Related papers: Upper limits on the robustness of Turing models an…
Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on…
A common problem to all applications of linear finite dynamical systems is analyzing the dynamics without enumerating every possible state transition. Of particular interest is the long term dynamical behaviour. In this paper, we study the…
In this paper, we consider the stability analysis of large-scale distributed networked control systems with random communication delays between linearly interconnected subsystems. The stability analysis is performed in the Markov jump…
This paper presents a novel method for stability analysis of a wide class of linear, time-delay systems (TDS), including retarded non-neutral ones, as well as those incorporating incommensurate and distributed delays. The proposed method is…
The population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the…
Methods exhibiting linear scaling with respect to the size of the system, so called O(N) methods, are an essential tool for the calculation of the electronic structure of large systems containing many atoms. They are based on algorithms…
We give a self-contained modern linear stability analysis of a system of n equal mass bodies in circular orbit about a single more massive body. Starting with the mathematical description of the dynamics of the system, we form the linear…
Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To…
In this paper, we present a novel framework for quantifying a lower bound on resilience in continuous-time (non)linear systems subject to external disturbances while ensuring satisfaction of signal temporal logic specifications. Unlike…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…
We consider the classical Turing instability in a reaction-diffusion system as the secend part of our study on pattern formation. We prove that nonlinear dynamics of a general perturbation of the Turing instability is determined by the…
The larger the distance to instability from a matrix is, the more robustly stable the associated autonomous dynamical system is in the presence of uncertainties and typically the less severe transient behavior its solution exhibits.…
We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem)…
In this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability…
This article introduces, and reviews recent work using, a simple optimisation technique for analysing the nonlinear stability of a state in a dynamical system. The technique can be used to identify the most efficient way to disturb a system…
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
This paper develops a computational method for studying stable/unstable manifolds attached to periodic orbits of differential equations. The method uses high order Chebyshev-Taylor series approximations in conjunction with the…
Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous applied and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which…
High-dimensional predictive models, those with more measurements than observations, require regularization to be well defined, perform well empirically, and possess theoretical guarantees. The amount of regularization, often determined by…
This paper derives two stabilizability theorems for a basic class of discrete-time nonlinear systems with multiple unknown parameters. First, we claim that a discrete-time multi-parameter system is stabilizable if its nonlinear growth rate…