Resonance and safety basin erosion of fractional order delay asymmetric Duffing-Mathieu system

Abstract

This paper focuses on the resonance and safety basin erosion of the fractional-order delayed asymmetric Duffing-Mathieu system. Its innovation compared with existing studies lies in: for the first time, integrating fractional calculus, time-delay effect and asymmetric stiffness characteristics into a coupled analysis framework, and introducing a memory characteristic correction term of fractional operators when deriving the amplitude-frequency relationship, which improves the accuracy of analytical modeling for non-integer order vibration systems. In the research, the improved averaging method is used to approximate the amplitude-frequency relationship and verify its accuracy, combined with the Jacobian matrix for stability analysis; the cell mapping method is adopted to capture the boundary of fractal attractive basins of coexisting attractors, and the potential function theory is used to quantify the erosion process of the safety basin, which is better than traditional methods in revealing the intrinsic mechanism. This system can simulate the dynamic response of asymmetric vibration structures containing viscoelastic materials under time-delay feedback control, and the research results can provide a theoretical basis for parameter design and safety early warning of related systems.