Halanay inequality involving Caputo-Hadamard fractional derivative and application

A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability.

Combination Synchronization of Fractional Systems Involving the Caputo–Hadamard Derivative

The main aim of this paper is to investigate the combination synchronization phenomena of various fractional-order systems using the scaling matrix. For this purpose, the combination synchronization is performed by considering two drive systems and one response system. We show that the combination synchronization phenomenon is achieved theoretically. Moreover, numerical simulations are carried out to confirm and validate the obtained theoretical results.

Applications of Fixed Point Theory to Investigate a System of Fractional Order Differential Equations

We investigate a nonlinear system of pantograph-type fractional differential equations (FDEs) via Caputo-Hadamard derivative (CHD). We establish the conditions for existence theory and Ulam-Hyers-type stability for the underlying boundary value system (BVS) of FDE. We use Krasnoselskii’s and Banach’s fixed point theorems to obtain the desired results for the existence of solution. Stability is an important aspect from a numerical point of view we investigate here. To justify the main work, relevant examples are provided.

Global bifurcation at isolated singular points of the Hadamard derivative

Consider F ∈ C ( R × X , Y ) such that F ( λ , 0) = 0 for all λ ∈ R , where X and Y are Banach spaces. Bifurcation from the line R × { 0 } of trivial solutions is investigated in cases where F ( λ , · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for μ to be a bifurcation point and yield global information about the connected component of { ( λ , u ) : F ( λ , u ) = 0 and u ≠ 0 } ∪ { ( μ , 0 ) } containing ( μ , 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Fréchet differentiability is not available. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Chaos Detection of the Chen System with Caputo–Hadamard Fractional Derivative

In this paper, we investigate the chaotic behaviors of the Chen system with Caputo–Hadamard derivative. First, we construct some practical numerical schemes for the Chen system with Caputo–Hadamard derivative. Then, by means of the variational equation, we estimate the bounds of the Lyapunov exponents for the considered system. Furthermore, we analyze the dynamical evolution of the Chen system with Caputo–Hadamard derivative based on the Lyapunov exponents, and we found that chaos does exist in the considered system. Some phase diagrams and Lyapunov exponent spectra are displayed to verify our analysis.