Piecewise-smooth dynamical systems: Bouncing, slipping and switching:

1. Introduction Chris Budd

Most of the present theory of dynamical systems deals with smooth systems

)(),( 1 nCxfdt

dx

)(),( 11

n

nn Cxfx

These systems are now ‘fairly well understood’

Flows

Maps

Can broadly explain the dynamics in terms of the omega-limit sets

•Fixed points

•Periodic orbits and tori

•Homoclinic orbits

•Chaotic strange attractors

And the bifurcations from these

•Fold/saddle-node

•Period-doubling/flip

•Hopf

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

0)( xH

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 …

Newton’s cradle

Beam impacting with a smooth rotating cam [di Bernardo et. al.]

)sin()( ttz

)(tu

Piecewise-smooth systems have behaviour which is quite

different from smooth systems and is induced by the

discontinuity

Eg. 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.

This course will illustrate the behaviour of piecewise smooth systems by looking at

• Some physical examples (Today)

• Piecewise-smooth Maps (Tomorrow)

• Hybrid impacting systems and piecewise-smooth flows

(Sunday)M di Bernardo et. al.

Bifurcations in Nonsmooth Dynamical Systems

SIAM Rev iew, 50, (2008), 629—701.

M di Bernardo et. al.

Piecewise-smooth Dynamical Systems: Theory and Applications

Springer Mathematical Sciences 163. (2008)

Example I: The Impact Oscillator: a canonical piecewise-smooth hybrid system

.,

,),cos(

xxrx

xtxxx

obstacle

)(),(),sin()(),cos()(,1

1

)()sin()()cos()(),;(

00002

0000000

tSStCCttSttC

tCttSvttCtvtx

Solution in free flight (undamped)

xx

Periodic dynamics Chaotic dynamics

Experimental

Analytic

x

dx/dt

008.08.2 r

Chaotic strange attractor

Complex domains of attraction of the periodic orbits

dx/dt

x

008.06.2 r

Regular and discontinuity induced bifurcations as

parameters vary

Regular and discontinuity induced bifurcations as parameters vary.

Period doubling

Grazing008.0 r

Grazing bifurcations occur when periodic orbits intersect the obstacle tanjentially:

see Sunday for a full explanation

x x

08.02 r

Grazing bifurcation

x

Partial period-adding

Robust chaos

08.02 r

Chaotic motion

x

dx/dtt

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 we study on Friday

Example II: The DC-DC Converter: a canonical piecewise-smooth flow

R

R

VVforLE

VVfor

L

VI

C

IV

RCV

/

0

1

Sliding flow

Sliding flow is also a characteristic of: III Friction Oscillators

3)sgn()(

),cos()1(

cubuuauC

tAyCyy

Coulomb friction

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

• Next two lectures will describe the analysis in more detail.