In this lecture-video (at about 37:17) on Hamiltonian dynamics, the instructor mentions that for an (Arnold-Liouville) integrable finite-dimensional Hamiltonian system one has the following:

  • Phase-space dimension: $2N$
  • Energy-hypersurface dimension: $2N-1$
  • $N$-torus solution-hypersurface dimension: $N$

He then recalls that for $N=2$ and for a given energy (i.e. for a given energy-hypersurface), the dimensionality of the solution-hypersurface (the $N$-torus) is $2$, so for a given solution-hypersurface one can separate the $3$-dimensional energy hypersurface into two parts (inside and outside). He then states that for $N>2$ such a separation isn't possible, because then the difference in dimensionality between a solution-hypersurface and an energy-hypersurface would be greater than $1$. As an example, remember that you can't separate a $3$-dimensional into two parts with a line.

After all this, he says that when $N>2$ the system's phase space becomes a "stochastic web", where solution-hypersurfaces for a given energy cover the corresponding energy in a "web"-like fashion. I would like to know more about this "stochastic web". Where can I find a proper definition of a stochastic web, and also further literature on the topic?

P.S: I didn't know if I should asked this question on MSE.


I believe the term "stochastic web" was first employed in this article by Zaslavsky, where he studied not the quasi-integrable stochasticity you're talking about, but stochastic processes in 1-dimensional systems. The reason is that most people can't see dimensions higher than three, and he was particularly interested in the fractal structure of chaos (perhaps that's why he coined the "stochastic web" term). The important point here is that "stochastic web" is not a very popular name, and it's not restricted to quasi-integrable systems: every chaotic system whose chaos doesn't cover phase space completely has stochastic webs associated to it; they're around the separatrix that covers the islands of stability. In his article Zaslavsky studies many chaotic systems and maps, trying hard to adjust initial conditions such that phase space not only presents chaotic features, but that those features be fractal in nature. The result is a paper where amazing phase space pictures can be found, even though some are not stochastic webs: the whole space space is densely covered.

Regarding chaos in integrable systems, it's elementary to prove each and every one-degree-of-freedom Hamiltonian system is integrable. This means that the only way to turn a 1-dimensional Hamiltonian system into something non-integrable is by adding perturbations, which means you have a choice to make: if you decide to add a time dependent perturbation and see the chaos that emerges, then you're absolutely sure the resultant chaotic system will not be integrable. In a 4-dimensional phase space you can take Hamiltonians that actually need no perturbation, since they are intrinsically chaotic, even though the Hamiltonian is a constant of motion. Now, in 6-dimensional phase space you get the effect you're talking about: chaos can "leak" in between two toruses and accumulate in a middle dimension, since the toruses cannot cover the whole phase space when indexed by energy... But I'd say this is a property associated to dimension, not to chaos. The definition of "stochastic web" does not change: it is a chaotic filament of phase space where dynamics is chaotic, but outside of it it is regular. I think the teacher only used the term "stochastic web" as an analogy for a region there chaos "spreads as a web", not referring to the technical jargon that was started by Zaslavsky.


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