Effects of Fractional Differentiation Orders on a Coupled Duffing Circuit

Abstract

Classical electrical circuits consist of resistors and capacitors and are governed by integer-order model. Circuits may have so-called fractance which represents an electrical element with fractional order impedance. Therefore, fractional order derivative is important to study the dynamical behaviors of circuits. This paper extends the classical coupled Duffing circuits to cover a new fractional order Duffing system consisting of two identical periodic forced circuits coupled by a linear resistor. The fundamental resonance responses under various paired and unpaired fractional orders are investigated in detail using the harmonic balance in combination with polynomial homotopy continuation. The approximate solutions having a high degree of accuracy in the steady state response are sought. There exist different shapes of frequency versus response and excitation amplitude versus response curves under various fractional orders. Multiple-valued solutions and nonclassical bifurcations are observed analytically and verified numerically. The influence of coupling intensity on the fundamental resonance response is also examined. New contributions include the innovative introduction of fractional order to the coupled Duffing circuits, the explicit integration of fractional order and the linear consideration of higher harmonics to improve the nonlinear solutions of the lower harmonics without increasing the computational complexity.