Dynamical analysis of fractional-order Mathieu equation

The dynamical characteristics of Mathieu equation with fractional-order derivative is analytically studied by the Lindstedt-Poincare method and the multiple-scale method. The stability boundaries and the corresponding periodic solutions on these boundaries for the constant stiffness δ0=n2 (n = 0, 1, 2, …), are analytically obtained. The effects of the fractional-order parameters on the stability boundaries and the corresponding periodic solutions, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient (ELDC) and the equivalent linear stiffness coefficient (ELSC). The comparisons between the transition curves on the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. The following analysis is focused on the effects of the fractional parameters on the stability boundaries located in the δ-e plane. It is found that the increase of the fractional order p could make the ELDC larger and ELSC smaller, which could result into the rightwards and upwards moving of the stability boundaries simultaneously. It could also be concluded the increase of the fractional coefficient K1 would make the ELDC and ELSC larger, which could move the transition curves to the left and upwards at the same time. These results are very helpful to design, analyze or control this kind of system, and could present beneficial reference to the similar fractional-order system.

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