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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. [ABSTRACT FROM AUTHOR]

This Letter analyzes the hyperchaotic dynamics of the fractional-order Rössler system from a time-domain point of view. The approach exploits the Adomian decomposition method (ADM), which generates series solution of the fractional differential equations. A remarkable finding of the Letter is that hyperchaos occurs in the fractional Rössler system with order as low as 3.12. This represents the lowest order reported in literature for any hyperchaotic system studied so far. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]