Piecewisesmooth dynamical systems: - PowerPoint PPT Presentation

Title: Piecewisesmooth dynamical systems:


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Piecewise-smooth dynamical systems Bouncing,
slipping and switching 1.
Introduction Chris Budd

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Most of the present theory of dynamical systems
deals with smooth systems
Flows
Maps
These systems are now fairly well understood
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• 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

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What is a piecewise-smooth system?
Map
Heartbeats or Poincare maps
Flow
Rocking block, friction, Chua circuit
Hybrid
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
Discontinuity set
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, Newtons
  cradle, friction, rattle, switching, control
  systems, DC-DC converters, gear boxes

Newtons cradle
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Beam impacting with a smooth rotating cam di
Bernardo et. al.
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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.
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• 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),
629701. M di Bernardo et. al. Piecewise-smooth
Dynamical Systems Theory and Applications Springe
r Mathematical Sciences 163. (2008)
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Example I The Impact Oscillator a canonical
piecewise-smooth hybrid system
obstacle
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Solution in free flight (undamped)
x
x
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Periodic dynamics Chaotic dynamics
Experimental Analytic
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Chaotic strange attractor
dx/dt
x
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Complex domains of attraction of the periodic
orbits
dx/dt
x
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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 bifurcations occur when periodic orbits
intersect the obstacle tanjentially see Sunday
for a full explanation
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x
Robust chaos
Grazing bifurcation
Partial period-adding
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Chaotic motion
x
t
dx/dt
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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 we study on Friday
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Example II The DC-DC Converter a canonical
piecewise-smooth flow
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Sliding flow
Sliding flow is also a characteristic of III
Friction Oscillators
Coulomb friction
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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
• Next two lectures will describe the analysis in
  more detail.

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