A stochastic controlled Schroedinger equation: convergence and robust stability for numerical solutions

In this paper we study the stochastic stability of numerical solutions of a stochastic controlled Schr¨odinger equation. We investigate the boundedness in second moment, the convergence and the stability of the zero solution for this equation, using two new definitions of almost sure exponential robust stability and asymptotic stability, for the Euler-Maruyama numerical scheme. Considering that the diffusion term is controlled, by using the method of Lyapunov functions and the corresponding diffusion operator associated, we apply techniques of X. Mao and A. Tsoi for achieve our task. Finally, we illustrate this method with a problem in Nuclear Magnetic Resonance (NMR).

Fixed points and stability of A class of Stochastic dynamical system driven by Brownian motion

In real life, many models and systems are affected by random phenomena. For this reason, experts and scholars propose to describe these stochastic processes with Brownian motion respectively. In this paper we consider a kind of stochastic Vollterra dynamical systems of nonconvolution type and give some new conditions to ensure that the zero solution is asymptotically stable in mean square by means of fixed point method. The theorems of asymptotically stability in mean square with a necessary conditions are proved. Some results of related papers are improved.

Qualitative Analysis of some Types of Neutral Delay Differential Equations

In this paper, we conduct some qualitative analysis that involves the global asymptotic stability (GAS) of the Neutral Differential Equation (NDE) with variable delay, by using Banach contraction mapping theorem, to give some necessary conditions to achieve the GAS of the zero solution.

BVPs with the nonlinearity depending on the fractional derivative

In this paper, we study the existence and the numerical estimates of the solutions for a set of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for the exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at origin and infinity. The novelty of this paper is to use variational methods to obtain the multiplicity of solutions of boundary value problems with the nonlinearity depending on the fractional derivative.

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

Stability of intuitionistic fuzzy nonlinear fractional differential equations

In this paper, we study the existence and uniqueness results of solution for the intuitionistic fuzzy nonlinear fractional differential equations involving the Caputo concepts of fractional derivative. In addition, we establish essentially the Mittag-Leffler stability result for the intuitionistic fuzzy nonlinear fractional differential equations by giving some sufficient criteria to guarantee the stability of the zero solution. Finally, some examples are presented to illustrate the proposed stability theorem.

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions

We consider a system of first order coupled mode equations in $${\mathbb {R}}^d$$ R d describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov–Schmidt decomposition in Fourier variables and a nested Banach fixed point argument. The reduced bifurcation equation is a perturbed stationary nonlinear Schrödinger equation. The existence of solitary waves follows in a symmetric subspace thanks to a spectral stability result. A numerical example of gap solitons in $${\mathbb {R}}^2$$ R 2 is provided.

Decomposition and stability of linear singularly perturbed systems with two small parameters

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, \varepsilon _{2} >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} x_{0} +A_{11} x_{1} +A_{12} x_{2},} \\ {\varepsilon _{1} \varepsilon _{2} \dot{x}_{2} =A_{20} x_{0} +A_{21} x_{1} +A_{22} x_{2},} \end{array}\right. \] where $x_{0} \in {\mathbb R}^{n_{0}}$, $x_{1} \in {\mathbb R}^{n_{1}}$, $x_{2} \in {\mathbb R}^{n_{2}}$. In this paper, schemes of decomposition and splitting of the system into independent subsystems by using the integral manifolds method of fast and slow variables are investigated. We give the conditions under which the reduction principle is truthful to study the stability of zero solution of the original system.

The Functional Equation max{χ(xy),χ(xy-1)}=χ(x)χ(y) on Groups and Related Results

This research paper focuses on the investigation of the solutions χ:G→R of the maximum functional equation max{χ(xy),χ(xy−1)}=χ(x)χ(y), for every x,y∈G, where G is any group. We determine that if a group G is divisible by two and three, then every non-zero solution is necessarily strictly positive; by the work of Toborg, we can then conclude that the solutions are exactly the e|α| for an additive function α:G→R. Moreover, our investigation yields reliable solutions to a functional equation on any group G, instead of being divisible by two and three. We also prove the existence of normal subgroups Zχ and Nχ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G/Nχ.