Canards II:Canards II:Hooks and AsymmetryHooks and Asymmetry

Reduced System of theReduced System of the

Forced van der Pol EquationForced van der Pol Equation

3D System: Symmetric solutions

What Would a Hook Look Like?

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

Half Return Map

Computing H(θ) Start at (θ, 2) Flow until x = 1

(slow subsystem) Jump to x=-2

(fast subsystem) Shift θ by ½

(symmetry)

Extended Half Return Map

Positive JumpH(θ) =

flow from positive jump forward to x=1,

shift by ½

Negative JumpH(θ) = flow from negative jump backwards to x=-2,

shift by ½

Trajectory H

0 = x’ = -x + a sin(2π θ)

θ = -½ π Sin-1(2/a)

Only exist for a>=2

a < 2 always hooks

max canard = endpoint a =>2 what is ω?

Asymmetric Solutions

Asymmetric Solutions Exist

Maximal canard touches H(θ) = θ

θm = θ1s

Bifurcation of symmetric and asymmetric solutions

Asymmetric Solution without Canards Exists Maximal Canard

“shoots” to preimage of canard

H(θ m) = θ 1s

Asymmetric solutions with canards also exist

Asymmetric Solutions Stop Existing Bottom of jump back

canard touches

H(θ) = θ θ1u = θ1s

Symmetric solutions with jump across exist

Where to?

Higher Asymmetric Periodic Solutions Horse shoe in Reduced System Parameterize the canards, looks like x2,x3

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