Strongly Nonlinear Responses in Coupled Duffing Oscillators: Homotopy-Enriched Multiple Scales and Branch-Resolved Continuation Benchmarking

Abstract

First-order classical multiple scales (MMS) linearizes frequency detuning and discards a quadratic correction that grows away from resonance. Homotopy-enriched multiple scales (EMMS) uses the forcing-frequency carrier and retains the full squared-frequency mismatch. We derive coupled modulation equations and explicit stability Jacobians for a two-degree-of-freedom Duffing system with a general cubic modal tensor, and compare its algebraic responses with AUTO2000 periodic-orbit continuation. The principal benchmark is a large-detuning stress test. On the 107 reference cells matched by both methods, the median amplitude errors are 1.28% for EMMS and 10.89% for classical MMS. EMMS matches 186 of 201 reference cells, and classical MMS 107. The lower-fold frequency errors are 0.011% and 0.503%, respectively; the EMMS and reference lower-fold frequencies both round to 1.108. Both methods recover a separate weak small-detuning control. Shooting of the full equations of motion shows that the maximum sampled peak–fundamental difference increases from 3.55% to 4.28% with forcing. Classical MMS predicts both secondary-lobe stability boundaries more closely, while EMMS predicts the interval width more closely. The symmetric coupling sweep tests consistency because the direct coupling cubic term cancels from the first modal equation; an asymmetric benchmark exercises the additional odd tensor coefficients.