Chattering and grazing in impact oscillators Chris Budd

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

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

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

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

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

Chattering occurs when an infinite number of impacts occur in a finite time v u u v

Poincare Maps

Complex domains of attraction of periodic orbits

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

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

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

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

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.