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III: Hybrid systems and the grazing bifurcation Chris Budd

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Hybrid system Impact or control systems

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Impact oscillator: a canonical hybrid system obstacle

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

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

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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

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

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

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Small perturbations of a non-impacting orbit v u

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Small perturbations of an orbit with a high velocity impact

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

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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

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G+G+ G G-G- A1A1 A2A2

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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

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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

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G

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

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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

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Newtons cradle w z u Mass ratio

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The square rotating cam

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Bifurcation diagram

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