Bifurcations in piecewise-smooth systems

Chris Budd

What is a piecewise-smooth system?

0)()(

0)()()(

2

1

xHifxF

xHifxFxfx

0)()(

0)()(

2

1

xHifxF

xHifxF

dt

dx

.0)()(

,0)()(

xHifxRx

xHifxFdt

dx

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

0)(: xHxx

Discontinuity set

)(),( 21 xFxF

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, 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.

Will illustrate the behaviour of piecewise smooth systems by looking at

• Maps

• Hybrid impacting systems

Some piecewise-smooth maps

0,1

0,)(

2

1

xx

xxxfx

Linear, discontinuous

.,

,,)(

xx

xxxfx

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

1,0 21

Fixed point

Dynamics of the piecewise-linear map

21 10

Period adding Farey sequence

Chaotic

x

.,

,,)(

xx

xxxfx

Square-root map

Map arises in the study of grazing bifurcations of flows and hybrid systems

Infinite stretching when

Fixed point at 00 ifx

4

10

3

2

4

1

Chaos

Period adding

13

2

Immediate jump to robust chaos

Partial period adding

Get similar behaviour in higher-dimensional square-root maps

.)(

,0)(,

,0)()(,

),(

xHy

xHByMAx

xHNMAxCMAx

xfx

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

.,

,),cos(

xxrx

xtxxx

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

x x

Observe grazing bifurcations identical to the dynamics of the two-dimensional square-root map

01.0

2

x

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

10 21

Parameter range for simple periodic orbits

Fractions 1/n Fractions (n-1)/n