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We illustrate how to extend the concept of structural stability through applying it to the front propagation speed selection problem. This consideration leads us to a renormalization group study of the problem. The study illustrates two…

We give necessary and/or sufficient conditions for stochastic stability of second-order linear autonomous systems with parameters, which are perturbed by a random process of the "white noise" type. The Ito's and Stratonovich's forms of…

We present a perturbation method for determining the moment stability of linear ordinary differential equations with parametric forcing by colored noise. In particular, the forcing arises from passing white noise through an $n$th order…

We study synchronization of locally coupled noisy phase oscillators which move diffusively in a one-dimensional ring. Together with the disordered and the globally synchronized states, the system also exhibits several wave-like states which…

The paper describes the robust algorithm for linear time-invariant plants under parametric uncertainties, external disturbances and high-frequency noises in measurements. The proposed algorithm allows one to reduce the noise impact on the…

The combination of bistability and noise is ubiquitous in complex systems, from biological to social interactions, and has important implications for their functioning and resilience. We analyze a simple three-state model for bistability in…

The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described which allows the probabilities with which the metastable states are…

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multiscale theory for such systems and apply it to the Kuramoto-Sivashinsky equation as a case study. We first…

Stochastic differential equations have proved to be a valuable governing framework for many real-world systems which exhibit ``noise'' or randomness in their evolution. One quality of interest in such systems is the shape of their…

The influence of noise on the generalized synchronization regime in the chaotic systems with dissipative coupling is considered. If attractors of the drive and response systems have an infinitely large basin of attraction, generalized…

A system of partial differential equations describing the spatial oscillations of an Euler-Bernoulli beam with a tip mass is considered. The linear system considered is actuated by two independent controls and separated into a pair of…

This paper introduces new techniques for using convex optimization to fit input-output data to a class of stable nonlinear dynamical models. We present an algorithm that guarantees consistent estimates of models in this class when a small…

This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order that that of the nominal…

We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of…

The state of a continuously monitored qubit evolves stochastically, exhibiting competition between coherent Hamiltonian dynamics and diffusive partial collapse dynamics that follow the measurement record. We couple these distinct types of…

We assume that to zero order in the Knudsen number the deviation of the entropy from a background value vansishes. We then show that adding a super-Burnett term we obtain a stable state of rest. The resulting equations have the same form as…

We analyse the non-equilibrium distribution in dissipative dynamical systems at finite noise intensities. The effect of finite noise is described in terms of topological changes in the pattern of optimal paths. Theoretical predictions are…

The effect of the beam-beam interactions on the stability of impedance mode is discussed. The detuning is evaluated by the means of single particle tracking in arbitrarily complex collision configurations, including lattice non-linearities,…

We consider a large class of nonlinear diffusive systems with nonlocal coupling. By using a non-perturbative analytical approach we are able to determine the convective and absolute instabilities of all the uniform states of these systems.…

Maintaining numerical stability in machine learning models is crucial for their reliability and performance. One approach to maintain stability of a network layer is to integrate the condition number of the weight matrix as a regularizing…