How does chaos arise in Hamiltonian systems?

Question

I have a question about how chaos arises in Hamiltonian systems. I've been taking a upper year undergrad class on dynamics and it has focused on chaos a lot. So far however we have only seen the more standard signatures of chaos. We've studied the Lorenz system and the damped driven pendulum which are both dissipative. As a result they have strange attractors and a negative sum to their Lyapounov exponents.

All this falls apart for me when I think about Hamiltonian systems and more specifically Liouville's Theorem. If volumes in phase space are conserved for Hamiltonian systems, how does chaos arise? The solutions cannot fall into a strange attractors and I expect the Lyapounov spectrum to be symmetric.

I have read answers that make references to KAM theory, but I don't understand how that is relevant. KAM theory seems to deal with when perturbations preserve toroidal phase spaces, but if the perturbed solutions become chaotic there still lies the question as to whether or not that is in fact a solution. Or am I missing something that has to do with perturbation theory?

Answer 1

Let me not claim that any system that can transit into Chaos must be "Hamiltonian". But there is a very large class of such systems, to which you probably intuitivley refer, and those are systems that can be represented at microscopic level by Fokker Planck equations. Many classical cases of Chaos are of this family, such as in fluid dynamics, pattern formation, chemical reaction, mechanics, laser theory...

The Fokker Planck equations, roughly said, resemble somehow partition functions encapsulating the Hamiltonian. This is when you could identify, for instance, a potential.

Let me suggest you two books that will help you to learn this aspects in detail:

Stochastic Methods: A Handbook for the Natural and Social Sciences

Synergetics: Introduction and Advanced Topics

While the first will help you to learn about the Fokker-Planck equations on microscopic level the second helps you to understand the connect to how Chaos arises in such Hamiltonian systems via phase transition. The second book however not only describes Hamiltonian systems.

In general to your question, how chaos arises in Hamiltonian systems, there ara several factors some of the we collected in the following discussions:

>>>here

and >>>here

What is very basic requirement, is that your non-linear system has at least 3 instable eigen-modes while all other eigen-modes stable. Once you vary the parameters (eigenvalues) of the instable eigen-modes phase transition and bifurcations of the stable solutions of the system can occur in some cases where system behaves then chaotic. That is if for instance a system before phase transition has two instable eigen-modes and after the phase transition one more parameter (eigenvalue) gets instable and the system changes to a phase where 3 eigen-modes are instable. However this is just a scetch, and you need to rigorousely read into the stuff.

Answer 2

I can offer some qualitative understanding presented in this Wiki article https://en.wikipedia.org/wiki/Measure-preserving_dynamical_system "In terms of physics, the measure-preserving dynamical system often describes a physical system that is in equilibrium, for example, thermodynamic equilibrium. One might ask: how did it get that way? Often, the answer is by stirring, mixing, turbulence, thermalization or other such processes. If a transformation map T describes this stirring, mixing, etc. then the (Hamiltonian) system is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator (see Perron–Frobenius theorem) that have eigenvalue less than one"

More precise definition can be given in the following way: We know that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and a wandering (dissipative) set. This is called Hopf decomposition (read about this on Wiki if u like). Now imagine that we have a system which eventually arrives to some kind of chaotic form in terms of strange attractor,e.g.,fluid in a tube which transitions to turbulence. Strange attractor has zero Lebegue measure in the initial phase space which it's embedded in (see more in this related topic What is the meaning of volume of dynamical system). In my opinion this strange attractor is defined in the 'Hamiltonian part' of the original system. And its measure is not changing once it's come there. So, turbulent fluid would be described as a Hamiltonian system and can be characterised as 'inviscid fluid' although viscocity has not gone away. Fluid which is only transitioning to turbulence is still a dissipative system.

Found one more example: "A commonplace informal example of Hopf decomposition is the mixing of two liquids (some textbooks mention rum and coke): The initial state, where the two liquids are not yet mixed, can never recur again after mixing; it is part of the dissipative set. Likewise any of the partially-mixed states. The result, after mixing (a cuba libre, in the canonical example), is stable, and forms the conservative set; further mixing does not alter it. In this example, the conservative set is also ergodic: if one added one more drop of liquid (say, lemon juice), it would not stay in one place, but would come to mix in everywhere."

Also I would add that classical Hamiltonian systems cannot exist in 'pure' form, because in nature we always have some dissipation. So all classical systems are dissipative. On the other hand, atoms and electrons are able to oscillate without any forcing and dissipations, so this systems are inherently Hamiltonian. But up to my knowledge this is about stochastic chaos and completely different story :)

p.s.: this is my first answer :) I would glad to see where I'm mistaken :)

UPD

Significantly explored my understanding since last time i've been here :) Very briefly: the key feature is Arnold diffusion and Chirikov's resonance overlap. Regarding the former: there is actually a conventional wisdom that KAM theory is a good starting point, when an integrable system is pertubed. So that as a result of this pertubation we are getting some 'stochastic layers' between invariant tori's (because some tori's are destroyed). While dimensionality of phase space is less than 5, those stochastic layers are isolated from each other (because of tori's). But when, say, the pertubation force is increasing more and more and the dimensionality is 5 or higher than the stochastic layers can 'diffuse' into each other, so chaos is able to develop (but with different speed!). https://en.wikipedia.org/wiki/Arnold_diffusion

Regarding resonances: article https://en.wikipedia.org/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem is a very good starting point. Than look this one Onorato, M.; Vozella, L.; Proment, D.; Lvov, Y. (2015). "Route to thermalization in the α-Fermi–Pasta–Ulam system" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 112 (14): 4208–4213. arXiv:1402.1603. Bibcode:2015PNAS..112.4208O. doi:10.1073/pnas.1404397112. PMC 4394280. PMID 25805822. External links