Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSemi-dynamical systems generated by autonomous Caputo fractional differential equations
81
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 functions $f:R^+rightarrow R^d$ 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 $mathbb{R}^d$.
rate research
Read More
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.
It is shown that the attractor of an autonomous Caputo fractional differential equation of order $alphain(0,1)$ in $mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of N. D. Cong and H.T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations. Journal of Integral Equations and Applications, 29 (2017), 1-24 which shows that no two solutions of such a Caputo FDE can intersect in finite time
We show that any two trajectories of solutions of a one-dimensional fractional differential equation (FDE) either coincide or do not intersect each other. In contrary, in the higher dimensional case, two different trajectories can meet. Furthermore, one-dimensional FDEs and triangular systems of FDEs generate nonlocal fractional dynamical systems, whereas a higher dimensional FDE does, in general, not generate a nonlocal dynamical system.
This manuscript investigates the existence and uniqueness of solutions to the first order fractional anti-periodic boundary value problem involving Caputo-Katugampola (CK) derivative. A variety of tools for analysis this paper through the integral equivalent equation of the given problem, fixed point theorems of Leray--Schauder, Krasnoselskiis, and Banach are used. Examples of the obtained results are also presented.
We present some distinct asymptotic properties of solutions to Caputo fractional differential equations (FDEs). First, we show that the non-trivial solutions to a FDE can not converge to the fixed points faster than $t^{-alpha}$, where $alpha$ is the order of the FDE. Then, we introduce the notion of Mittag-Leffler stability which is suitable for systems of fractional-order. Next, we use this notion to describe the asymptotical behavior of solutions to FDEs by two approaches: Lyapunovs first method and Lyapunovs second method. Finally, we give a discussion on the relation between Lipschitz condition, stability and speed of decay, separation of trajectories to scalar FDEs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
