piecewise-smooth dynamical systems: bouncing, slipping and switching: 1. introduction chris budd
TRANSCRIPT
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.