1. University of Delhi, Delhi, India
Route to chaos
Awadhesh Prasad
University of Delhi, Delhi, India

2. -- Models
-- Functions
-- Objects

3. Length of lightening?
Detection of direction of information flow in coupled complex systems

4. Length of a tree?
Detection of direction of information flow in coupled complex systems

5. Length of coastline

6. Area of surface?

7. Examples Linear: Point, Line, Plane, Cube
Nonlinear: Sphere, length of a tree
Detection of direction of information flow in coupled complex systems

8. Linear vs Nonlinear
Detection of direction of information flow in coupled complex systems

9. Differential equation:
Linear
Nonlinear

10. Differential equation:
Linear
Nonlinear

11. Pendulum
Individual/single

12. Pendulum: Linear vs Nonlinear
T0=time at vertical which th=0
sn Jacobian Elliptic function

13. Pendulum: Linear vs Nonlinear
T0=time at vertical which th=0

14. Center Linear Pendulum Damped Undamped Stable
T0=time at vertical which th=0
Center
Stable

15. Nonlinear pendulum Undapmed Damped Center/saddle Stable & Unstable
T0=time at vertical which th=0
Center/saddle
Stable & Unstable

16. Solutions
= ? +
Dissipation
Nonlinearity

17. Fixed/stationary points
1D Systems
Fixed/stationary points

18. Fixed/stationary points
1D Systems
Fixed/stationary points U S U S U S U S

19. 2D Systems

20. >2D Systems

21. Solutions
Invariants
Attractors

22. Solutions
-- Fixed point
-- Periodic
-- Quasiperiodic
-- Chaotic

23. Solutions
-- Fixed point
Nonlinear
Linear
-- Periodic
-- Quasiperiodic
-- Chaotic

24. Fixed point Solutions
l=a ib

25. Periodic Solutions
Unstable fixed points

26. Periodic solutions

27. Quasiperiodic

28. Quasiperiodic

29. Stable/unstable periodic orbits
Chaos!!

30. Chaotic Solutions

31. Chaotic Solutions

32. Properties of chaos
-- geometrically strange
-- temporally irregular
-- sensitive to initial conditions
Due to UPO?

33. Unstable Periodic Orbits (UPOs)

34. Bifurcation Qualitative change in dynamics as parameter varies
Sand piling Heartbeat

35. Chaos to Periodic: Heart Attack
Christini D J et al. PNAS 98, 5827(2001)

36. Montage

37. Onset of a temporal lobe epileptic seizure
Ref. L. Iasemidis

38. EEG: Epileptic Patient (temporal lobe epilepsy)
Preictal
ictal
postictal
minutes

39. Aim
Periodic
Chaotic
Routes?
Chaotic
Chaos

41. Period doubling: Laser

42. End