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We study the evolution of observables of dynamical systems. For linear systems, we show that observables satisfy a closed differential equation whose minimal order is determined by the dynamical system and observation operator. This yields…
A method for sequential Bayesian inference of the static parameters of a dynamic state space model is proposed. The method is based on the observation that many dynamic state space models have a relatively small number of static parameters…
To assess the quality of a probabilistic prediction for stochastic dynamical systems (SDSs), scoring rules assign a numerical score based on the predictive distribution and the measured state. In this paper, we propose an…
Open dynamical systems are mathematical models of machines that take input, change their internal state, and produce output. For example, one may model anything from neurons to robots in this way. Several open dynamical systems can be…
Sequential dynamical systems (SDS) are used to model a wide range of processes occurring on graphs or networks. The dynamics of such discrete dynamical systems is completely encoded by their phase space, a directed graph whose vertices and…
In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply…
In a dynamical system $(X,f)$, with $X$ a compact metric space, the chain components, the fundamental building blocks in the Conley decomposition of dynamics, have a natural partial order induced by the chain relation between points.…
End-to-end learning of dynamical systems with black-box models, such as neural ordinary differential equations (ODEs), provides a flexible framework for learning dynamics from data without prescribing a mathematical model for the dynamics.…
In this article, we focus on extending the notion of lattice linearity to self-stabilizing programs. Lattice linearity allows a node to execute its actions with old information about the state of other nodes and still preserve correctness.…
Saddle dynamics is a time continuous dynamics to efficiently compute the any-index saddle points and construct the solution landscape. In practice, the saddle dynamics needs to be discretized for numerical computations, while the…
This chapter reviews the fundamentals of continuous and discrete Linear Time-Invariant (LTI) systems with Single Input-Single Output (SISO). We start from the general notions of signals and systems, the signal representation problem and the…
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An iterative scheme is constructed for…
Stochastic differential equations (SDEs) are a fundamental tool for modelling dynamic processes, including gene regulatory networks (GRNs), contaminant transport, financial markets, and image generation. However, learning the underlying SDE…
Traditional deterministic subgrid-scale (SGS) models are often dissipative and unstable, especially in regions of chaotic and turbulent flow. Ongoing work in climate science and ocean modeling motivates the use of stochastic SGS models for…
This work is concerned with the finite-horizon optimal covariance steering of networked systems governed by discrete-time stochastic linear dynamics. In contrast with existing work that has only considered systems with dynamically decoupled…
Encoding time-series with Linear Dynamical Systems (LDSs) leads to rich models with applications ranging from dynamical texture recognition to video segmentation to name a few. In this paper, we propose to represent LDSs with…
This article deals with stabilizing discrete-time switched linear systems. Our contributions are threefold: Firstly, given a family of linear systems possibly containing unstable dynamics, we propose a large class of switching signals that…
Stochastic differential equations (SDEs) are one of the most important representations of dynamical systems. They are notable for the ability to include a deterministic component of the system and a stochastic one to represent random…
In this paper, we investigate the M{\"o}bius function $\mu\_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study…
This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction and integrability. In particular, we show that an SDS which is diffusion-wise symmetric with…