Fractional-order Chua’s circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos. (English) Zbl 1147.34302

Cafagna, Donato; Grassi, Giuseppe

Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, No. 3, 615-639 (2008).

Summary: 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.

MSC:

34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34C23	Bifurcation theory for ordinary differential equations
34C28	Complex behavior and chaotic systems of ordinary differential equations
26A33	Fractional derivatives and integrals
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
37-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory

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