Structured time-delay models for dynamical systems with connections to Frenet–Serret frame

Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet–Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet–Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modelling procedure on data from several nonlinear synthetic and real-world examples.

Deep Learning for Nonlinear Time Series: Examples for Inferring Slow Driving Forces

Records for observing dynamics are usually complied by a form of time series. However, time series can be a challenging type of dataset for deep neural networks to learn. In deep neural networks, pairs of inputs and outputs are usually fed for constructive mapping. Such inputs are typically prepared as static images in successful applications. And so, here we propose two methods to prepare such inputs for learning the dynamical properties behind time series. In the first method, we simply array a time series in the shape of a rectangle as an image. In the second method, we convert a time series into a distance matrix using delay coordinates, or an unthresholded recurrence plot. We demonstrate that the second method performs well in inferring a slow driving force from observations of a forced system within which there are symmetry and almost invariant subsets.

Predicting Spatio-temporal Time Series Using Dimension Reduced Local States

We present a method for both cross-estimation and iterated time series prediction of spatio-temporal dynamics based on local modelling and dimension reduction techniques. Assuming homogeneity of the underlying dynamics, we construct delay coordinates of local states and then further reduce their dimensionality through Principle Component Analysis. The prediction uses nearest neighbour methods in the space of dimension reduced states to either cross-estimate or iteratively predict the future of a given frame. The effectiveness of this approach is shown for (noisy) data from a (cubic) Barkley model, the Bueno-Orovio–Cherry–Fenton model, and the Kuramoto–Sivashinsky model.

Unfolding the Phase Space Structure of Noisy Time Series by means of Angular First-Return Maps

A new approach which uses the joint probability matrix computation of noisy time series is proposed to construct a phase space portrait which reflects the orbit visitation frequency of the different regions of the phase space. The resulting representation provides a clear cut of the dynamical reconstructed attractor giving both quantitative information and qualitative information about the attractor structure. The orbital distribution recovered in the map is studied by an angular first-return map where the orbital time used for the reconstruction is obtained from the magnitude information of the complex representation of the data belonging to the probability phase portrait. The resulting phase delay coordinates serve to identify phase intermittency. The Lorenz-like Shimizu-Morioka model and the Rossler model are used to present the methodology. Finally, some experimental pressure time series measured on gas-solid fluidized beds operated at different dynamical regimes are presented to analyze the reliability of the proposed method to deal with experimental noise time series.