I have a problem characterizing the solution of a dynamic system. It is a system of four nonlinear differential equations of first order. The considered solution shows:
- For the selected initial conditions a stable behavior (very long simulation durations show a limited set).
- However, the calculated Lyapunov spectrum shows only positive exponents (+69,+59,+41,+31).
- This autocorrelation:
- I know, the Poincaré–Bendixson theorem is only valid for two dimensional systems, but nevertheless at initial conditions (with a much larger value than the assumed to be limited set), the trajectory converges to the limited set.
Assuming the calculation of the Lyapunov spectrum (I validated the code with known systems) and the autocorrelation are correct, what does this mean for the solution? Does this mean that it is not a limited set, so for $t$ against infinity the amplitude would increase? The Lyapunov exponents seem very large to me, shouldn't one see such a rise in a long simulation time?
Your observations contradict each other. A system with only positive Lyapunov exponents should exhibit an unbounded dynamics. Given the time scale on which your dynamics appears to act (as per the autocorrelation), even one Lyapunov exponent of the order of magnitude you are reporting seems questionably high, as it should lead to a quick effects within a few time units.
Moreover your plot of the autocorrelation function does not allow to make any conclusions as to whether it decays (chaos) or not (periodic or quasiperiodic dynamics).