III: Hybrid systems and the grazing bifurcation Chris Budd

Hybrid system Impact or control systems

Impact oscillator: a canonical hybrid system obstacle

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

Grazing occurs when periodic orbits intersect the obstacle tanjentially This is highly destabilising

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

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

Now give an explanation for this observed behaviour. To do this we need to construct a Poincare map related to the flow

Small perturbations of a non-impacting orbit v u

Small perturbations of an orbit with a high velocity impact

Small perturbations of a non-impacting orbit Flow matrices Saltation matrix to allow for the impact

v Small perturbations of a grazing orbit (v = 0) u-sigma S breaks down! G: Initial data leading to a graze … v = 0 Large perturbation

G+G+ G G-G- A1A1 A2A2

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

If A has complex eigenvalues we see discontinuous transitions between periodic orbits similar to the piecewise-linear case. If A has real eigenvalues we see similar behaviour to the 1D map

G

Complex domains of attraction of the periodic orbits dx/dt x

Systems of impacting oscillators can have even more exotic behaviour which arises when there are multiple collisions. This can be described by looking at the behaviour of discontinuous maps

Newtons cradle w z u Mass ratio

The square rotating cam

Bifurcation diagram