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• We introduce for the first time in the literature a fractional-calculus-based artificial macroeconomic model. • A new system of fractional difference-order equations is proposed for the Goodwin model. • Using bifurcation diagrams, phase portraits and Lyapunov exponents among other measures, the characteristics of the new system are depicted. • We implement extensive simulations via a new hardware laboratory realization set up. In this novel research, through dynamical analysis, we introduce for the first time a fractional-calculus based artificial macroeconomic model, actually implemented in the Laboratory via a new hardware set up. Firstly, we propose a new model of a discrete-time macroeconomic system where fractional derivatives are incorporated into the system of equations. Using well-known tools and methods, including bifurcation diagrams and Lyapunov exponents, the characteristics of the system are disclosed, and the importance of the fractional-order derivative in the modeling of the system is shown. After that, a laboratory hardware realization is also carried out for the proposed system that provides further insight and a better understanding of the properties of the system. For the hardware realization an Arduino Due™ is chosen in which possess two analog output pins. Experimental results conspicuously illustrate the chaotic behavior of the system. Through the results of the hardware realization, phase portraits and bifurcation diagram of the system are demonstrated, and the effects of the parameters and fractional derivatives are studied. We believe the presented study and its results pave the way for future studies on the incorporation of fractional calculus into macroeconomic models. [ABSTRACT FROM AUTHOR]

• A new scheme for solving a type of singularly perturbed boundary value problem with a second order delay differential equation is presented. • Getting through the solution, Reproducing Kernel Hilbert Space (RKHS) method is implemented. • A construction of the reproducing kernel in Hilbert space satisfying the homogeneous nonlinear boundary conditions of the considered problem is presented. The present paper aims to carry out a new scheme for solving a type of singularly perturbed boundary value problem with a second order delay differential equation. Getting through the solution, we used Reproducing Kernel Hilbert Space (RKHS) method as an efficient approach to obtain the analytical solution for ordinary or partial differential equations that appear in vast areas of science and engineering. A key of this method is to keeping the continuous form of problems. Indeed, without discretizing the continuous problem, we change it to an equivalent iterative form and proving its convergence. Also, we will present a construction of the reproducing kernel in Hilbert space that satisfying the homogeneous nonlinear boundary conditions of the considered problem. Accuracy amount of absolute error with respect to different parameters of singularity has been studied for the performance of this method by solving several hardly nonlinear problems. Error estimation and convergence analysis show that the approximate results have uniform convergence to the continuous solutions. [ABSTRACT FROM AUTHOR]

• A new type-2 fuzzy disturbance-observer-based active control. • Robustness against unknown time-varying uncertainties and disturbances. • Investigation of the chaotic behavior of a fractional-order time-delayed system. • Synchronization of a fractional-order time-delayed financial system. In the present article, as a new approach, a fuzzy disturbance observer is combined with an active controller for the synchronization of fractional-order time-delayed systems. Since the type-2 fuzzy logic system shows better performances than the type-1 fuzzy logic system on handling the uncertainties and disturbances, the type-2 fuzzy disturbance observer is utilized in this study. The stability of the proposed method is performed using the Lyapunov stability theorem and active control concept. Then, a fractional-order time-delayed financial system has been investigated, and some features of this chaotic system have been studied. Finally, by applying the proposed control scheme, synchronization results of the fractional-order time-delayed financial system when there exist time-varying disturbances are presented. Numerical simulations demonstrate the robustness and effectiveness of the proposed control technique. [ABSTRACT FROM AUTHOR]

• Finance and economics are complex nonlinear systems, affected by various external factors. • We study the dynamics and complexity in a fractional-order financial system with time delays. • We observe fascinating transitions to deterministic chaos, including cascading period doubling. • Complexity is particularly high in response to variations of derivative orders. • Derivative orders are identified as key system parameters. Finance and economics are complex nonlinear systems that are affected by various external factors, including of course human action, bilateral relations, conflicts, and policy. Time delays in a financial system take into account the amount of time that passes from a particular policy or decision being made to it actually taking effect. It is thus important to consider time delays as an integral part of modeling in this field. Moreover, many features of financial systems cannot be expressed sufficiently precisely by means of integer-order calculus. Fractional-order calculus alleviates these shortcomings. The aim of this paper is therefore to study the dynamics and complexity in a fractional-order financial system with time delays. We observe fascinating transitions to deterministic chaos, including cascading period doubling, as well as high levels of complexity. This is particularly true in response to variations of derivative orders, which are thus identified as key system parameters. [ABSTRACT FROM AUTHOR]