Figure 4

Screening for emergent nonlinear-deterministic dynamics in rill signals. (a) We reconstructed low-dimensional shadow attractors from all five in-situ rill signals, and from eight of the ten flume rill signals whose flow discharge levels exceeded 1000 Lh⁻¹ (black trajectories). The two failed shadow attractors had too few orbits to adequately sample an underlying attractor, possibly indicating failure of the associated flume rill profiles to develop step-pool units. All reconstructed attractors required at least from two to four embedding dimensions (Table 2), indicating low-dimensional nonlinear dynamics. The plots of shadow attractors have a cyclical appearance composed of aperiodic non-repeating oscillations. In a demonstration of dynamic correspondence, state-space trajectories reconstructed from echo state neural network out-of-sample forecasts (red trajectories) largely rest on shadow attractors reconstructed from in-sample rill signals (black trajectories). Echo state neural network models fit to in-situ rill signals 5 sl–2500 Lh⁻¹ and 5 sl–5000 Lh⁻¹ were explosive for wide ranges of sampled hyperparameter values, and thus not used for forecasting. (b) The visual geometric structure of shadow attractors stands out in contrast to a random scattering of points resulting from reconstruction of a uniform random time series. Surrogate data results soundly reject the null hypothesis that this geometric structure can be attributed to mimicking linear-stochastic dynamics (Table 2).