UK Nonlinear News , November 1997.

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

GRAZING AND SLIDING Nonlinear Phenomena in Piecewise Smooth Systems

Mario's picture The study of piecewise smooth systems has been the subject of much ongoing research, see for instance [1] for some early results on impacting systems. The past few years have seen a dramatic increase of interest in both the academic and industrial world. In fact, piecewise smooth models are used in many different branches of applied science and engineering. Examples include impacting machines, systems oscillating under the effect of an earthquake, power electronic circuits and many others.

These systems can be described by a set of ordinary differential equations (ODEs) or map. The phase space of a general piecewise smooth system or map can then be divided in countably many regions; in each region the system having a different smooth functional form. At the boundaries of these regions the system functional may be either discontinuous or have discontinuous first derivatives. Many interesting bifurcations occur when a part of a system trajectory (e.g. a periodic orbit) becomes tangent to one of this boundary as one or more system parameters are varied.

There is now a rapidly expanding literature characterising the seemingly exotic dynamical transitions that are unique to piecewise smooth system. These phenomena occur in many systems of relevance in applications [2,3,4,5]. It has been shown for instance, that chaos and bifurcations are organising the evolution of many power converters which are widely used in electrical applications, (see my contribution in UK-NLN Feb. 1997 issue for more details).

As often happens in Nonlinear Dynamics, phenomena exhibited by a particular type of system are also found in many others. In pioneering work on impact oscillators, for example, it has been shown that their dynamics are crucially affected by grazing between periodic orbits and the boundary of the physical domain. Grazing bifurcations have been shown to be a universal phenomenon in characterising the route to chaos of many different piecewise smooth systems [6,7,8].

Also, Border Collision bifurcations (called C-bifurcations in the Russian Literature), introduced and classified by Feigin and more recently by Yorke et al. are an important clue to understand the structure of the bifurcation diagrams exhibited by piecewise smooth systems [9,10,11]. They occur when the map or flow is continuous but non-smooth across the boundary between different regions of the phase space.

In recent work with Dr. Alan Champneys, University of Bristol and Prof. Chris Budd, University of Bath we came across the importance of the so-called sliding solutions in determining the nature and structure of chaotic attractors exhibited by piecewise smooth systems (see preprint). These are solutions moving within the discontinuity set of a piecewise smooth system and their detailed analysis and characterisation can be found in [12]. Although they are often used in mathematical control theory to achieve the control of a dynamical system (for instance through sliding or Variable Structure controllers [13]), they are almost never mentioned in the nonlinear dynamics literature.

Studying the occurrence of nonlinear phenomena in DC/DC converters, together with groups in Napoli and Barcelona, we found out that sliding is very important to understand their dynamics. Moreover, a peculiar ``route to sliding'' was detected. Namely, periodic orbits characterised by different numbers of discontinuous events (switchings) accumulate onto a sliding orbit, i.e. a periodic solution having infinite switchings, following a specific bifurcation diagram.

Figure 1

Figure 1: "Double Spiral Bifurcation Diagram": as a parameter is varied branches of three-periodic orbits characterised by different numbers of switchings accumulate onto a sliding orbit, lying in the middle of the spiral. (M. di Bernardo, E. Fossas, G. Olivar & F. Vasca, "Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter", to appear in International Journal of Bifurcations and Chaos, November 1997)

This ``double spiral bifurcation diagram'', depicted in Fig. 1, has certain interesting features. Its corners, for instance, are grazing bifurcation points and their occurrence follows a specific scaling law (for more information see [14]). It is possible to understand this structure analytically and it seems that such a structure might occur also in a more general class of systems.

Clearly there is a lot of rich structure unique to piecewise smooth dynamics and it is an enticing topic both because of the ease with which such structure can be analysed in piecewise linear systems and due to the wealth of realistic applications.

I would like to use this space on UK-Nonlinear News to suggest to researchers interested in the study and analysis of nonlinear phenomena in piecewise smooth systems to contact me, with a view to forming an informal research network. I believe there are many exciting open questions regarding these systems that can only be answered through the discussion and integration of results obtained in different areas of Science and Engineering.

For any comment, suggestion or advice please write me an e-mail:


  1. J.M.T. Thompson, and H.B. Stewart, Nonlinear Dynamics and Chaos, John Wiley and Sons, 1986.
  2. E. Fossas, and G. Olivar, "Study of Chaos in the Buck Converter", IEEE Trans. on Circuits and Systems-I, to appear.
  3. D.C. Hamill, J.H.B. Deane, and D.J. Jefferies, "Modeling of Chaotic DC-DC Converters by Iterated Nonlinear Mappings", IEEE Trans. on Power Electronics, 7:1, pp. 25-36, 1992.
  4. S.H. Doole and S.J. Hogan, "A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation", Dynamics and Stability of Systems, 11, pp. 19-29, 1996.
  5. C. Budd, F. Dux, A. Cliffe, "The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators", Journal of Sound and Vibration, 184:3, pp. 475-502, 1995.
  6. C.J. Budd, and A.G. Lee, "Double Impact Orbits of a Single-Degree-of-Freedom Impact Oscillator Subject to Periodic Forcing of Odd Frequency", Proc. Royal Soc. London A.
  7. A. B. Nordmark, "Non-Periodic Motion Caused by Grazing Incidence in an Impact Oscillator", Journal of Sound and Vibration, 145:2, pp. 279-297, 1991.
  8. S. Foale and S.R. Bishop, "Dynamical Complexities of forced impacting systems", Phil. Trans. Royal Society London A, 338, pp. 547-556, 1992.
  9. H.E. Nusse, and J.A. Yorke, "Border Collision bifurcation: an explanation for observed bifurcation phenomen", Physical Review E, 49, pp. 1073-1076, 1994.
  10. H.E. Nusse, and J.A. Yorke, "Border Collision bifurcations including 'period two- to period three' for piecewise smooth systems", Physica D, 57, pp. 39-57, 1992.
  11. M.I. Feigin, "The increasingly complex structure of the bifurcation tree of a piecewise-smooth system", Journal of Applied Maths and Mechanics, 59, pp. 853-863, 1995
  12. A.F. Filippov, Differential Equations with Discontinuous Righthand sides, Kluwier Academic Press, 1988.
  13. V.I. Utkin, Sliding Modes and their Application in Variable Structure Systems, MIR, Moscow, 1978.
  14. M. di Bernardo, A.R. Champneys, and C.J. Budd, "Grazing, skipping and sliding: analysis of the non-smooth dynamics of the DC/DC buck converter", submitted to Nonlinearity, June 1997.

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