1. Chattering and grazing in impact oscillators
Chris Budd

2. Look at exceptional types of dynamics in piecewise-smooth systems
Hybrid systems
Maps

3. Key idea …
The functions or one of their nth derivatives, differ when
Discontinuity set
Interesting discontinuity induced bifurcations occur when limit sets of the flow/map intersect the discontinuity set

4. Impact oscillators: a canonical hybrid system
u(t)
g(t)
obstacle

5. ‘Standard’ dynamics
Periodic dynamics Chaotic dynamics v
Experimental
Analytic u v u

6. Grazing dynamics
Grazing occurs when (periodic) orbits intersect the obstacle tanjentially v

7. Chattering occurs when an infinite number of impacts occur in a finite timev u u v

8. Poincare Maps

15. Complex domains of attraction of periodic orbits

16. Observe grazing bifurcations identical to the dynamics of the two-dimensional square-root map
Transition to a periodic orbit
Non-impacting
orbit
Period-adding

17. Local analysis of a Poincare map associated with a grazing periodic orbit shows that this map has a locally square-root form, hence the observed period-adding and similar behaviour
Poincare map associated with a grazing periodic orbit of a piecewise-smooth flow typically is smoother (eg. Locally order 3/2 or higher) giving more regular behaviour

18. CONCLUSIONS Piecewise-smooth systems have interesting dynamics
Some (but not all) of this dynamics can be understood and analysed
Many applications and much still to be discovered

19. Parameter range for simple periodic orbits
Fractions 1/n
Fractions (n-1)/n

21. Why are we interested in them?
Lots of important physical systems are piecewise-smooth: bouncing balls, Newton’s cradle, friction, rattle, switching, control systems, DC-DC converters, gear boxes …
Piecewise-smooth systems have behaviour which is quite different from smooth systems and is induced by the discontinuity: period adding
Much of this behaviour can be analysed, and new forms of discontinuity induced bifurcations can be studied: border collisions, grazing bifurcations, corner collisions.