An autonomous Caputo fractional differential equation of order $alphain(0,1)$ in $mathbb{R}^d$ whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space $mathfrak{C}$ of continuous fu
nctions $f:R^+rightarrow R^d$ with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong & Tuan cite{cong}, which showed that such equations do not, in general, generate a dynamical system on the space $mathbb{R}^d$.
In this paper, we show that under a generic condition of the coefficient of a stochastic phase oscillator the Lyapunov exponent of the linearization along an arbitrary solution is always negative. Consequently, the generated random dynamical system exhibits a synchronization.
Using a temporally weighted norm we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order $alphain(frac{1}{2},1)$ whose coefficients satisfy a standard Lipschitz
condition. For this class of systems we then show that the asymptotic distance between two distinct solutions is greater than $t^{-frac{1-alpha}{2alpha}-eps}$ as $t to infty$ for any $eps>0$. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.
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 $$left{lambdainCsetm
inus{0}:|arg{(lambda)}|<frac{alpha pi}{2}right},$$ where $alphain (0,1)$ is the order of the fractional differential systems, then the equilibrium of the nonlinear systems is unstable.
We give a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution. We also obtain an explicit description for the set of all bounded (or decay) solutions for these systems.
Our aim in this paper is to investigate the asymptotic behavior of solutions of the perturbed linear fractional differential system. We show that if the original linear autonomous system is asymptotically stable then under the action of small (either
linear or nonlinear) nonautonomous perturbations the trivial solution of the perturbed system is also asymptotically stable.
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 equ
ilibrium is asymptotically stable. As a consequence we extend Lyapunovs first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector ${lambda in C : |arg lambda| > frac{alpha pi}{2}}$ where $alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.
