Continuation of global bifurcations using collocation technique
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Continuation of global bifurcations using
collocation technique
George van Voorn3th March 2006Schoorl
In cooperation with:Bob Kooi, Yuri Kuznetsov (UU), Bas Kooijman
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Overview• Recent biological experimental examples of:Local bifurcations (Hopf)Chaotic behaviour
• Role of global bifurcations (globif’s)• Techniques finding and continuation global
connecting orbits • Find global bifurcations
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Bifurcation analysis
• Tool for analysis of non-linear (biological) systems: bifurcation analysis
• By default: analysis of stability of equilibria (X(t), t ∞) under parameter variation
• Bifurcation point = critical parameter value where switch of stability takes place
• Local: linearisation around point
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Biological application
• Biologically local bifurcation analysis allows one to distinguish between:
Stable (X = 0 or X > 0)Periodic (unstable X )Chaotic
• Switches at bifurcation points
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Hopf bifurcation
• Switch stability of equilibrium at α = αH
• But stable cycle persistence of species
time
Bio
mas
s
α < αH α > αH
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Hopf in experimentsFussman, G.F. et al. 2000.Crossing the Hopf Bifurcation ina Live Predator-Prey System. Science 290: 1358 – 1360.
a: Extinction food shortageb: Coexistence at equilibriumc: Coexistence on stable limit cycled: Extinction cycling
Measurement point
Chemostatpredator-prey system
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Chaotic behaviour
• Chaotic behaviour: no attracting equilibrium or stable periodic solution
• Yet bounded orbits [X(t)min, X(t)max]
• Sensitive dependence on initial conditions
• Prevalence of species (not all cases!)
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Experimental results
Becks, L. et al. 2005. Experimental demonstration of chaos in a microbial food web. Nature 435: 1226 – 1229.
0.90
0.75
0.50
0.45
Dilution rate d (day -1)
Brevundimonas
Pedobacter
Tetrahymena (predator)
Chaotic behaviour
Chemostat predator-two-prey system
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Boundaries of chaosExample: Rozenzweig-MacArthur next-minimum map
unstable equilibrium X3
Minima X3 cycles
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Boundaries of chaosExample: Rozenzweig-MacArthur next-minimum map
X3
No existence X3
Possible existence
X3
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Boundaries of chaos
• Chaotic regions bounded
• Birth of chaos: e.g. period doubling
• Flip bifurcation (manifold twisted)
• Destruction boundaries • Unbounded orbits • No prevalence of species
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Global bifurcations
• Chaotic regions are “cut off” by global bifurcations (globifs)
• Localisation globifs by finding orbits that:• Connect the same saddle equilibrium or
cycle (homoclinic)• Connect two different saddle cycles and/or
equilibria (heteroclinic)
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Global bifurcations
Minima homoclinic cycle-
to-cycle
Example: Rozenzweig-MacArthur next-minimum map
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Global bifurcations
Minima heteroclinic point-to-cycle
Example: Rozenzweig-MacArthur next-minimum map
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Localising connecting orbits
• Difficulties:
• Nearly impossible connection
• Orbit must enter exactly on stable manifold
• Infinite time
• Numerical inaccuracy
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Shooting method
• Boer et al., Dieci & Rebaza (2004)• Numerical integration (“trial-and-error”)• Piling up of error; often fails• Very small integration step required
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Shooting method
X3
X2
X1
d1 = 0.26, d2 = 1.25·10-2
Example error shooting:Rozenzweig-MacArthur modelDefault integration step
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Collocation technique
• Doedel et al. (software AUTO)
• Partitioning orbit, solve pieces exactly
• More robust, larger integration step
• Division of error over pieces
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Collocation technique
• Separate boundary value problems (BVP’s) for:
• Limit cycles/equilibria
• Eigenfunction linearised manifolds
• Connection
• Put together
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Equilibrium BVP
v = eigenvectorλ = eigenvaluefx = Jacobian matrixIn practice computer program (Maple, Mathematica) is used to find equilibrium f(ξ,α)Continuation parameters:Saddle equilibrium, eigenvalues, eigenvectors
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Limit cycle BVP
T = period of cycle, parameterx(0) = starting point cyclex(1) = end point cycleΨ = phase
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Eigenfunction BVP
T = same period as cycleμ = multiplier (FM) w = eigenvectorФ = phaseFinds entry and exit points of stable and unstable limit cycles
w(0)
w(0) μ
Wu
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Margin of error
ε
Connection BVP
ν
T1 = period connection +/– ∞Truncated (numerical)
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Case 1: RM model
X3
X2 X1
d1 = 0.26, d2 = 1.25·10-2
Saddle limit cycle
X3
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Case 1: RM model
X3
X2 X1
Wu
Unstable manifold
μu = 1.5050
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Case 1: RM model
Ws
X3
X2 X1
Stable manifold
μs = 2.307·10-3
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Case 1: RM model
Heteroclinic point-to-cycle
connection
X3
X2 X1
Ws
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Case 2: Monod model
X3
X2X1
Xr = 200, D = 0.085
Saddle limit cycle
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Case 2: Monod model
X3
X2X1
Wu
μs too small
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Case 2: Monod model
X3
X2X1
Heteroclinic point-to-cycle
connection
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Case 2: Monod model
X3
X2X1
Homoclinic cycle-to-cycle
connection
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Case 2: Monod model
X3
X2X1
Second saddle limit cycle
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Case 2: Monod model
X3
X2X1
Wu
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Case 2: Monod model
X3
X2X1
Homoclinicconnection
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Future work
• Difficult to find starting points
• Recalculate global homoclinic and heteroclinic bifurcations in models by M. Boer et al.
• Find and continue globifs in other biological models (DEB, Kooijman)
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Thank you for your attention!
[email protected] Primary references:
Boer, M.P. and Kooi, B.W. 1999. Homoclinic and heteroclinic orbits to a cycle ina tri-trophic food chain. J. Math. Biol. 39: 19-38.
Dieci, L. and Rebaza, J. 2004. Point-to-periodic and periodic-to-periodic connections.BIT Numerical Mathematics 44: 41–62.
Supported by
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Case 1: RM model
X3
X2
X1
Integration step 10-3 good approximation, but:
Time consuming Not robust