Continuation of global bifurcations using

collocation technique

George van Voorn3th March 2006Schoorl

In cooperation with:Bob Kooi, Yuri Kuznetsov (UU), Bas Kooijman

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

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

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

Hopf bifurcation

• Switch stability of equilibrium at α = αH

• But stable cycle persistence of species

time

Bio

mas

s

α < αH α > αH

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

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!)

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

Boundaries of chaosExample: Rozenzweig-MacArthur next-minimum map

unstable equilibrium X3

Minima X3 cycles

Boundaries of chaosExample: Rozenzweig-MacArthur next-minimum map

X3

No existence X3

Possible existence

X3

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

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)

Global bifurcations

Minima homoclinic cycle-

to-cycle

Example: Rozenzweig-MacArthur next-minimum map

Global bifurcations

Minima heteroclinic point-to-cycle

Example: Rozenzweig-MacArthur next-minimum map

Localising connecting orbits

• Difficulties:

• Nearly impossible connection

• Orbit must enter exactly on stable manifold

• Infinite time

• Numerical inaccuracy

Shooting method

• Boer et al., Dieci & Rebaza (2004)• Numerical integration (“trial-and-error”)• Piling up of error; often fails• Very small integration step required

Shooting method

X3

X2

X1

d1 = 0.26, d2 = 1.25·10-2

Example error shooting:Rozenzweig-MacArthur modelDefault integration step

Collocation technique

• Doedel et al. (software AUTO)

• Partitioning orbit, solve pieces exactly

• More robust, larger integration step

• Division of error over pieces

Collocation technique

• Separate boundary value problems (BVP’s) for:

• Limit cycles/equilibria

• Eigenfunction linearised manifolds

• Connection

• Put together

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

Limit cycle BVP

T = period of cycle, parameterx(0) = starting point cyclex(1) = end point cycleΨ = phase

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

Margin of error

ε

Connection BVP

ν

T1 = period connection +/– ∞Truncated (numerical)

Case 1: RM model

X3

X2 X1

d1 = 0.26, d2 = 1.25·10-2

Saddle limit cycle

X3

Case 1: RM model

X3

X2 X1

Wu

Unstable manifold

μu = 1.5050

Case 1: RM model

Ws

X3

X2 X1

Stable manifold

μs = 2.307·10-3

Case 1: RM model

Heteroclinic point-to-cycle

connection

X3

X2 X1

Ws

Case 2: Monod model

X3

X2X1

Xr = 200, D = 0.085

Saddle limit cycle

Case 2: Monod model

X3

X2X1

Wu

μs too small

Case 2: Monod model

X3

X2X1

Heteroclinic point-to-cycle

connection

Case 2: Monod model

X3

X2X1

Homoclinic cycle-to-cycle

connection

Case 2: Monod model

X3

X2X1

Second saddle limit cycle

Case 2: Monod model

X3

X2X1

Wu

Case 2: Monod model

X3

X2X1

Homoclinicconnection

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)

Thank you for your attention!

george.van.voorn@falw.vu.nl 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

Case 1: RM model

X3

X2

X1

Integration step 10-3 good approximation, but:

Time consuming Not robust