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Mini-course bifurcation theory George van Voorn Part three: bifurcations of 2D systems
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Bifurcations Bifurcations of equilibria in 1D –Transcritical (λ = 0) –Tangent (λ = 0) In 2D –Transcritical (One λ = 0) –Tangent (One λ = 0)
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Example 2D ODE Rosenzweig-MacArthur (1963) R = intrinsic growth rate K = carrying capacity A/B = searching and handling C = yield D = death rate
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Example At K = 0 x = 0, y = 0 0 0, y = 0 At K = 6 x = K, y = 0 6 0 K = 0 and K = 6 transcritical Invasion criterion species x,y
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Hopf bifurcation Andronov-Hopf bifurcation: Transition from stable spiral to unstable spiral (equilibrium becomes unstable) Or vice versa Periodic orbit (limit cycle), stable Or vice verse, respectively Conditions:
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Hopf bifurcation α x,yx,y
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Example At K ≈ 20 x = K, y > 0 Limit cycle Change in dynamics of species x,y
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Paradox of enrichment Isoclines RM model x = K, for all K > 6 increase only in y
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Paradox of enrichment One-parameter bifurcation diagram
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Paradox of enrichment One-parameter bifurcation diagram Maximal values x and y Minimal values x and y Equilibria
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Paradox of enrichment Oscillations increase in size for larger K
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Paradox of enrichment Large oscillations extinction probabilities Paradox: –Increase in food availability K –No benefit for prey x –Increased probability of extinction system
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Limit cycle bifurcations For limit cycles same bifurcations as for equilibria Imagine cross section
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Limit cycle bifurcations Transcritical Cycle on axis (mostly) Tangent Birth or destruction cycle(s)
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Limit cycle bifurcations Hopf (called Neimark-Sacker) Torus
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Limit cycle bifurcations Flip bifurcation Manifold around cycle
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Flip bifurcations Manifold twisted
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Flip bifurcations Flip bifurcation of limit cycle –Manifold twisted (Möbius ribbon) –Period doubling
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Codim 2 points Bifurcation points can be continued in two- parameter space = bifurcation curve Continuation can result in: Bifurcation points of higher co-dimension
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Codim 2 points Bogdanov-Takens Cusp Generalised Hopf (Bautin)
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Example Bazykin model Calculate equilibrium Vary one parameter until a bifurcation is encountered
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Bazykin Continuation in two-parameter space x*,y*x*,y*
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Bazykin: dynamics Stable node: coexistence Stable cycle: coexistence No positive equilibria: extinction Unstable equilibria: extinction
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Bazykin: BT point Bogdanov-Takens point tangent & Hopf
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Bazykin: GH point Bautin point transition Hopf from stable to unstable point
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Bazykin: cusp point Cusp point collision two tangent points
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Question Stable cycle: coexistenceUnstable equilibria: extinction What happens here?
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Global bifurcations BT point: origin of homoclinic bifurcation
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Bazykin: homoclinic Starting at Hopf continue cycle. What happens?
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Bazykin: homoclinic Limit cycle period to infinity. Why?
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Homoclinic connection WsWs WuWu Homoclinic connecting orbit: W u = W s Time to infinity near equilibrium
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Heteroclinic connection ŴsŴs WuWu Heteroclinic connecting orbit: W u = Ŵ s
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Bazykin: homoclinic
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