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I have a high dimensional dynamical system, and I was wondering if there is a method to identify the various attractors of the system i.e, a way of mapping the energy landscape?

I was thinking of a naive computational approach to have an ensemble of particles, to evolve them and analyse their trajectories. Is there a known, formal way to solve this problem?

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In the broad sense in which you state your question, and if you are looking for rigorous results, then the answer is surely no.

Keep in mind that the existence of the Lorenz attractor was only proved formally (using a computer-assisted proof) less than twenty years ago. This shows two things:

  • In particular cases, it can be possible to show the existence of attractors by formal calculation, and probably a heuristic approach as you are describing would be a good start to develop an idea of where the attractors are
  • It is highly unlikely that there is a general-purpose method that will work in all situations.

A few further comments: By the Newhouse phenomenon, in higher-dimensional systems, you will find places where topologically generic systems have infinitely many attractors, so you will never find "all" of them. Moreover, the structure of attractors will change radically when moving parameters. This shows that there is no general method of the type that you are talking about.

On the other hand, the Palis conjecture says that, at least for a dense set of systems, you will have finitely many attractors that you will detect from a positive-measure set of starting values. So you might not really expect to "see" Newhouse-type behaviour in actual applications.

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