Can a molecular dynamics simulation enter attractors like stable limit cycle?

Question

In my rough understanding Molecular Dynamics using Classical Newtonian mechanics is a 6N dimensional non linear system. 6N dimension because we have 3 position vectors and 3 momentum vectors for each N particles. Nonlinearity because of the terms in force fields as far as I understand. In principle this system can exhibit chaos. So my question is the following: Is there any scenario where by the system enters a stable/ semi stable limit cycle or periodic trajectory? I have read that the system cannot go into a point attractor since the volume of probability density function must be conserved and cannot become zero. If the system can enter periodic oscillations or limit cycle what does that imply?

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UPDATE : Can someone give me reference suggestions like books, papers, video lectures or any other document links related to this topic? I also wanted to know whether the assumption of ergodicity in the system prohibits limit cycles ?

( @Wrzlprmft answered my question about ergodicity in chats. I am copying it here :

In case of a system with attractors, you can consider ergodicity as follows: On the one hand you take one arbitrary condition, evolve it for some time (to discard transients) and then start pooling the ensuing trajectory for the time average. On the other hand you take a bunch of arbitrary initial conditions, evolve them for some time (to discard transients) and pool the resulting states (one per initial condition) to obtain the ensemble average. In case of a single attractor, the time and ensemble average are the same, namely the average over all states on the attractor. In case of two or more attractors, the time average only reflects the states of one attractor, while the ensemble average reflects the states of all attractors. Hence such systems are not ergodic. )

Answer 1

Is there any scenario where by the system enters a stable/ semi stable limit cycle or periodic trajectory?

A limit cycle is a periodic trajectory.

That being said, despite all non-linearity, we are still looking at a Hamiltonian system here. As you already mentioned, phase-space volume is conserved in such systems (Liouville’s theorem). Now, a defining quality of an attractor is that it has a basin of attraction that becomes the attractor through time evolution. As the basin of attraction has to be a true and open superset of the attractor, it’s volume inevitably shrinks upon time evolution. This contradicts Liouville’s theorem and hence there cannot be any attractors in Hamiltonian systems – they only exist in dissipative systems. (Also see this answer of mine.)

For periodic or fixed-point solutions, there is another simple argument via time symmetry, which applies to these systems. If the system converging to a periodic or fixed-point attractor is a solution, then the time-reversal, i.e., the solution becoming aperiodic or starting motion, must also be a solution. But then the respective solutions cannot be attractors anymore. Every attractor would have to be a repellor when time is reversed.

There are periodic solutions to such systems, though. Consider, e.g., a solid state where all molecules are on a lattice and perform oscillations where one half of all oscillators oscillate in an identical fashion and anti-phase to the other half.