A model- and derivative-free arclength continuation method for tracing bifurcation diagrams in nonlinear systems

Nonlinear mechanical systems can exhibit multiple coexisting periodic responses, including both stable and unstable solutions. These responses encompass fundamental, superharmonic, and subharmonic resonances, which are essential for understanding the full system dynamics but are often overlooked in classical experimental studies. Traditional open loop testing methods are unable to track unstable branches, resulting in missed solutions, sudden jumps at bifurcations, and incomplete bifurcation diagrams. Isolated branches, in particular, are challenging to access without advanced control strategies. Control-based continuation methods have emerged as powerful experimental tools. This study presents applications of an extended version of a derivative- and model- free arclength control based continuation method (x-ACBC), which enables the experimental identification of complete bifurcation diagrams. By combining a double sweep strategy with temporarily invasive control, x-ACBC successfully accesses unstable and isolated periodic responses.