The term ``Hamiltonian for dissipative systems'' is contradictory, since the Hamiltonian is the generator of time translations, makes sense if the system is time translation invariant and the term  ``dissipative'' means that the system isn't time translation invariant. IF there exists a stationary configuration, i.e. that is, itself, invariant under time translations, for the system, that isn't, just, a point, then the generator of time translations in that subset may be able to play the role of a Hamiltonian, provided the usual conditions can be adapted to this case. 

In the paper on the Helmholtz oscillator, for instance, the system is, in fact, Hamiltonian-that's what the existence of the  change of variables *means*.  The dissipative properties, are, thus,  coordinate artifacts, if a change of variables exists that makes the system Hamiltonian. So the question, really, is, what are the perturbations of Hamiltonian systems, that are, in fact, coordinate artifacts.

This is, also, further, explored here: http://arxiv.org/abs/1106.6034

The properties of exact solvability and of integrability of the equations of motion are not equivalent, but, quite,  distinct and don't have anything, beyond history, to do with one another.  The fundamental concept is that of integrability, while that of exact solvability just implies that the solution can be expressed in terms of certain functions, that don't enjoy any particular mathematical property, beyond familiarity. 

You can find sets of parameters for which some simple nonlinear oscillators such as Duffing oscillator or Helmholtz oscillator with dissipative terms are integrable. Similarly the Lorenz system can be integrable for special choices of parameters.

It could perhaps be more helpful to search on internet for non-Hamiltonian integrable systems instead of dissiplative integrable systems. There are many examples, for instance, integrable cases of the Lorenz systems (see e.g. the first link below and references therein), or system (19) from the second link below. For some general results in this direction see e.g. third,  fourth and fifth links below.

Another broad class of examples of dissipative integrable systems is provided by the integrable Lotka--Volterra systems, see e.g. fifth, sixth and seventh links below and references therein.

Article Another integrable case in the Lorenz model

Article Coupling constant metamorphosis as an integrability-preservi...

Article Integrability and Nonintegrability of Dynamical Systems / A. Goriely.

Article Geometry of integrable non-Hamiltonian systems

Article On the complete and partial integrability of non-Hamiltonian systems

Article Integrable Lotka-Volterra systems

Article The Integration of Three-Dimensional Lotka-Volterra Systems

However, one should be careful: if the system is integrable, it isn't dissipative-and vice versa.  So it would be more precise to state that such systems depend on parameters and, when these are varied, undergo phase transitions from a phase, where dissipative properties characterize the physics to a phase where conserved quantities in involution characterize the physics. 

Hi Stam,

What you say can be misleading. I was taught to think like that as an student. I found that assuming some conceptualizations about hamiltonian, conservative and dissipative systems was not of much use to me. And to my surprise I learned later that first, you can construct hamiltonians for dissipative systems (in other works, there is a common confusion about being hamiltonian and being conservarive) and furthermore many "dissipative" dynamical systems, or if you prefer, dynamical systems with dissipative terms (as friction terms in the case of nonlinear oscillators) or systems with a negative divergence like the Lorenz system can be integrable (of course not for all choices of parameters).

See:

https://www.researchgate.net/publication/258277174_Integrals_of_motion_for_the_Lorenz_system

http://arxiv.org/pdf/nlin/0404001.pdf

https://www.researchgate.net/publication/231019631_Integrability_and_symmetries_for_the_Helmholtz_oscillator_with_friction

https://www.researchgate.net/publication/231181354_Comments_on_the_Hamiltonian_formulation_for_linear_and_non-linear_oscillators_including_dissipation

Best wishes,

Miguel AF Sanjuan

Article Integrals of motion for the Lorenz system

Article Integrability and symmetries for the Helmholtz oscillator wi...

Article Comments on the Hamiltonian formulation for linear and non-l...

The term ``Hamiltonian for dissipative systems'' is contradictory, since the Hamiltonian is the generator of time translations, makes sense if the system is time translation invariant and the term  ``dissipative'' means that the system isn't time translation invariant. IF there exists a stationary configuration, i.e. that is, itself, invariant under time translations, for the system, that isn't, just, a point, then the generator of time translations in that subset may be able to play the role of a Hamiltonian, provided the usual conditions can be adapted to this case. 

In the paper on the Helmholtz oscillator, for instance, the system is, in fact, Hamiltonian-that's what the existence of the  change of variables *means*.  The dissipative properties, are, thus,  coordinate artifacts, if a change of variables exists that makes the system Hamiltonian. So the question, really, is, what are the perturbations of Hamiltonian systems, that are, in fact, coordinate artifacts.

This is, also, further, explored here: http://arxiv.org/abs/1106.6034

The properties of exact solvability and of integrability of the equations of motion are not equivalent, but, quite,  distinct and don't have anything, beyond history, to do with one another.  The fundamental concept is that of integrability, while that of exact solvability just implies that the solution can be expressed in terms of certain functions, that don't enjoy any particular mathematical property, beyond familiarity. 

I do not quite get the question, if the system is dissipative, the energy is not a constant of motion, you are not supposed to have an integrable system, at least not one where the energy is a constant of motion.(agree with stam)

Perhaps you mean solvable.

Some dynamic projectile  problems with certain forms of air resistance are exactly solvable.Try simple forms for the friction, say proportional to v as part of the force law.

You can derive things like terminal speed , for fall in the atmosphere.