Tripled Fixed Points and Existence Study to a Tripled Impulsive Fractional Differential System via Measures of Noncompactness

In this paper, a tripled fractional differential system is introduced as three associated impulsive equations. The existence investigation of the solution is based on contraction principle and measures of noncompactness in terms of tripled fixed point and modulus of continuity. Our results are valid for both Kuratowski and Hausdorff measures of noncompactness. As an application, we apply the obtained results to a control problem.

Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

Normal Form of Bifurcation for Caputo-Hadamard Fractional Differential System With a Parameter

This paper focuses on the normal form computation of the pitchfork bifurcation for the Caputo-Hadamard fractional differential system with a parameter. By using Taylor’s expansion and Implicit Function Theorem, we derive the normal form of the pitchfork bifurcation for the Caputo-Hadamard fractional differential system with a parameter.

On the Stability Analysis of Linear Unperturbed Non-integer Differential Systems

In this paper, the stability of non-integer differential system is studied using Riemann-Liouville and Caputo derivatives. The stability notion for determining the stability/asymptotic stability or otherwise fractional differential system is given. Example is provided to demonstrate the effectiveness of the result.

Stability Analysis of Perturbed Linear Non-integer Differential Systems

In this work, the effect of perturbation on linear fractional differential system is studied. The analysis is done using Riemann-Liouville derivative and the conclusion extended to using Caputo derivative since the result is similar. Conditions for determining the stability and asymptotic stability of perturbed linear fractional differential system are given.

Successive approximations for random coupled Hilfer fractional differential systems

In this paper, we study the global convergence of successive approximations as well as the uniqueness of the random solution of a coupled random Hilfer fractional differential system. We prove a theorem on the global convergence of successive approximations to the unique solution of our problem. In the last section, we present an illustrative example.

An Estimate of the Bound of the Lyapunov Exponents for Caputo-Hadamard Fractional Differential System

This paper is devoted to estimating the bound of the Lyapunov exponents for the Caputo-Hadamard fractional differential system. First, using the Gronwall inequality, we analyze the continuous dependence of the solution to the Caputo-Hadamard fractional initial value problem. Then, we define the Lyapunov exponents for the Caputo-Hadamard fractional differential system and estimate their bounds. Besides, numerical examples are displayed which support the theoretical results.