Bifurcations in Non-smooth Dynamical Systems

Author: Remco Leine

Physical phenomena such as impact, dry friction and backlash in mechanical systems are often studied by means of mathematical models with some kind of discontinuity or non-smoothness. Non-smooth dynamical systems can be divided in three types according to their degree of non-smoothness: non-smooth continuous systems, differential inclusions (Fillipov systems) and impulsive systems with a state reset (e.g. mechanical systems with instantaneous impact).

It is often desirable to know how the equilibria and periodic solutions of a system change when a parameter of the system is changed. Such parameter studies are usually conducted by means of path-following (or continuation) techniques where a branch of equilibria or periodic solutions is followed while varying a parameter. The number and type of equilibria and periodic solutions (being stable or unstable) can change at a certain parameter value. This qualitative change in the structural behaviour of the system is loosely called bifurcation. Many practical problems in engineering are related to vibrations caused or influenced by transitions between modes (e.g. stick, slip, contact, no contact). Depending on the way of modelling, a mathematical model of the physical system may belong to one of the three classes of non-smooth dynamical systems mentioned above. This urges for a description of the bifurcation behaviour of non-smooth dynamical systems. The objective of this research project is to investigate different aspects of bifurcations in non-smooth dynamical systems. Non-smooth dynamical systems expose non-classical bifurcations, being different from bifurcations occurring in smooth dynamical systems. In this work, topics from non-smooth analysis and Floquet theory are combined to give more insight into bifurcations of periodic solutions of non-smooth dynamical systems.

phase plane of a non-smooth Hopf bifurcation

Publications:

Leine, R.I. & Nijmeijer, H. Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics Vol. 18, Berlin Heidelberg New-York, Springer-Verlag, 2004.

Leine, R.I., "Bifurcations of equilibria in non-smooth continuous systems", Physica D, Vol 223, pp. 121-137, 2006. PDF(2343kb)

Leine, R.I. and van Campen, D.H., "Bifurcation phenomena in non-smooth dynamical systems", European Journal of Mechanics A/Solids, Vol. 25, pp. 595-616, 2006. PDF(625kb)

Leine, R.I. and Van Campen, D.H., "Discontinuous fold bifurcations", Journal on Systems Analysis Modeling Simulation, Vol 43, No 3., pp. 321-332, 2003.

Leine, R.I. and van Campen, D.H., "Discontinuous bifurcations of periodic solutions", Mathematical and Computer Modelling, Vol. 36,3, pp. 259-273, 2002. PDF(1207kb)

Leine, R.I. and van Campen, D.H., "Discontinuous fold bifurcations in mechanical systems" Archive of Applied Mechanics, Vol. 72, pp. 138-146, 2002. PDF (125kb)

Leine, R.I., Van Campen, D.H. and van de Vrande, B.L., "Bifurcations in nonlinear discontinuous systems", Nonlinear Dynamics, Vol. 23, No. 2, pp.105-164, 2000. PDF (488kb)