![Page 1: Bifurcations in piecewise-smooth systems Chris Budd](https://reader035.vdocument.in/reader035/viewer/2022062619/5515df47550346dd6f8b4c4d/html5/thumbnails/1.jpg)
Bifurcations in piecewise-smooth systems
Chris Budd
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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
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PWS Flow PWS Sliding Flow Hybrid
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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
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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.
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Will illustrate the behaviour of piecewise smooth systems by looking at
• Maps
• Hybrid impacting systems
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Some piecewise-smooth maps
0,1
0,)(
2
1
xx
xxxfx
Linear, discontinuous
.,
,,)(
xx
xxxfx
Square-root, continuous
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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
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Dynamics of the piecewise-linear map
Period adding Farey sequence
Fixed point
Homoclinic orbit
1,0 21
Fixed point
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Dynamics of the piecewise-linear map
21 10
Period adding Farey sequence
Chaotic
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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
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4
10
3
2
4
1
Chaos
Period adding
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13
2
Immediate jump to robust chaos
Partial period adding
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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.
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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
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Impact oscillators: a canonical hybrid system
.,
,),cos(
xxrx
xtxxx
obstacle
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Periodic dynamics Chaotic dynamics
Experimental
Analytic
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Complex domains of attraction of periodic orbits
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Regular and discontinuity induced bifurcations as
parameters vary
Regular and discontinuity induced bifurcations as parameters vary.
Period doubling
Grazing
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Grazing occurs when periodic orbits intersect the obstacle tanjentially
x x
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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
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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
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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
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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
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10 21
Parameter range for simple periodic orbits
Fractions 1/n Fractions (n-1)/n
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