Stability of a torus in a dynamic system

Question

I have a nonlinear dynamic system with $n$ degrees of freedom:

$\dot{x} = f(x,t)$

In simulations a stable attractor can be found. I assume it is a torus because:

1. In the frequency spectrum it has two significant frequencies.
2. The Poincaré map shows an area of intersections.
3. I could create a hypersphere and for a long simulation time the solutions would constantly stay in that sphere.

By now I only know how to determine the stability of limit cycles using Poincaré maps by calculating the Jacobi and monodromy matrix resulting in Lyapunov exponents. This does not seem to be valid for tori!

Do you know other possibilities to determine the stability of a torus by Lyapunov by other means? Pleas note that the dynamic system is so non-linear it can only be solved numerically.

Answer 1

I am not quite exactly sure what exactly you want to show, but the following may help:

• The signs of the first Lyapunov exponents of a (stable) quasiperiodic dynamics (torus) should be 00−, 000−, etc. In your case, I would expect 00−. A periodic dynamics would have 0−; a chaotic one would have +0− (or ++0−, +00−, or similar).

• The autocorrelation function of a quasiperiodic dynamics does not decay. (For a chaotic dynamics, it does.)

• If you do not see any changes in the characteristics of a dynamics over a long simulation, this is a good indicator that it is stable.