From Linear

1. IAPT Workshop 2nd August ISP, CUSAT From Linear Nonlinear Physics To V P N Nampoori nampoori@gmail.com

2. Simple pendulum Oscillate Rotate

3. Simple pendulum Oscillate Rotate Double pendulum

6. Taking the amplitude small θ

7. The hallmark of linear equations. We can predict state of the simple pendulum at any future time Linearlty helps us to predict future.

9. Alternate representation of damped motion of simple pendulum Phase space plot

10. Second order nonlinear differential equation Superposition principle is not valid Prediction is not possible

11. Linear dynamics – prediction possible

12. Traffic jam –Nonlinear dynamics - prediction is impossible

13. One-dimensional maps One-dimensional maps, definition: - a set V (e.g. real numbers between 0 and 1) - a map of the kind f:VV Linear maps: - a and b are constants - linear maps are invertible with no ambiguity Non-linear maps: The logistic map

14. Motivation: Discretization of the logistic equation for the dynamics of a biological population x b: birth rate (assumed constant) cx: death rate depends on population (competition for food, …) How do we explore the logistic map? One-dimensional maps Non-linear maps: The logistic map with

15. Evolution of the logistic map fixed point ? Geometric representation 1 Evolution of a map: 1) Choose initial conditions 2) Proceed vertically until you hit f(x) 3) Proceed horizontally until you hit y=x 4) Repeat 2) 5) Repeat 3) . : x f(x) 0 0.5 1

16. 1 1 y=x y=x f(x) f(x) 0 0.5 1 0 0.5 1 1 1 0 0.5 1 0 0.5 1 b) a) Phenomenology of the logistic map fixed point fixed point c) d) chaos? 2-cycle? What’s going on? Analyze first a)  b) b)  c) , …

17. 1 1 x x f(x) f(x) 0 0.5 1 0 0.5 1 Geometrical representation Evolution of the logistic map fixed point How do we analyze the existence/stability of a fixed point?

19. Stability - Define the distance of from the fixed point Taylor expansion - Consider a neighborhood of - The requirement implies Logistic map? - Condition for existence: Fixed points - Logistic map: - Notice: since the second fixed point exists only for

20. 1 1 x x f(x) f(x) 0 1 0.5 0 0.5 1 - Stability condition: Stability and the Logistic Map - First fixed point: stable (attractor) for - Second fixed point: stable (attractor) for • No coexistence of 2 stable fixed points for these parameters • (transcritical biforcation) What about ?

21. 1 Observations: Period doubling x 1) The map oscillates between two values of x f(x) 0 0.5 1 0 0.5 1 Evolution of the logistic map 2) Period doubling: What is it happening?

22. 0 0.5 1 These points form a 2-cycle for However, the relation suggests they are fixed points for the iterated map Stability analysis for : and thus: For , loss of stability and bifurcation to a 4-cycle Now, graphically.. • - At the fixed point becomes unstable, • since • Observation: an attracting 2-cycle starts  • (flip)-bifurcation • The points are found solving the equations > Period doubling and thus: Why do these points appear?

23. Plot of fixed points vs Bifurcation diagram

24. International Relations and Logistic Map Let A and B are two neighbouring countries Both countries look each other with enmity. Country A has x1 fraction of the budget for the Defence for year 1 Country B has same fraction in its budget as soon as A’s budget session Is over Next year A has increased budget allocation x2 Budget allocation goes on increasing. If complete budget is for Defence , it is not possible since no funds for Other areas

25. 1 x f(x) 0 0.5 1 Fund allocation for subsequent years for the country A As time progresses, budget allocation for defense decreases. Peace time . A and B are friends.

26. Parameter μ is called enmity parameter. Now let a third country C intervenes In the region to modulate the enmity parameter and μ = μ(t)

27. 1 1 y=x y=x f(x) f(x) 0 0.5 1 0 0.5 1 1 1 0 0.5 1 0 0.5 1 b) a) Phenomenology of the logistic map fixed point fixed point c) d) chaos? 2-cycle? What’s going on? Analyze first a)  b) b)  c) , …

28. 1 1 x x f(x) f(x) 0 1 0.5 0 0.5 1 Budget allocation stabilises To a fixed value. Caution time. Yellow Budget allocation decreases and goes to zero Full peace time Green

29. 1 Observations: Period doubling x 1) The map oscillates between two values of x f(x) 0 0.5 1 0 0.5 1 Evolution of the logistic map 2) Period doubling: What is it happening?

