Decomposing Long-Time Behavior of Dynamical Systems through Linear Regression

Short trajectories of a dynamical system are a bridge between the governing equation and long-time behavior of the system. Trajectories are inherently one-dimensional objects. Despite this, the neighborhood of a trajectory extends to the full dimension of the system's state space. We use simple linear regression to determine the contribution of the individual trajectory neighborhoods to the long-time state-space probability distribution of the system. Constructing this probability distribution directly allows prediction of long-time averages of observable quantities. Better yet, the regression matrix and regression weights tell us which combinations of trajectories are relevant to the long-time behavior.

Periodic Orbit Theory (POT) provides exact expressions for the same predictions but requires careful theoretical analysis and computational effort specific to the system under consideration. Worse, it suffers from truncation error when only finitely many periodic orbits are known. We compare the performance of our method against leading alternatives in the Lorenz system. When tuned properly, our scheme is more precise than POT and other schemes when provided with the same trajectories as input.