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By using a time-domain approach, this paper analyzes the chaotic dynamics of the fractional-order system obtained by coupling two fractional Lorenz systems. The work exploits the Adomian decomposition method (ADM), shows that the proposed fractional system with order as low as 2.88 exhibits a novel four-wing chaotic attractor. The existence of chaos is also confirmed by the application of the "0-1 test" for chaos. Finally, an analysis of the system equilibria is conducted, which shows that the fractional-order four-wing attractor is truly the counterpart of the recently introduced integer-order four-wing attractor.

This tutorial investigates bifurcation and chaos in the fractional-order Chen system from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which allows the solution of the fractional differential equations to be written in closed form. By taking advantage of the capabilities given by the decomposition method, the paper illustrates two remarkable findings: (i) chaos exists in the fractional Chen system with order as low as 0.24, which represents the smallest value ever reported in literature for any chaotic system studied so far; (ii) it is feasible to show the occurrence of pitchfork bifurcations and period-doubling routes to chaos in the fractional Chen system, by virtue of a systematic time-domain analysis of its dynamics.

In this tutorial the chaotic behavior of the fractional-order Chua's circuit is investigated from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which enables the solution of the fractional differential equations to be found in closed form. By exploiting the capabilities offered by the decomposition method, the paper presents two remarkable findings. The first result is that a novel bifurcation parameter is identified, that is, the fractional-order q of the derivative. The second result is that chaos exists in the fractional Chua's circuit with order q = 1.05, which is the lowest order reported in literature for such circuits. Finally, a reliable and efficient binary test for chaos (called "0–1 test") is utilized to detect the presence of chaotic attractors in the system dynamics.

This paper focuses on the numerical study of chaotic dynamics via the Adomian decomposition method. The approach, which provides series solutions of the system equations, is first applied to Chua's circuit and Chua's oscillator, both with cubic nonlinearity. Successively, the method is utilized for obtaining hyperchaotic multiscroll attractors in a ring of three Chua's circuits, where the smooth nonlinearities are Hermite interpolating polynomials. The reported examples show that the approach presents two main features, i.e. the system nonlinearity is preserved and the chaotic solution is provided in a closed form.

This paper illustrates a reliable binary test for detecting the presence of chaos in nonlinear systems described by fractional-order differential equations. The method, which is inspired by the technique proposed in [Gottwald & Melbourne, 2004] for integer-order differential equations, does not require phase space reconstruction of the fractional system. It consists of obtaining the data series, constructing a random walk-type process and studying how the variance of the random walk scales with time. In order to show the capabilities of the approach, the test is successfully applied to three well-known dynamical systems, i.e. fractional Chua, Chen and Lorenz systems.

By considering a chain of Chua's circuits, this paper focuses on the numerical study of hyperchaotic 3D-scroll attractors via the Adomian decomposition method. The objective is achieved by designing a new type of nonlinearity based on the Hermite interpolating polynomials. Several examples of 3D-scroll attractors highlight the potential of the proposed approach.

In this paper the attention is focused on the numerical study of hyperchaotic 2D-scroll attractors via the Adomian decomposition method. The approach, which provides series solutions of the system equations, is first applied to weakly-coupled Chua's circuits with Hermite interpolating polynomials. Then the method is successfully utilized for achieving hyperchaos synchronization of two coupled Chua's circuits. The reported examples show that the approach presents two main features, i.e., the system nonlinearity is preserved and the chaotic solution is provided in a closed form.