Continuation of global bifurcations using collocation technique

1
Continuation of global bifurcations using
collocation technique
In cooperation with Bob Kooi, Yuri Kuznetsov
(UU), Bas Kooijman
George van Voorn 3th March 2006 Schoorl
2
Overview

• Recent biological experimental examples of
• Local bifurcations (Hopf)
• Chaotic behaviour
• Role of global bifurcations (globifs)
• Techniques finding and continuation global
  connecting orbits
• Find global bifurcations

3
Bifurcation analysis

• Tool for analysis of non-linear (biological)
  systems bifurcation analysis
• By default analysis of stability of equilibria
  (X(t), t ? 8) under parameter variation
• Bifurcation point critical parameter value
  where switch of stability takes place
• Local linearisation around point

4
Biological application

• Biologically local bifurcation analysis allows
  one to distinguish between
• Stable (X 0 or X gt 0)
• Periodic (unstable X )
• Chaotic
• Switches at bifurcation points

5
Hopf bifurcation

• Switch stability of equilibrium at a aH
• But stable cycle ? persistence of species

Biomass
time
a lt aH
a gt aH
6
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
7
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!)

8
Experimental results
Dilution rate d (day -1)
Becks, L. et al. 2005. Experimental demonstration
of chaos in a microbial food web. Nature 435
1226 1229.
0.90
Chemostat predator-two-prey system
0.75
Brevundimonas
0.50
Pedobacter
Tetrahymena (predator)
0.45
Chaotic behaviour
9
Boundaries of chaos
Example Rozenzweig-MacArthur next-minimum map
Minima X3 cycles
10
Boundaries of chaos
Example Rozenzweig-MacArthur next-minimum map
Possible existence X3
No existence X3
11
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

12
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)

13
Global bifurcations
Example Rozenzweig-MacArthur next-minimum map
Minima homoclinic cycle-to-cycle
14
Global bifurcations
Example Rozenzweig-MacArthur next-minimum map
Minima heteroclinic point-to-cycle
15
Localising connecting orbits

• Difficulties
• Nearly impossible connection
• Orbit must enter exactly on stable manifold
• Infinite time
• Numerical inaccuracy

16
Shooting method

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

17
Shooting method
Example error shooting Rozenzweig-MacArthur
model Default integration step
X3
X1
X2
d1 0.26, d2 1.2510-2
18
Collocation technique

• Doedel et al. (software AUTO)
• Partitioning orbit, solve pieces exactly
• More robust, larger integration step
• Division of error over pieces

19
Collocation technique

• Separate boundary value problems (BVPs) for
• Limit cycles/equilibria
• Eigenfunction ? linearised manifolds
• Connection
• Put together

20
Equilibrium BVP
v eigenvector ? eigenvalue fx Jacobian
matrix In practice computer program (Maple,
Mathematica) is used to find equilibrium
f(?,a) Continuation parameters Saddle
equilibrium, eigenvalues, eigenvectors
21
Limit cycle BVP
T period of cycle, parameter x(0) starting
point cycle x(1) end point cycle ? phase
22
Eigenfunction BVP
Wu
w(0)
T same period as cycle µ multiplier (FM) w
eigenvector ? phase Finds entry and exit points
of stable and unstable limit cycles
w(0) µ
23
Connection BVP
T1 period connection ? / 8 Truncated
(numerical)
Margin of error
e
?
24
Case 1 RM model
d1 0.26, d2 1.2510-2
X3
Saddle limit cycle
X2
X1
25
Case 1 RM model
X3
Wu
Unstable manifold
µu 1.5050
X2
X1
26
Case 1 RM model
X3
Stable manifold
Ws
µs 2.30710-3
X2
X1
27
Case 1 RM model
X3
Heteroclinic point-to-cycle connection
Ws
X2
X1
28
Case 2 Monod model
Xr 200, D 0.085
X3
Saddle limit cycle
X1
X2
29
Case 2 Monod model
X3
Wu
µs too small
X1
X2
30
Case 2 Monod model
X3
Heteroclinic point-to-cycle connection
X1
X2
31
Case 2 Monod model
X3
Homoclinic cycle-to-cycle connection
X1
X2
32
Case 2 Monod model
X3
Second saddle limit cycle
X1
X2
33
Case 2 Monod model
Wu
X3
X1
X2
34
Case 2 Monod model
X3
Homoclinic connection
X1
X2
35
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)

36
Supported by
Thank you for your attention!
george.van.voorn_at_falw.vu.nl
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 4162.
37
Case 1 RM model

• Integration step 10-3 ? good approximation, but
• Time consuming
• Not robust

X3
X1
X2