Abstract
A key feature of many nonlinear dynamical systems is the presence of co-existing solutions, i.e, nonlinear systems are often sensitive to initial conditions. While there have been many studies to explore this behavior from a numerical perspective, in which case it is trivial to prescribe initial conditions (for example using a regular grid), this is more challenging from an experimental perspective. This paper will discuss the basins of attraction in a simple mechanical experiment. By applying both small and large stochastic perturbations to steady-state behavior, it is possible to interrogate the initial condition space and map-out basins of attraction as system parameters are changed. This tends to provide a more complete picture of possible behavior than conventional bifurcation diagrams with their focus on local steady-state behavior.
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References
Thompson JMT, Stewart HB (2002) Nonlinear dynamics and chaos: geometrical methods for engineers and scientists, 2nd edn. Wiley, Chichester
McDonald SW, Grebogi C, Ott E, Yorke JA (1985) Fractal basin boundaries. Physica D 17:125
Murphy KD, Virgin LN, Rizzi SA (1996) Experimental snap-through boundaries for acoustically excited, thermally buckled plates. J Exp Mech 36:312–317
Virgin LN (2000) Introduction to experimental nonlinear dynamics: a case study in mechanical vibration. Cambridge University Press, New York
Waite JJ, Virgin LN, Wiebe R (2013) Competing responses in a discrete mechanical system. Int J Bifurcat Chaos (to appear)
Wiebe R, Virgin LN, Stanciulescu I, Spottswood SM, Eason TG (2013) Characterizing dynamic transitions associated with snap-through: a discrete system. J Comput Nonlinear Dyn 8. doi:10.1115/1.4006201
Krantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, Cambridge
Cusumano JP, Kimble BW (1995) A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: application to a two-well oscillator. Nonlinear Dyn 8:213–235
Murphy KD, Bayly PV, Virgin LN, Gottwald JA (1994) Measuring the stability of periodic attractors using perturbation-induced transients: applications to two nonlinear oscillators. J Sound Vib 172:85–102
Acknowledgements
This work was partially supported by the US Air Force, AFOSR Grant no. FA9550-09-1-0201, and NSF grant 0927186 (Dynamical Systems).
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© 2014 The Society for Experimental Mechanics, Inc.
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Virgin, L.N., Waite, J.J., Wiebe, R. (2014). Co-existing Responses in a Harmonically-Excited Nonlinear Structural System. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 2. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-04522-1_1
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DOI: https://doi.org/10.1007/978-3-319-04522-1_1
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