How does chaos arise in Hamiltonian systems? 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 Lyaponov exponents. 

All this falls apart for me when I think about Hamiltonian systems and more specifically Louisville'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 Lyaponov 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 toriodal 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?
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.
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.