Measuring nonlinear resonances using extended arclength control-based continuation

Nonlinear systems can resonate not only at their natural frequencies but also at integer and fractional multiples thereof, giving rise to primary, superharmonic, subharmonic, and ultra-subharmonic resonances. Measuring this rich resonance structure remains challenging, as conventional frequency sweeps reveal only stable responses and often miss isolated branches. In this paper, we introduce an experimental framework that uncovers the complete resonance landscape of nonlinear dynamical systems without requiring a mathematical model. The approach relies on extended arclength control-based continuation (x-ACBC), which combines real-time feedback control with arclength continuation to track both stable and unstable multiharmonic periodic responses. The method is demonstrated experimentally using an electronic circuit emulating a Duffing oscillator. Beyond the primary resonance, x-ACBC reveals numerous secondary resonances, including several that appear as isolated branches. These results provide a robust framework for the experimental characterization of complex resonance phenomena in nonlinear systems.