Canards II: Hooks and Asymmetry

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Canards IIHooks and Asymmetry

• Reduced System of the
• Forced van der Pol Equation

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3D System Symmetric solutions
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What Would a Hook Look Like?
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Existence of Hooks

• Does there exist a trajectory tangent to a canard
  jump? NO
• (Canard jump not a trajectory)
• Proof
• Compare slopes
• HRM explains Hooks

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Half Return Map

• Computing H(?)
• Start at (?, 2)
• Flow until x 1
• (slow subsystem)
• Jump to x-2
• (fast subsystem)
• Shift ? by ½
• (symmetry)

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Extended Half Return Map

• Positive Jump
• H(?)
• flow from positive jump forward to x1,
• shift by ½
• Negative Jump
• H(?) flow from negative jump backwards to
  x-2,
• shift by ½

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Trajectory H

• 0 x -x a sin(2p ?)
• ? -½ p Sin-1(2/a)
• Only exist for agt2
• a lt 2 always hooks
• max canard endpoint
• a gt2 what is ??

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Asymmetric Solutions
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Asymmetric Solutions Exist

• Maximal canard touches H(?) ?
• ?m ?1s
• Bifurcation of symmetric and asymmetric solutions

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Asymmetric Solution without Canards Exists

• Maximal Canard shoots to preimage of canard
• H(? m) ? 1s
• Asymmetric solutions with canards also exist

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Asymmetric Solutions Stop Existing

• Bottom of jump back canard touches
• H(?) ?
• ?1u ?1s
• Symmetric solutions with jump across exist

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Where to?

• Higher Asymmetric Periodic Solutions
• Horse shoe in Reduced System
• Parameterize the canards, looks like x2,x3
• Mathfest!
• www.mathlab.cornell.edu/katybold