Continuation of global
bifurcations using
collocation technique
In cooperation with:
Bob Kooi, Yuri Kuznetsov (UU), Bas Kooijman
George van Voorn
3th March 2006
Schoorl
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
Biomass
• Switch stability of equilibrium at α = αH
• But stable cycle  persistence of species
α < αH
time
α > αH
Hopf in experiments
Fussman, G.F. et al. 2000.
Crossing the Hopf Bifurcation in
a Live Predator-Prey System.
Science 290: 1358 – 1360.
Chemostat
predator-prey system
a: Extinction food shortage
b: Coexistence at
equilibrium
c: Coexistence on stable
limit cycle
d: Extinction cycling
Measurement point
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
Dilution rate
d (day -1)
0.90
0.75
0.50
Becks, L. et al. 2005.
Experimental demonstration of
chaos in a microbial food web.
Nature 435: 1226 – 1229.
Chemostat predatortwo-prey system
Brevundimonas
Pedobacter
0.45
Tetrahymena (predator)
Chaotic behaviour
Boundaries of chaos
Example: Rozenzweig-MacArthur next-minimum map
unstable equilibrium X3
Minima X3 cycles
Boundaries of chaos
Example: Rozenzweig-MacArthur next-minimum map
X3
Possible
existence
X3
No 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
Example: Rozenzweig-MacArthur next-minimum map
Minima homoclinic
cycle-to-cycle
Global bifurcations
Example: Rozenzweig-MacArthur next-minimum map
Minima heteroclinic
point-to-cycle
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
Example error shooting:
Rozenzweig-MacArthur model
Default integration step
X3
X1
X2
d1 = 0.26, d2 = 1.25·10-2
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 = eigenvector
λ = eigenvalue
fx = Jacobian matrix
In 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, parameter
x(0) = starting point cycle
x(1) = end point cycle
Ψ = phase
Eigenfunction BVP
Wu
w(0)
T = same period as cycle
μ = multiplier (FM)
w = eigenvector
Ф = phase
Finds entry and exit points of
stable and unstable limit cycles
w(0) μ
Connection BVP
T1 = period connection
 +/– ∞
Truncated (numerical)
Margin of error
ε
ν
Case 1: RM model
X3
X3
d1 = 0.26, d2 = 1.25·10-2
Saddle
limit cycle
X2
X1
Case 1: RM model
X3
Wu
Unstable
manifold
μu = 1.5050
X2
X1
Case 1: RM model
X3
Ws
Stable
manifold
μs = 2.307·10-3
X2
X1
Case 1: RM model
X3
Ws
X2
Heteroclinic
point-to-cycle
connection
X1
Case 2: Monod model
Xr = 200, D = 0.085
X3
Saddle
limit cycle
X2
X1
Case 2: Monod model
X3
Wu
μs too small
X2
X1
Case 2: Monod model
X3
Heteroclinic
point-to-cycle
connection
X2
X1
Case 2: Monod model
X3
Homoclinic
cycle-to-cycle
connection
X2
X1
Case 2: Monod model
X3
Second saddle
limit cycle
X2
X1
Case 2: Monod model
Wu
X3
X2
X1
Case 2: Monod model
X3
Homoclinic
connection
X2
X1
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)
Supported by
Thank you for your attention!
[email protected]
Primary references:
Boer, M.P. and Kooi, B.W. 1999. Homoclinic and heteroclinic orbits to a cycle in
a 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.
Case 1: RM model
Integration step 10-3 
good approximation, but:
X3
 Time consuming
 Not robust
X1
X2