Arclength Control-Based Continuation of Systems Featuring Internally-Resonant Modes

Arclength control-based continuation (ACBC) enables the experimental identification of frequency response curves (FRC) in nonlinear systems, including their unstable branches. So far, its application has been limited to systems exhibiting isolated nonlinear modes. This work extends the ACBC methodology to nonlinear systems featuring internally-resonant modes. Specifically, we consider the case where an internal resonance induces a Neimark–Sacker bifurcation, leading to an unstable branch of periodic responses and quasiperiodic oscillations. We demonstrate that ACBC can stabilize the unstable branch through two examples, namely a system of two coupled Duffing oscillators exhibiting
a 1:1 internal resonance and a nonlinear cantilever beam exhibiting a 3:1 internal resonance between its first two modes.