Partial Multistep Prediction of Chaotic Time Series: What Remains Predictable beyond the Horizon of Predictability?

Abstract

Sensitive dependence on initial conditions limits long-horizon prediction of chaotic systems even for deterministic dynamics, yet some dynamical information may remain recoverable after precise state information is lost. Recursive multistep methods are fragile because they reuse each intermediate prediction as input, allowing local errors to propagate through the rollout. We investigate whether partial multistep prediction can recover predictive information beyond the predictability horizon by omitting intermediate positions instead of assigning them unreliable values. The method uses a lightweight pattern-based local predictor. We analyze partial trajectories generated by different omission patterns and quantify the trade-off between accuracy on retained positions and the fraction classified as non-predictable. Experiments on the Lorenz system show that admissible masks contain trajectories that remain accurate far beyond the horizon of predictability, and an agreement-based ensemble recovers part of this information without future observations at the cost of a growing rejected fraction. At short horizons a broad accepted subset retains accuracy at moderate rejection, whereas at long horizons positive geometric skill requires increasingly strong selection. In this regime retained predictions no longer localize the state within a Lorenz scroll but still identify the occupied scroll, so topological structure outlasts geometric accuracy. A noise-based control reproduces this survival only after tuning of its perturbation scale, unlike masking. Across a heterogeneous collection of chaotic systems the ensemble strictly extends the positive-$R^2$ horizon of the recursive baseline for every retained system.