UK Nonlinear News , February 1998.

Chaos, Nonlinearity and Control

Mario di Bernardo's regular column

Mario di Bernardo has published a number of papers on the analysis and control of chaotic systems. He is currently a member of the Applied Nonlinear Mathematics Group, Department of Engineering Mathematics , University of Bristol. He writes quarterly columns for us from the frontiers of nonlinearity in Control Engineering and Electronics.

We are looking for additional columnists to report back from the different corners of the nonlinear world. If you would like to become another of them, do contact the editors at

Report: Analysis of Piece-wise Smooth Systems and Their Applications

Mario's picture Last autumn a series of informal workshops on the analysis of piecewise smooth systems and their applications was organised by the Applied Nonlinear Mathematics Group at the Department of Engineering Mathematics of the University of Bristol. During these workshops several distinguished speakers, working in different areas of Science, ranging from Pure Mathematics to Electronic Engineering, discussed their ideas on the subject.

Dr. David Chillingworth, Institute for Sounds and Vibration, University of Southampton, focused on the global phase space structure for an impact oscillator. Following a topological approach, he sketched a full picture of all the possible bifurcations exhibited by this system. He showed how these bifurcations can be classified in the three main types of transitions (named after their geometrical structure as lips, beaks and swallow tails) using singularity theory.

In the following talk several applications were presented by Professor Vladimir Babitsky, School of Mechanical Engineering, University of Loughborough. He discussed the occurrence of impact dynamics in mechanical rigs, showing how phenomena such as resonance that have been usually avoided in classical engineering design, can be exploited to achieve greater flexibility and control. In particular, he detailed the case of a resonant robot, showing a video produced with his students at Moscow State Technical University. Professor. Babitsky also reported on some of his theoretical results on lattices of impacting systems where he derived the occurrence of coherent nonlinear behaviour, i.e. stable solutions characterised by all the systems in the lattice impacting simultaneously.

Piece-wise smooth models do not arise only in engineering, they are also common in theoretical physics as Dr. Prellberg, Department of Theoretical Physics, UMIST, outlined in his presentation on the dynamics of a single particle in a horizontally shaken box. This problem, which is a first step towards the understanding the dynamics of more complicated systems (such as shaken granular media), presents many interesting characteristics. Chattering and other non-smooth phenomena take place and the impact map contains intricate KAM theorem structures, showing surprising similarities with other non-smooth systems.

Finally, applications to Power Electronics were presented by Dr. Jonathan Deane, Department of Electronic and Electrical Engineering, Univerity of Surrey during the last workshop of the series. Dr. Deane spoke on the problem of studying the spectral properties of the output of a chaotic ``boost'' DC/DC converter (see my article in UK Nonlinear News 7 for more details). The possibility of linking the spectral properties of the circuit output to the parameters of the map describing its dynamics was outlined. Namely, using chaos the energy could be spread over the entire waveform spectrum instead of having few energy peaks in it. This would be an advantage for EMC (electromagnetic compatibility) problems, suggesting that gaining a deeper insight into the spectral properties of a chaotic waveform could be of immediate relevance in applications.

All the themes presented generated a lot of debate and discussion and outlined the use of non-smooth models in many diverse areas of science and engineering.

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