III: Hybrid systems and the grazing bifurcation Chris Budd

.0)()(

,0)()(

xHifxRx

xHifxFdt

dx

Hybrid system

Impact or control systems

Impact oscillator: a canonical hybrid system

.,

,),cos(

uuru

utuuu

obstacle

1

)cos()(, vut

v

xF

t

v

u

x

t

rv

u

xRuxH )(,)(

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

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

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

S S

Small perturbations of a non-impacting orbit

x

y

xAy

v

u

Small perturbations of an orbit with a high velocity impact

y

x

)(xR

),( v

xASAy 21

21, AA

S

100

0

00

rv

arar

S 0v

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-

GxbxAxAy

GxxAy

xxxGx gg

,

,

,

1

A1

A2

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

008.06.2 r

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

.,,0

,,,0

)sin()(

222

2

212

2

zudt

dzvu

dt

ud

wzdt

dzyz

dt

zd

ttw

m

M

Newton’s cradle

v

y

u

z

rr

rr

v

y

u

z

1

1

1

)1(00

1

1

100

0010

0001

w z u

Mass ratio

The square rotating cam

Bifurcation diagram