1. Practical Bifurcation Theory1 John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute

2. Practical Bifurcation Theory2 References Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley) Kuznetsov, Elements of Applied Bifurcation Theory (Springer) XPP-AUT www.math.pitt.edu/~bard/xppwww.math.pitt.edu/~bard/xpp Oscill8 http://oscill8.sourceforge.nethttp://oscill8.sourceforge.net

3. Practical Bifurcation Theory3 The Dynamical Perspective Molec Genetics Biochemistry Cell Biology Kinetic Equations State Space, Vector Field Attractors, Transients, Repellors Bifurcation Diagrams Molecular Mechanism Signal-Response Curves The Curse of Parameter Space

4. Practical Bifurcation Theory4

5. 5 Generic Bifurcations of Dynamical Systems Saddle-Node Hopf Cyclic Fold Saddle Loop Saddle-Node Invariant Circle Rene Thom (1972), “Structural Stability and Morphogenesis”

6. Practical Bifurcation Theory6

7. 7

8. 8

9. 9

10. 10

11. Practical Bifurcation Theory11

12. Practical Bifurcation Theory12

13. Practical Bifurcation Theory13

14. Practical Bifurcation Theory14

15. Practical Bifurcation Theory15

16. Practical Bifurcation Theory16

17. Practical Bifurcation Theory17

18. Practical Bifurcation Theory18

19. Practical Bifurcation Theory19

20. Practical Bifurcation Theory20

21. Practical Bifurcation Theory21

22. Practical Bifurcation Theory22

23. Practical Bifurcation Theory23

24. Practical Bifurcation Theory24

25. Practical Bifurcation Theory25

26. Practical Bifurcation Theory26

27. Practical Bifurcation Theory27

28. Practical Bifurcation Theory28

29. Practical Bifurcation Theory29

30. Practical Bifurcation Theory30

31. Practical Bifurcation Theory31

32. Practical Bifurcation Theory32

33. Practical Bifurcation Theory33

34. Practical Bifurcation Theory34

35. Practical Bifurcation Theory35

36. Practical Bifurcation Theory36

37. Practical Bifurcation Theory37

38. Practical Bifurcation Theory38

39. Practical Bifurcation Theory39

40. Practical Bifurcation Theory40

41. Practical Bifurcation Theory41

42. Practical Bifurcation Theory42

43. Practical Bifurcation Theory43

44. Practical Bifurcation Theory44

45. Practical Bifurcation Theory45

46. Practical Bifurcation Theory46

47. Practical Bifurcation Theory47

48. Practical Bifurcation Theory48

49. Practical Bifurcation Theory49

50. Practical Bifurcation Theory50

51. Practical Bifurcation Theory51

52. Practical Bifurcation Theory52

53. Practical Bifurcation Theory53

54. Practical Bifurcation Theory54

55. Practical Bifurcation Theory55 Torus Heteroclinic

56. Practical Bifurcation Theory56

57. Practical Bifurcation Theory57 Torus ?

58. Practical Bifurcation Theory58

59. Practical Bifurcation Theory59

60. Practical Bifurcation Theory60 Trajectories for the Fold-Hopf bifurcations as illustrated in the Kuznetsov’s book.

61. Practical Bifurcation Theory61

62. Practical Bifurcation Theory62

63. Practical Bifurcation Theory63