Lyapunov functions for nabla discrete fractional order systems

Dual identities in fractional difference calculus within Riemann

We Investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences we have to use both the nabla and delta operators. The solution representation for higher order Riemann fractional difference equation is obtained as well.

带分数噪声的随机偏微分方程的有限时间Lyapunov指数

Finite-time Lyapunov exponents for SPDEs with fractional noise

本文研究由分数布朗运动驱动的随机偏微分方程在叉形分岔附近的有限时间Lyapunov指数，刻画了其为正的区域并揭示稳定性变化，结果对分数噪声扰动的SDE亦属首次。

Lyapunov Functions and Stability Analysis of Fractional-Order Systems

This study presents new estimates for fractional derivatives without singular kernels defined by some specific functions. Based on obtained inequalities, we give a useful method to establish the global stability of steady states for fractional-order systems and generalize some works existing in the literature. Finally, we apply our results to prove the global stability of a fractional-order SEIR model with a general incidence rate.

Lyapunov functions for fractional-order systems in biology: methods and applications

We prove new estimates of the Caputo derivative of order $α\in (0,1]$ for some specific functions. The estimations are shown useful to construct Lyapunov functions for systems of fractional differential equations in biology, based on those known for ordinary differential equations, and therefore useful to determine the global stability of the equilibrium points for fractional systems. To illustrate the usefulness of our theoretical results, a fractional HIV population model and a fractional cellular model are studied. More precisely, we construct suitable Lyapunov functionals to demonstrate th…

Chaos suppression in a Gompertz-like discrete system of fractional order

In this paper we introduce the fractional-order variant of a Gompertz-like discrete system. The chaotic behavior is suppressed with an impulsive control algorithm. The numerical integration and the Lyapunov exponent are obtained by means of the discrete fractional calculus. To verify numerically the obtained results, beside the Lyapunov exponent, the tools offered by the 0-1 test are used.

Mittag-Leffler函数的离散延迟扰动及其在线性分数阶延迟差分系统中的应用

Discrete delayed perturbation of Mittag-Leffler function and its application to linear fractional delayed difference system

本文引入常系数线性非齐次分数阶差分系统，通过提出nabla Mittag-Leffler型函数的离散延迟扰动获得显式解，并讨论了相关特殊情况。

基于微集合的局部Lyapunov分析：复值BAM神经网络中的有限时间Lyapunov指数估计与KNN预测比较

Local Lyapunov analysis via micro-ensembles: finite-time Lyapunov exponent estimation and KNN-based predictive comparison in complex-valued BAM neural networks

针对分数阶复值BAM神经网络的瞬态不稳定性，提出基于微集合的有限时间Lyapunov指数估计器和KNN预测误差代理，实现稳定性分析与预测性能评估。

Fractional differences and sums with binomial coefficients

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Litnikov fractional derivatives. In this article we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the d…

Sturm Liouville Equations in the frame of fractional operators with Mittag-Leffler kernels and their discrete versions

Very recently, some authors have studied new types of fractional derivatives whose kernels are nonsingular. In this article, we study Sturm-Liouville Equations ($SLEs$) in the frame of fractional operators with Mittag-Leffler kernels. We formulate some Fractional Sturm-Liouville Problems ($FSLPs$) with the diffferential part containing the left and right sided derivatives. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional Sturm-Liouville Operators ($FSLOs$) by using fractional integration by parts formulas. The nabla discrete version o…

On fractional-order maps and their synchronization

We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map $f(x)=ax$ is stable for $a_c(α)<a<1$ where $α$ is a fractional order parameter and $a_c(α)\approx -α$. For coupled linear fractional maps, we can obtain `normal modes' and reduce the evolution to effectively one-dimensional system.…

一类带分数噪声的抛物型SPDE光滑稳定流形的存在性

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

本文针对由非线性乘性分数噪声驱动的抛物型SPDE，通过插值理论构造合适函数空间并利用离散化Lyapunov-Perron型算子的唯一不动点，证明了其局部稳定流形的存在性与光滑性，从而推进了Lu-Schmalfuß猜想的研究。

Necessary condition on Lyapunov functions corresponding to the globally asymptotically stable equilibrium point

It is well known that, the existence of a Lyapunov function is a sufficient condition for stability, asymptotic stability, or global asymptotic stability of an equilibrium point of an autonomous system $\dot{\mathbf{x}} = f(\mathbf{x})$. In variants of Lyapunov theorems, the condition for a Lyapunov candidate $V$ (continuously differentiable and positive definite function) to be a Lyapunov function is that its time derivative along system trajectories must be negative semi-definite or negative definite. Numerically checking positive definiteness of $V$ is very difficult; checking negative defi…

二阶梯度系统的渐进行为 via Lyapunov 方法

Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method

本文通过 Lyapunov 方法研究了阻尼二阶梯度系统的渐进行为，证明了均匀渐近稳定性和指数衰减，并揭示了全局吸引子的存在。