scholarly journals Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

2021 ◽  
Vol 24 (1) ◽  
pp. 225-253
Author(s):  
Oana Brandibur ◽  
Eva Kaslik
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Order ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Fractional Order Differential Equations ◽  
Stability And Instability ◽  
Instability Regions ◽  
Necessary And Sufficient

Abstract Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

scholarly journals The Oscillation of a Class of the Fractional-Order Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qianli Lu ◽  
Feng Cen
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Constant Coefficients ◽  
Delay Differential ◽  
Necessary And Sufficient

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

scholarly journals Stability of invariant sets of Itô stochastic differential equations with Markovian switching

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
Jiaowan Luo
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Markovian Switching ◽  
Invariant Sets ◽  
Stochastic Asymptotic Stability ◽  
Stability And Instability

Consider the nonlinear Itô stochastic differential equations with Markovian switching, some sufficient conditions for the invariance, stochastic stability, stochastic asymptotic stability, and instability of invariant sets of the equations are derived.

Stability and Bifurcations in 2D Spatiotemporal Discrete Systems

2018 ◽  
Vol 28 (08) ◽  
pp. 1830026
Author(s):  
Mohamed Lamine Sahari ◽  
Abdel-Kaddous Taha ◽  
Louis Randriamihamison
Keyword(s):  
Asymptotic Stability ◽  
Current Literature ◽  
Sufficient Conditions ◽  
Discrete Systems ◽  
Necessary And Sufficient Conditions ◽  
Two Dimensional ◽  
Local Bifurcations ◽  
Stability And Bifurcations ◽  
Necessary And Sufficient

This paper deals with stability and local bifurcations of two-dimensional (2D) spatiotemporal discrete systems. Necessary and sufficient conditions for asymptotic stability of the systems are obtained. They prove to be more accurate than those in the current literature. Some definitions for the bifurcations of 2D spatiotemporal discrete systems are also given, and an illustrative example is provided to explain the new results.

scholarly journals Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1751
Author(s):  
Oana Brandibur ◽  
Eva Kaslik ◽  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Discrete Version ◽  
Model Parameters ◽  
Step Size ◽  
Order Difference ◽  
Stability And Instability ◽  
Fractional Orders

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences.

On the ξ-Exponentially Asymptotic Stability of Linear Systems of Generalized Ordinary Differential Equations

2001 ◽  
Vol 8 (4) ◽  
pp. 645-664
Author(s):  
M. Ashordia ◽  
N. Kekelia
Keyword(s):  
Asymptotic Stability ◽  
Linear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Nondecreasing Function ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations

Abstract Necessary and sufficient conditions and effective sufficient conditions are established for the so-called ξ-exponentially asymptotic stability of the linear system 𝑑𝑥(𝑡) = 𝑑𝘈(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are respectively matrix- and vector-functions with bounded variation components, on every closed interval from [0, +∞[ and ξ : [0, +∞[ → [0, +∞[ is a nondecreasing function such that .

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