Criteria of the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay

1998 ◽  
Vol 50 (8) ◽  
pp. 1224-1232 ◽  
Author(s):  
D. G. Korenevskii
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Continuous Time ◽  
Mean Square ◽  
Stability Of Solutions ◽  
Stochastic Difference Equations ◽  
The Mean

scholarly journals Stochastic Memristive Quaternion-Valued Neural Networks with Time Delays: An Analysis on Mean Square Exponential Input-to-State Stability

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 815 ◽  
Author(s):  
Usa Humphries ◽  
Grienggrai Rajchakit ◽  
Pramet Kaewmesri ◽  
Pharunyou Chanthorn ◽  
Ramalingam Sriraman ◽  
...  
Keyword(s):  
Neural Networks ◽  
Continuous Time ◽  
Lyapunov Functional ◽  
Time Delays ◽  
System Model ◽  
Mean Square ◽  
New Class ◽  
Quaternion Multiplication ◽  
The Mean ◽  
Theoretical Results

In this paper, we study the mean-square exponential input-to-state stability (exp-ISS) problem for a new class of neural network (NN) models, i.e., continuous-time stochastic memristive quaternion-valued neural networks (SMQVNNs) with time delays. Firstly, in order to overcome the difficulties posed by non-commutative quaternion multiplication, we decompose the original SMQVNNs into four real-valued models. Secondly, by constructing suitable Lyapunov functional and applying It o ^ ’s formula, Dynkin’s formula as well as inequity techniques, we prove that the considered system model is mean-square exp-ISS. In comparison with the conventional research on stability, we derive a new mean-square exp-ISS criterion for SMQVNNs. The results obtained in this paper are the general case of previously known results in complex and real fields. Finally, a numerical example has been provided to show the effectiveness of the obtained theoretical results.

scholarly journals Generalized Diffusion Equation Associated with a Power-Law Correlated Continuous Time Random Walk

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Long Shi
Keyword(s):  
Random Walk ◽  
Diffusion Equation ◽  
Power Law ◽  
Continuous Time ◽  
Continuum Limit ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Continuous Time Random Walk ◽  
Mean Square ◽  
The Mean

In this work, a generalization of continuous time random walk is considered, where the waiting times among the subsequent jumps are power-law correlated with kernel function M(t)=tρ(ρ>-1). In a continuum limit, the correlated continuous time random walk converges in distribution a subordinated process. The mean square displacement of the proposed process is computed, which is of the form 〈x2(t)〉∝tH=t1/(1+ρ+1/α). The anomy exponent H varies from α to α/(1+α) when -1<ρ<0 and from α/(1+α) to 0 when ρ>0. The generalized diffusion equation of the process is also derived, which has a unified form for the above two cases.

scholarly journals Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Long Shi ◽  
Aiguo Xiao
Keyword(s):  
Brownian Motion ◽  
Continuous Time ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Weak Ergodicity ◽  
Normal Diffusion ◽  
Weak Ergodicity Breaking ◽  
The Mean ◽  
Ergodicity Breaking

We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse α-stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when 0<α<1/3, normal diffusion when α=1/3, and superdiffusion when 1/3<α<1. The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.

scholarly journals Asymptotic Stability Results for Nonlinear Fractional Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Fulai Chen ◽  
Zhigang Liu
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Difference Equations ◽  
Stability Of Solutions ◽  
Fractional Difference ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Difference Equations ◽  
Stability Results

We present some results for the asymptotic stability of solutions for nonlinear fractional difference equations involvingRiemann-Liouville-likedifference operator. The results are obtained by using Krasnoselskii's fixed point theorem and discrete Arzela-Ascoli's theorem. Three examples are also provided to illustrate our main results.

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