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We consider several models of State Dependent Delay Differential Equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions,…
We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, exponential stability and norm…
We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs). As a consequence, we can show that the $\omega$-limit sets of all pullback trajectories of…
A network of noisy bistable elements with global time-delayed couplings is considered. A dichotomous mean field model has recently been developed describing the collective dynamics in such systems with uniform time delays near the…
Numerical methods for stochastic differential equations with non-globally Lipschitz coefficients are currently studied intensively. This article gives an overview of our work for the case that the drift coefficient is potentially…
This paper studies path stabilities of the solution to stochastic differential equations (SDE) driven by time-changed L\'evy noise. The conditions for the solution of time-changed SDE to be path stable and exponentially path stable are…
A general framework for recovering drift and diffusion dynamics from sampled trajectories is presented for the first time for stochastic delay differential equations. The core relies on the well-established SINDy algorithm for the sparse…
In this paper, we introduce a new type of backward stochastic differential equations (BSDEs) with infinite anticipation, where the generator depends on the entire future values of the solution in infinite horizon. We show that the new BSDEs…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
For the delay differential equations $$ \ddot{x}(t) +a(t)\dot{x}(g(t))+b(t)x(h(t))=0, g(t)\leq t, h(t)\leq t, $$ and $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)+a_1(t)\dot{x}(g(t))+b_1(t)x(h(t))=0 $$ explicit exponential stability conditions…
We present a novel model Graph Neural Stochastic Differential Equations (Graph Neural SDEs). This technique enhances the Graph Neural Ordinary Differential Equations (Graph Neural ODEs) by embedding randomness into data representation using…
The problem considered in the paper is exponential stability of linear equations and global attractivity of nonlinear non-autonomous equations which include a non-delay term and one or more delayed terms. First, we demonstrate that…
We consider a class of reflected backward doubly stochastic differential equations with time delayed generator (in short RBDSDE with time delayed generator), in this case generator at time $t$ can depend on the values of a solution in the…
Fourth-order accurate compact schemes for variable coefficient convection diffusion equations are considered. A sufficient condition for the stability of the fully discrete problem is derived using a difference equation based approach. The…
We consider stochastic differential equations on $\mathbb R^d$ with coefficients depending on the path and distribution for the whole history. Under a local integrability condition on the time-spatial singular drift, the well-posedness and…
In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Levy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order…
Data-driven methodologies are nowadays ubiquitous. Their rapid development and spread have led to applications even beyond the traditional fields of science. As far as dynamical systems and differential equations are concerned, neural…
We describe a method to model nonlinear dynamical systems using periodic solutions of delay-differential equations. We show that any finite-time trajectory of a nonlinear dynamical system can be loaded approximately into the initial…