1. Chaos Theory MS Electrical Engineering Department of Engineering
GC University Lahore

2. Bifurcations
Qualitative changes in the dynamics are called bifurcations,
The parameter values at which they occur are called bifurcation points.
Bifurcations are important scientifically-they provide models of transitions and instabilities as some control parameter is varied.

3. Saddle-Node Bifurcation
Fixed-Points are created and destroyed

4. Saddle-Node Bifurcation

5. Saddle-Node Bifurcation

6. Bifurcation Diagram
Fold Bifurcation, Turning-Point Bifurcation, Blue Sky Bifurcation

7. Saddle-Node Bifurcation

9. Saddle-Node Bifurcation

10. Transcritical Bifurcation
At least one fixed-point always exists
It may change its stability with change in parameter

11. Transcritical Bifurcation

12. Laser Threshold Solid-State Laser
Each Atom is a tiny radiating antenna
Below threshold – random phases
After threshold – in-phase oscillations

13. Laser Threshold Number of photons – n(t)
Gain is due to stimulated emission
G > 0 – Gain coefficent
N(t) – Number of excited atoms
K>0 – rate constant that is reciprocal of typical lifetime of a photon in laser

14. Laser Threshold
N decreases by emission of photons
Thus

15. Laser Threshold

17. Pitchfork Bifurcation
Supercritical Pitchfork Bifurcation
Subcritical Pitchfork Bifurcation

18. Supercritical Pitchfork Bifurcation

19. Supercritical Pitchfork Bifurcation

21. Subcritical Pitchfork Bifurcation

22. Subcritical Pitchfork Bifurcation
Stabilizing term

23. Overdamped Bead on a Rotating Hoop

24. Overdamped Bead on a Rotating Hoop

25. Imperfect Bifurcations and Catastrophe
Imperfection parameter

26. Imperfect Bifurcations and Catastrophe

27. Imperfect Bifurcations and Catastrophe

28. Cusp Catastrophe

29. Cusp Catastrophe

30. Insect Outbreak spruce budworm vs balsam fir tree
Budworms – Fivefold in a year (Characteristic Time Scale in Months)
Trees – Replace their foliage in 7-10 years (Lifespan of years without budworms)

31. Model

32. p(N)

33. Complete Model

34. Dimensionless Formulation
Divide by B and let x = N/A

35. Analysis of Fixed Points
Fixed Point at x = 0
Other Fixed Points:

36. Fixed Points
Refuge vs Outbreak

37. Bifurcation Curves
Condition for Saddle-Node Bifurcation is that the line r(1-x/k) intersects the curve x/(1+x2) tangentially. Thus,

38. Bifurcation Curves