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Eigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid…
The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by…
In this article, we focus on the global stabilizability problem for a class of second order uncertain stochastic control systems, where both the drift term and the diffusion term are nonlinear functions of the state variables and the…
For dispersive Hamiltonian partial differential equations of order 2N+1, N integer, there are two criteria to analyse to examine the stability of small-amplitude, periodic travelling wave solutions to high-frequency perturbations. The first…
We prove the existence of local stable, unstable, and center manifolds for stochastic semiflows induced by rough differential equations driven by rough paths valued stochastic processes around random fixed points of the equation. Examples…
This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices, and the stability of a corresponding random transition matrix (or exponential…
We study the long time statistics of a class of semi--linear wave equations modeling the motions of a particle suspended in continuous media while being subjected to random perturbations via an additive Gaussian noise. By comparison with…
Current questions in ecology revolve around instabilities in the dynamics on spatial networks and particularly the effect of node heterogeneity. We extend the Master Stability Function formalism to inhomogeneous biregular networks having…
A two-dimensional extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities is derived. Prior work has demonstrated that the emergent dynamics of these systems is characterized by a…
We present a non-uniform analogue of the classical Datko-Pazy theorem. Our main result shows that an integrability condition imposed on orbits originating in a fractional domain of the generator (as opposed to all orbits) implies polynomial…
Polynomial stability of exact solution and modified truncated Euler-Maruyama method for stochastic differential equations with time-dependent delay are investigated in this paper. By using the well known discrete semimartingale convergence…
Uncertainty quantification seeks to provide a quantitative means to understand complex systems that are impacted by parametric uncertainty. The polynomial chaos method is a computational approach to solve stochastic partial differential…
This article aims to investigate sufficient conditions for the stability of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic…
In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting…
Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a…
In this paper, we investigate a class of nonlinear impulsive stochastic differential evolution equations with infinite delay in Banach space. Based on the Krasnoselskii's fixed point theorem, sufficient conditions of the existence of the…
For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise,…
We consider the internal dynamics of the polymer molecule which is injected in the chaotic flow with strong mean shear component. The flow geometry corresponds to the recent experiments on the elastic turbulence (Groisman, Steinberg 2000).…
We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition…
Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) $m$-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint…