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# Mini-course bifurcation theory

Mini-course bifurcation theory. Part four: chaos. George van Voorn. Bifurcations. Bifurcations in 3 and higher D ODE models Chaos (requires at least 3D) Example: 3D RM model. Rosenzweig-MacArthur. The 3D RM model is written as. Where X = prey, Y = predator, Z = top predator.

## Mini-course bifurcation theory

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### Presentation Transcript

1. Mini-course bifurcation theory Part four: chaos George van Voorn

2. Bifurcations • Bifurcations in 3 and higher D ODE models • Chaos (requires at least 3D) • Example: 3D RM model

3. Rosenzweig-MacArthur The 3D RM model is written as Where X = prey, Y = predator, Z = top predator

4. 3D R: rescaling The rescaled version is written as Scaled functional responses

5. 3D RM: equilibria

6. 3D RM: bifurcations • Primary bifurcation parameters d1 and d2 • Displays a whole range of bifurcation curves • Point M of higher co-dimension • Tangent of equilibrium (Te) • Transcritical of equilibrium (TCe) • Hopf of 2D system equilibrium (Hp) • Hopf of non-trivial equilibrium (H+) • Transcritical of limit cycle (TCc)

7. 3D RM: bifurcations Maximum x3 Minimum x3 d1 = 0.5

8. 3D RM: bifurcations Separatrix (3D) 2 attractors

9. 3D RM: bifurcations

10. 3D RM: chaos • Flip bifurcations after each other • Period doubling 1,2,4,8,16 to infinity

11. 3D RM: chaos d1 = 0.5

12. 3D RM: chaos Pattern *2 *4 *8

13. unstable equilibrium X3 Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map Minima x3 cycles

14. X3 Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map Possible existence x3 No existence x3

15. Boundaries of chaos • Chaos born through flip bifurcations (possible route) • Chaos bounded by global bifurcations (work by Martin Boer)

16. The end (for now) Any questions?

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