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In a Poincaré map, when quasi-periodicity is exhibited by the dynamical system, what does it mean in terms of stability for the dynamical system?. Why is it so that as Maximum Lyapunov exponent (MLE) becomes equal to 1 the poin care map describes approx. a circular cross-section.

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If by "stability of the system" it's meant its structural stability (i.e., whether a change in the value of a parameter brings about an abrupt, qualitative change in the system behavior), then just by knowing that there is a quasiperiodic orbit one can't really tell much.

However, we do know some things that, overall, point out to instability. For instance:

  • quasiperiodic orbits are common in conservative systems, which are generally not structurally stable;
  • some bifurcation sequences include quasiperiodic behavior, such as the Ruelle–Takens–Newhouse route to chaos and, in this way, can also be an indication of instability; and
  • there's evidence that quasiperiodicity is vanishingly unlikely in high-dimensional systems - which implies that the detection of a quasiperiodic orbit should indicate that the measured system can be well described with a few variables: this, on its turn, could mean that the system is sensitive to a perturbation equivalent to the introduction of a new degree of freedom.

If the question is about the stability of a specific given orbit, then again we cannot say anything in general without more information, but most often it'll be stable, in the sense, for instance, of the KAM theorem and its extensions.

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