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

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 d 1 and d 2 Displays a whole range of bifurcation curves Point M of higher co-dimension –Tangent of equilibrium (T e ) –Transcritical of equilibrium (TC e ) –Hopf of 2D system equilibrium (H p ) –Hopf of non-trivial equilibrium (H + ) –Transcritical of limit cycle (TC c )

7. 3D RM: bifurcations d 1 = 0.5 Maximum x 3 Minimum x 3

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

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 d 1 = 0.5

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

13. Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map unstable equilibrium X 3 Minima x 3 cycles

14. Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map X3X3 No existence x 3 Possible existence x 3

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?