Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations

By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately

Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations

Halanay-type theory in the context of evolutionary equations with time-lag

We consider extensions and modifications of a theory due to Halanay, and the context in which such results may be applied. Our emphasis is on a mathematical framework for Halanay-type analysis of problems with time lag and simulations using discrete versions or numerical formulae. We present selected (linear and nonlinear, discrete and continuous) results of Halanay type that can be used in the study of systems of evolutionary equations with various types of delayed argument, and the relevance and application of our results is illustrated, by reference to delay-differential equations, difference equations, and methods

Periodic solutions in next generation neural field models

We consider a next generation neural field model which describes the dynamics of a network of theta neurons on a ring. For some parameters the network supports stable time-periodic solutions. Using the fact that the dynamics at each spatial location are described by a complex-valued Riccati equation we derive a self-consistency equation that such periodic solutions must satisfy. We determine the stability of these solutions, and present numerical results to illustrate the usefulness of this technique. The generality of this approach is demonstrated through its application to several other systems involving delays, two-population architecture and networks of Winfree oscillators.Comment: 15 pages, 11 figure

Spontaneous pulse formation in bistable systems

This thesis considers localized spontaneous pulse formation in nonlinear, dissipative systems that are far from equilibrium and which exhibit bistability. It is shown that such pulses can form in systems that are dominated by the combined effects of: (1) a saturable amplifying or gain region, (2) a saturable absorbing or loss region, and (3) cavity effects. Analysis is based upon novel models for both an inertialess material in which the absorber responds instantaneously and inertial material in which there is temporal delay in the response. Additionally, we include the situation where the material does not fully relax between pulses, i.e. memory effects. The results are shown to be generic but direct application is made to pulse formation and stability as observed and exploited in a colliding pulse mode-locked (CPM) dye laser in which the saturable gain and absorber are spatially localized. Bifurcation from a steady, pulsing state to one of several possible other states (laser dropout phenomena) is observed to occur in these systems and will also be addressed. Key results arising from the inclusion of memory effects are as follows: the existence of highly degenerate bifurcation scenarios, implying hysteresis-like behavior in drop-out/drop-in transitions; damped period-two oscillations; and much lower frequency damped oscillations---reminiscent of breathing modes

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

From Crossref journal articles via Jisc Publications RouterHistory: epub 2024-01-23, issued 2024-01-23Article version: VoRPublication status: PublishedWe consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory

Stabilization via delay feedback for highly nonlinear stochastic time-varying delay systems with Markovian switching and Poisson jump

Little work seems to be known about stabilization results of highly nonlinear stochastic time-varying delay systems (STVDSs) with Markovian switching and Poisson jump. This paper is concerned with the stabilization problem for a class of STVDSs with Markovian switching and Poisson jump. The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. The aim of this paper is to design a delay feedback controller to make an unstable highly nonlinear STVDSs with Markovian switching and Poisson jump H∞-stable and asymptotically stable. Besides, an illustrative example is provided to support the theoretical results