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Sequence models, and particularly Linear Recurrent Neural Networks (LRNNs) of the form $\mathbf{h}_{k+1} = \mathbf{W} \mathbf{h}_{k} + \mathbf{y}_k + \mathbf{b}$, are widely applicable in time-series analysis for dynamical systems, yet, as…
Investigating the network stability or synchronization dynamics of multi-agent systems with time delays is of significant importance in numerous real-world applications. Such investigations often rely on solving the transcendental…
A central challenge in data-driven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens' theorem provides conditions for when it is possible to augment these…
Understanding how time delays impact the stability of a delay differential equation is important for modeling many natural and technological systems that experience time delays. Here we introduce a new stability criterion for…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
In nonlinear time series analysis and dynamical systems theory, Takens' embedding theorem states that the sliding window embedding of a generic observation along trajectories in a state space, recovers the region traversed by the dynamics.…
A common problem in time series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens theorem proves a time delayed embedding of the…
Experimental measurements of physical systems often have a limited number of independent channels, causing essential dynamical variables to remain unobserved. However, many popular methods for unsupervised inference of latent dynamics from…
We present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a…
Dynamic mode decomposition (DMD) is a leading tool for equation-free analysis of high-dimensional dynamical systems from observations. In this work, we focus on a combination of delay-coordinates embedding and DMD, i.e., delay-coordinates…
The time delay (or Sliding Window) embedding is a technique from dynamical systems to reconstruct attractors from time series data. Recently, descriptors from Topological Data Analysis (TDA) -- specifically, persistence diagrams -- have…
Complex nonlinear dynamics are ubiquitous in many fields. Moreover, we rarely have access to all of the relevant state variables governing the dynamics. Delay embedding allows us, in principle, to account for unobserved state variables.…
Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the…
Let $x_{j+1}=\phi(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $\phi$ being a diffeomorphism. Although the state vector $x_{j}$ is often unobservable, the dynamics can be recovered from the delay vector…
Cartograms are popular for visualizing numerical data for map regions. Maintaining correct adjacencies is a primary quality criterion for cartograms. When there are multiple data values per region (over time or different datasets) shown as…
Delay-coordinate reconstruction is a proven modeling strategy for building effective forecasts of nonlinear time series. The first step in this process is the estimation of good values for two parameters, the time delay and the embedding…
We study the stability of the fixed-point solution of an array of mutually coupled logistic maps, focusing on the influence of the delay times, $\tau_{ij}$, of the interaction between the $i$th and $j$th maps. Two of us recently reported…
In this paper, stability analysis of time delay systems is considered based on decomposition of the systems to subsystems. The decomposition process needs matrices of these systems to be simultaneously block triangularize. We show that a…
Identifying the qualitative changes in time-series data provides insights into the dynamics associated with such data. Such qualitative changes can be detected through topological approaches, which first embed the data into a…
This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state…