82 Alexandre Spits et. al. revealing distinct branches in regions with multiple solutions. The jump phenomenon manifests when the system transitions from multiple possible solutions to a single solution. Close to fold bifurcations, identification becomes more challenging because the Fourier coefficients of the merging branches are closer in value, suggesting that a reduced integral gain may be needed for the non-fundamental control. Conclusion This work introduced an extended version of the ACBC method, a novel online, derivative-free method for experimental continuation. By controlling the non-fundamental resonant harmonic, x-ACBC can detect and identify complex superharmonic and subharmonic resonance branches. The method was demonstrated experimentally using an electronic Duffing system. Beside the fundamental resonance, three secondary resonances, namely 3:1, 2:1 and 1:3 resonances, could be completely characterized. Acknowledgments Alexandre Spits and Ghislain Raze are a research fellow and a postdoctoral researcher, respectively, of the Fonds de la Recherche Scientifique - FNRS, which is gratefully acknowledged. References 1. Peter, S. and Leine, R.I. “Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked loop excitation”. Mechanical Systems and Signal Processing, 96:139–158 (2017) 2. Barton, D. and Sieber, J. “Systematic experimental exploration of bifurcations with noninvasive control”. Physical review E., 87:052916 (2012) 3. Volvert, M. and Kerschen, G. “Phase resonance nonlinear modes of mechanical systems”. Journal of Sound and Vibration, 511:116355 (2021) 4. Zhou, T. and Kerschen, G. “Identification of secondary resonances of nonlinear systems using phase-locked loop testing”. Journal of Sound and Vibration, 590:118549 (2024) 5. Abeloos, G. Control-based methods for the identification of nonlinear structures. PhD thesis, University of Lie`ge (2022) 6. Sieber, J. and Krauskopf, B. “Control-based continuation of periodic orbits with a time-delayed difference scheme”. International Journal of Bifurcation and Chaos, 17(08):2579–2593 (2007) 7. Raze, G., Abeloos, G., and Kerschen, G. “Experimental continuation in nonlinear dynamics: recent advances and future challenges”. arXiv (2024) 8. Spits, A. “Derivative-less control-based nonlinear vibration testing”. Master’s thesis, University of Lie`ge (2024) 9. Bureau, E., Schilder, F., Elmega˚rd, M., Santos, I.F., Thomsen, J.J., and Starke, J. “Experimental bifurcation analysis of an impact oscillator – tuning a non-invasive control scheme”. Journal of Sound and Vibration, 333:5464–5474 (2014)
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