Are there any experimental or statistical tests to check deterministicity or stochasticity of a dynamical system?

Question

Are there any simple experimental or statistical tests to check whether a dynamical system is deterministic?

Answer 1

There are several tests for determinism which (as far as I know) all boil down to the following, closely related ideas:

• If a system is deterministic, close-by trajectories in phase space tend to point in the same direction.
• In a deterministic system, trajectories should not intersect.
• A deterministic system has a continuous phase-space flow.

There are two general ways to test this:

1. Compute the scalar product of nearby tangents to the trajectory. The larger the better.

2. Consider trajectories that are close to each other at some time point and check how far they have diverged later. This may seem identical to computing the largest Lyapunov exponent at first glance, but it ignores the alignment of a tangent vector to Lyapunov vectors, etc.

The former approach has some advantages as it can also handle low-pass-filtered noise (which in some sense has a deterministic component), however the latter approach is more robust in some cases.

To do any of this, you first somehow need to access your phase space. If you have sufficiently many observables that you can be sure that they uniquely describe the state of your system (should it be deterministic), you can just use these as a state space and that’s it. Otherwise, you first need to perform a Takens reconstruction.

Then, depending on the size of your dataset and the computational power you have available, you can just do a brute-force approach and see whether there is a correlation between the distance of trajectory segments and direction (first approach) or divergence (second approach). For example, Binder et al. describe a simple test based on the second approach. If you have the luxury of more data, you need to first determine what are reasonably close elements, for example by partitioning the phase space. For instance, Kaplan and Glass, Wayland et al, and Salvino and Cawley propose methods using this.

Whatever you do, you should compare the outcome to an appropriate null model (time-series surrogate) of the data. What is appropriate, depends on your exact scientific hypothesis. For example, a very simple hypothesis could be that the order of your data is not arbitrary. Thus, if you randomise your data points, your measure for determinism should be lower. (Note that this hypothesis would be satisfied by low-pass-filtered noise.) Keep in mind that most real dynamics are neither completely stochastic nor deterministic: Even an Ohrnstein–Uhlenbeck process has some drift to the mean and there is always is some dynamic noise (and measurement noise).