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In this lecture I will examine nature of subtle phenomenon such grazing bifurcations occurring in non-smooth systems. I will start with linear oscillators undergoing impacts with secondary elastic supports, which have been studied experimentally and analytically for near-grazing conditions . We discovered a narrow band of chaos close to the grazing condition and this phenomenon was observed experimentally for a range of system parameters. Through stability analysis, we argue that this abrupt onset to chaos is caused by a dangerous bifurcation in which two unstable period-3 orbits, created at "invisible" grazing collide .
The experimentally observed bifurcations are explained theoretically using mapping solutions between locally smooth subspaces. Smooth as well as non-smooth bifurcations are observed, and the resulting bifurcations are often as an interplay between them. In order to understand the observed bifurcation scenarios, a global analysis has been undertaken to investigate the influence of stable and unstable orbits which are born in distant bifurcations but become important at the near-grazing conditions . A good degree of correspondence between the experiment and theory fully justifies the adopted modelling approach.
Similar phenomena were observed for a rotor system with bearing clearances, which was analysed numerically  and experimentally . To gain further insight into the system dynamics we have used a path following method to unveil complex bifurcation structures often featuring dangerous co-existing attractors.
- Ing, J., Pavlovskaia, E.E., Wiercigroch, M. and Banerjee, S. 2008 Philosophical Transactions of the Royal Society – Part A 366, 679-704. Experimental study of impact oscillator with one sided elastic constraint.
- Banerjee, S., Ing, J., Pavlovskaia, E., Wiercigroch, M. and Reddy, R. 2009 Physical Review E 79, 037201. Invisible grazing and dangerous bifurcations in impacting systems.
- Ing, J., Pavlovskaia, E., Wiercigroch, M. and Banerjee, S. 2010 International Journal of Bifurcation and Chaos 20(11), 3801-3817. Complex dynamics of bilinear oscillator close to grazing.
- Páez Chávez, J. and Wiercigroch, M. 2013 Communications in Nonlinear Science and Numerical Simulation 18, 2571–2580.Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model.
- Páez Chávez, J., Vaziri Hamaneh, V. and Wiercigroch, M. 2015 Journal of Sound and Vibration 334, 86-97. Modelling and experimental verification of an asymmetric Jeffcott rotor with radial clearance.
- Professor Marian Wiercigroch, Director of Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen
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