Damping efficiency of the Duffing system with additional fractional terms

• High efficiency intermittent bistable states emerge from fractionally. • Combination of different fractional terms leads to new dynamics characteristics. • Fractional Duffing system investigations using differential transform method. • The energy dissipation depends on different combinations of fractional terms. • Fractionality of the system modifies energy dissipation efficiency. Fractional derivatives appear to be a convenient and effective tool for describing complex processes, systems, and material characteristics. In this study these mathematical operations are used to modify an exemplary non-linear system (the Duffing system) by adding additional fractional components. Numerical analyses were performed to check how the orders of derivatives in fractional terms affect the energy efficiency of the modified system. A numerical differential transform method ( D T M ) was applied to solve fractional-order differential equations quickly and effectively. This analysis focuses on fractional terms with small intensity coefficients and low values of the fractional orders of derivatives (i.e. close to integer values). The results show that in some cases the fractional elements clearly modify the system dynamics and significantly increase the system energy efficiency. Different fractional derivatives can have markedly different qualitative effects and we show that introducing multiple fractional terms can stabilise changes to the energy efficiency with, for example, high efficiency intermittent bistable solutions.