arXiv subjects

S. Siegmund

Publications and source records attributed to S. Siegmund.

3 recordsLinked to original sources

arXiv2018

Stability of scalar nonlinear fractional differential equations with linearly dominated delay

In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.

math.CA↗

arXiv2016

An instability theorem for nonlinear fractional differential systems

In this paper, we give a criterion on instability of an equilibrium of nonlinear Caputo fractional differential systems. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector $$\left\{λ\in\C\setminus\{0\}:|\arg{(λ)}|<\frac{απ}{2}\right\},$$ where $α\in (0,1)$ is the order of the fractional differential systems, then the equilibrium of the nonlinear systems is unstable.

math.CA↗

arXiv2016

Linearized Asymptotic Stability for Fractional Differential Equations

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector $\{λ\in \C : |\arg λ| > \frac{απ}{2}\}$ where $α> 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.

math.DS↗