues to perturb a KAM torus, it reaches a

stage where the nearby phase space [2.4] becomes self-similar (has fractal

structure [3.2]). At this point the torus is "critical," and any increase in

the perturbation destroys it. In a remarkable sequence of papers, Aubry and

Mather showed that there are still quasiperiodic orbits that exist beyond this

point, but instead of tori they cover cantor sets [3.5]. Percival actually

discovered these for an example in 1979 and named them "cantori."

Mathematicians tend to call them "Aubry-Mather" sets. These play an important

role in limiting the rate of transport through chaotic regions.

Thus, the transition to chaos in Hamiltonian systems can be thought of as the

destruction of invariant tori, and the creation of cantori. Chirikov was the

first to realize that this transition to "global chaos" was an important

physical phenomena. Local chaos also occurs in Hamiltonian systems (in the

regions between the KAM tori), and is caused by the intersection of stable and

unstable manifolds in what PoincarĂ© called the "homoclinic trellis."

To learn more: See the introductory article by Berry, the text by Percival and

Richards and the collection of articles on Hamiltonian systems by MacKay and

Meiss [4.1]. There are a number of excellent advanced texts on Hamiltonian

dynamics, some of which are listed in [4.1], but we also mention

Meyer, K. R. and G. R. Hall (1992), Introduction to Hamiltonian dynamical

systems and the N-body problem (New York, Springer-Verlag).

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