I was simulating a really complicated dynamical system $\dot x = f(x(t))$ in 3D and the solution curve resulted in something like this
(After 10 time steps - Showing starting points)
(After 50 time steps)
(After 500 time steps)
I also have a plot of the norm of $x(t)$ at 500 time steps
Note that the trajectory is constrained on a plane described by $\{x \in \mathbb{R}^n| \sum_i x_i = 1\}$, it is not protruding out. Can anyone describe this type of chaotic behavior?
Guess: strange attractor. Can we say more? I feel like we can because this type of chaotic behavior is always around some circle/polygon shape.
From a visual inspection, I would guess that your time series is quasiperiodic: It has a periodic envelope with non-periodic contents. The phase-space structure looks like a slightly warped torus. If your plot of the time series suffers from considerable aliasing, it may also be periodic with a high period length.
As you do not have access to the differential equations, I suggest the following analyses:
You can look at the autocorrelation function. For chaotic dynamics, it decays; for quasiperiodic systems, it doesn’t.
You can calculate the Lyapunov exponents, preferably by analysing how small perturbations of your state grow over time. If you cannot even perturb your system, you can still apply pure time-series analysis. If your maximum Lyapunov exponent is zero (within statistical fluctuations), the dynamics is not chaotic. If there is a further zero Lyapunov exponent, you have a quasiperiodic dynamics; otherwise it’s periodic.
Usually, you can exclude a periodic dynamics (with a reasonably short period length) by a close look at the time series. If you cannot and you need to know whether it’s periodic or quasiperiodic, you can apply more sophisticated statistical tests.