Bifurcations in piecewise-smooth systems Chris Budd

1. Bifurcations in piecewise-smooth systems Chris Budd

2. What is a piecewise-smooth system? Map Heartbeats or Poincare maps Flow Rocking block, friction, Chua circuit Hybrid Impact or control systems

3. PWS FlowPWS Sliding FlowHybrid

4. 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

5. 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.

6. Will illustrate the behaviour of piecewise smooth systems by looking at • Maps • Hybrid impacting systems

7. Some piecewise-smooth maps Linear, discontinuous Square-root, continuous

8. Both maps have fixed points over certain ranges of Border collision bifurcations occur when for certain parameter values the fixed points intersect with the discontinuity set Get exotic dynamics close to these parameter values

9. Dynamics of the piecewise-linear map Homoclinic orbit Fixed point Fixed point Period addingFarey sequence

10. Dynamics of the piecewise-linear map Period addingFarey sequence Chaotic

11. Square-root map Map arises in the study of grazing bifurcations of flows and hybrid systems Infinite stretching when Fixed point at

12. Period adding Chaos

13. Immediate jump to robust chaos Partial period adding

14. Get similar behaviour in higher-dimensional square-root maps Map [Nordmark] also arises naturally in the study of grazing in flows and hybrid systems.

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

16. Impact oscillators: a canonical hybrid system obstacle

17. Periodic dynamics Chaotic dynamics Experimental Analytic

18. Complex domains of attraction of periodic orbits

19. Regular and discontinuity induced bifurcations as parameters vary. Regular and discontinuity induced bifurcations as parameters vary Period doubling Grazing

20. Grazing occurs when periodic orbits intersect the obstacle tanjentially

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

22. 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

23. 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 the discontinuous maps

24. 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

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