Related papers: Stability and chaos in real polynomial maps
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the…
The usual definition of the stability region of implicit multistep methods often implies that there are some isolated points of stability within the region of instability of the numerical method. These isolated stable points may appear when…
Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical…
Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction…
When a system has more than one stable state, how can the stability of these states be compared? This deceptively simple question has important consequences for ecosystems, because systems with alternative stable states can undergo dramatic…
A canonical system of basic invariants is a system of invariants satisfying a set of differential equations. The properties of a canonical system are related to the mean value property for polytopes. In this article, we naturally identify…
This paper initiates a systematic study of connections between undirected colored graphs and associated two-variable stable polynomials obtained via Cauchy transform-type formulas. Examples of such stable polynomials have played crucial…
In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem…
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…
A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by…
We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy…
We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the…
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…
Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the…
Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we…
We prove twisted homological stability for handlebody mapping class groups. Using the categorical framework developed by Randal-Williams and Wahl, we establish that the homology of the handlebody groups stabilises with respect to both genus…
For a discrete dynamical system on $\R$ generated by a quadratic function, we show, using elementary computations, that the existence, number, and stability of 3-cycles are determined by a single parameter depending on the coefficients of…
Given a ternary homogeneous polynomial, the fixed points of the map from $\mathbb{P}^2$ to itself defined by its gradient are called its eigenpoints. We focus on cubic polynomials, and analyze configurations of eigenpoints that admit one or…
A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix…
In this paper, a new polynomial chaos based framework for analyzing linear systems with probabilistic parameters is presented. Stability analysis and synthesis of optimal quadratically stabilizing controllers for such systems are presented…