Bifurcations in piecewise-smooth systems Chris Budd

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

PWS Flow PWS Sliding Flow Hybrid

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

Why are we interested in them? Lots of important physical systems are piecewise-smooth: bouncing balls, Newtons 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.

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

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

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

Dynamics of the piecewise-linear map Period adding Farey sequence Fixed point Homoclinic orbit Fixed point

Dynamics of the piecewise-linear map Period adding Farey sequence Chaotic

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

Chaos Period adding

Immediate jump to robust chaos Partial period adding

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

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

Impact oscillators: a canonical hybrid system obstacle

Periodic dynamics Chaotic dynamics Experimental Analytic

Complex domains of attraction of periodic orbits

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

Grazing occurs when periodic orbits intersect the obstacle tanjentially

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

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

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

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/nFractions (n-1)/n