1. Mini-course bifurcation theory George van Voorn Part three: bifurcations of 2D systems

2. Bifurcations Bifurcations of equilibria in 1D –Transcritical (λ = 0) –Tangent (λ = 0) In 2D –Transcritical (One λ = 0) –Tangent (One λ = 0)

3. Example 2D ODE Rosenzweig-MacArthur (1963) R = intrinsic growth rate K = carrying capacity A/B = searching and handling C = yield D = death rate

4. Example At K = 0  x = 0, y = 0 0 0, y = 0 At K = 6  x = K, y = 0 6 0 K = 0 and K = 6  transcritical Invasion criterion species x,y

5. Hopf bifurcation Andronov-Hopf bifurcation: Transition from stable spiral to unstable spiral (equilibrium becomes unstable) Or vice versa Periodic orbit (limit cycle), stable Or vice verse, respectively Conditions:

6. Hopf bifurcation α x,yx,y

7. Example At K ≈ 20  x = K, y > 0 Limit cycle Change in dynamics of species x,y

8. Paradox of enrichment Isoclines RM model x = K, for all K > 6  increase only in y

9. Paradox of enrichment One-parameter bifurcation diagram

10. Paradox of enrichment One-parameter bifurcation diagram Maximal values x and y Minimal values x and y Equilibria

11. Paradox of enrichment Oscillations increase in size for larger K

12. Paradox of enrichment Large oscillations  extinction probabilities Paradox: –Increase in food availability K  –No benefit for prey x –Increased probability of extinction system

13. Limit cycle bifurcations For limit cycles same bifurcations as for equilibria Imagine cross section

14. Limit cycle bifurcations Transcritical  Cycle on axis (mostly) Tangent  Birth or destruction cycle(s)

15. Limit cycle bifurcations Hopf (called Neimark-Sacker)  Torus

16. Limit cycle bifurcations Flip bifurcation Manifold around cycle

17. Flip bifurcations Manifold twisted

18. Flip bifurcations Flip bifurcation of limit cycle –Manifold twisted (Möbius ribbon) –Period doubling

19. Codim 2 points Bifurcation points can be continued in two- parameter space = bifurcation curve Continuation can result in: Bifurcation points of higher co-dimension

20. Codim 2 points Bogdanov-Takens Cusp Generalised Hopf (Bautin)

21. Example Bazykin model  Calculate equilibrium  Vary one parameter until a bifurcation is encountered

22. Bazykin Continuation in two-parameter space x*,y*x*,y*

23. Bazykin: dynamics Stable node: coexistence Stable cycle: coexistence No positive equilibria: extinction Unstable equilibria: extinction

24. Bazykin: BT point Bogdanov-Takens point  tangent & Hopf

25. Bazykin: GH point Bautin point  transition Hopf from stable to unstable point

26. Bazykin: cusp point Cusp point  collision two tangent points

27. Question Stable cycle: coexistenceUnstable equilibria: extinction What happens here?

28. Global bifurcations BT point: origin of homoclinic bifurcation

29. Bazykin: homoclinic Starting at Hopf continue cycle. What happens?

30. Bazykin: homoclinic Limit cycle period to infinity. Why?

31. Homoclinic connection WsWs WuWu Homoclinic connecting orbit: W u = W s Time to infinity near equilibrium

32. Heteroclinic connection ŴsŴs WuWu Heteroclinic connecting orbit: W u = Ŵ s

33. Bazykin: homoclinic