Steady state response analysis for fractional dynamic systems based on memory-free principle and harmonic balancing

• A method is proposed to study steady responses of fractional dynamic systems.
• It is based on YA memory-free principle and harmonic balancing.
• It provides periodic and triple-period solutions accurately.
• It is capable of analyzing both stable and unstable periodic responses.
• Bifurcations of fractional Duffing oscillator are investigated in detail.

A semi-analytic approach is proposed to analyze steady state responses of dynamic systems containing fractional derivatives. A major purpose is to efficiently combine the harmonic balancing (HB) technique and Yuan–Agrawal (YA) memory-free principle. As steady solutions being expressed by truncated Fourier series, a simple yet efficient way is suggested based on the YA principle to explicitly separate the Caputo fractional derivative as periodic and decaying non-periodic parts. Neglecting the decaying terms and applying HB procedures result into a set of algebraic equations in the Fourier coefficients. The linear algebraic equations are solved exactly for linear systems, and the non-linear ones are solved by Newton–Raphson plus arc-length continuation algorithm for non-linear problems. Both periodic and triple-periodic solutions obtained by the presented method are in excellent agreement with those by either predictor–corrector (PC) or YA method. Importantly, the presented method is capable of detecting both stable and unstable periodic solutions, whereas time-stepping integration techniques such as YA and PC can only track stable ones. Together with the Floquet theory, therefore, the presented method allows us to address the bifurcations in detail of the steady responses of fractional Duffing oscillator. Symmetry breakings and cyclic-fold bifurcations are found and discussed for both periodic and triple-periodic solutions.