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A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 5: Two-Parameter Bifurcation Diagrams John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute Click on icon to start audio

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One-Parameter Bifurcation Diagrams Parameter, p Variable, x SN Parameter, p Variable, x HB

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One-Parameter Bifurcation Diagrams Parameter, p Variable, x x y

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One-Parameter Bifurcation Diagrams Parameter, p Variable, x x y

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One-Parameter Bifurcation Diagrams Parameter, p Variable, x x y

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One-Parameter Bifurcation Diagrams Parameter, p Variable, x

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Numerical Bifurcation Theory Two equations in three unknowns. Fix p = p o ; solve for (x o, y o ). Expand using Taylor’s Theorem: = 0

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This is perfectly generalizable to any number of variables. As long as With this equation, we can follow a steady state as p changes. As we follow a locus of steady states, we can monitor Whenever D = 0, we have located a SN bifurcation point, and whenever R = 0, we have located a Hopf bifurcation.

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Parameter, p Variable, x SN Parameter, p Variable, x HBSN

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Two-parameter Bifurcation Diagram Three equations in four unknowns. Fix p = p o ; solve for (x o, y o, q o ). Follow the solution using… Parameter, p Parameter, q three ss one ss D = det(J) fold linecusp point “codimension-one” “codimension-two”

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Two-parameter Bifurcation Diagram Parameter, p Parameter, q Parameter, p Variable, x s sxs s

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Two-parameter Bifurcation Diagram Parameter, p Parameter, q Parameter, p Variable, x s sxs s

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Two-parameter Bifurcation Diagram Parameter, p Parameter, q Parameter, p Variable, x “codimension-two” “universal unfolding” s sxs s

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x p

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x pq

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LOW HIGH x pq

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Bistable (High or Low) High xLow x Medium x x p q p q

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Two-parameter Bifurcation Diagram Three equations in four unknowns. Fix p = p o ; solve for (x o, y o, q o ). Follow the solution using… Parameter, p Parameter, q one uss one slc one sss R = Re( 1,2 ) Parameter, p Variable, x

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Sub-critical Hopf Bifurcation Parameter, p Variable, x subHB CF supHB

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Two-parameter Bifurcation Diagram Parameter, p Parameter, q degenerate HB subHB supHB CF degenerate HB s u s Parameter, p Variable, x

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