30. Plot of fixed points vs Tension builds up Bifurcation diagram Peace time WAR!!!

31. Evolution of International Relationships between three countries Two countries are at enmity and the third is the controlling country From Peace time to War time

32. Interaction leads to modification of dynamics. A, B and C are three components of a system with two states YES (1) or NO ( 0) Case 1 A, B and C are non interacting Following are the 8 equal probable states of the system 1=( 000), 2=( 001), 3=(010), 4=( 100),5=(110),6=(101),7=(011), 8=(111) Probability of occurrence is 1/8 for all the states. States evolve randomly. Case II Let A obeys AND logic gate while B and C obey OR logic gate

33. T2 T3 T1 1/8 I (000) (000) (000) (000) (001) (010) (001) (010) II bistable (010) 2/8 (001) (001) (010) (111) (111) (100) (011) (111) (111) (011) III 5/8 (110) (111) (111) Evolution to Fixed state Blissful state!!! (101) (011) (111) (111) (111) (011) (111) (111) (111) (111)

34. Bistable picture

35. Bistable picture

36. Rabbit and duck – bistable state

37. Photograph of melting ice landscape – Face of Jesus Christ - evolution leading to fixed state

38. Evolution – Esher’s painting

39. Nonlinearity in Optics

40. Linear Optics Maxwell’s Equations : Light -- Matter Interaction Maxwell’s equations for charge free, nonmagnetic medium .D = 0 .B = 0 XE = -B/t XH = D/t D = 0E + P and B = 0H In vacuum, P = 0 and on combining above eqns 2E - 00 2 E/t2 = 0 or 2E - 1/c22 E/t2 = 0 In a medium, 2E - 00 2 E/t2 - 02P/t2 = 0 writing P = 0 E, 2E - 1/v22 E/t2 = 0 where, v = (0 )-1/2 Defining c/v = n, 2E - n2/c22 E/t2 = 0 we get a second order linear diffl eqn describing what is called, Linear Optics

41. P=(1) E + (2) E2 + (3)E3 +….. (n)/ (n+1) << 1 For isotropic medium, (n) will be scalar. (n) represents nth order non linear optical coefficient

42. Polarisation of a medium P = PL +PNL where,PL = (1) E andPNL = (2)E2 + (3)E3+….. On substituting P in the Maxwell’s eqns, we get 2E - n2/c22 E/t2 = 02PNL/t2 This is nonlinear differential equation and describes various types of Nonlinear optical phenomena. Type of NL effects exhibited by the medium depend the order of nonlinear optical coefficient. E Polarisation Maxwell’s eqns feedback

43.  2 Optical second harmonic generation 1 3 2 Sum (difference) frequency generation OPC by DFWM

44. Consequence of 3rd order optical nonlinearity intensity dependent complex refractive index

45. One of the consequences of 3rd order NL PNL = (1)E + (3)E3 =( (1) + (3)E2)E = (n1 + n2 I)E We have n(I) = (n1 + n2 I) n2< 0 n n1 n2> 0 I Intensity dependent ref.index has applications in self induced transparency, self focussing, optical limiting and in optical computing.

46. α T I Saturable absorbers– Materials which become transparent above threshold intense light pulses Materials become transparent at high intensity Absorption decreases with intensity Is

47. It I Optical Limiters : Materials which are opaque above a threshold laser intensity Materials become opaque at higher intensity Is

48. Materials become opaque at higher pump intensity – optical limiter Materials become transparent at higher pump intensity- saturable absorber

49. Optical limiters are used in…… • Protection of eyes and sensitive devices from intense light pulses • Laser mode locking • Optical pulse shaping • Optical signal processing and computing .

50. Intensity dependent refractive index Laser beam :Gaussian beam I(r ) = I0exp(-2r2/w2) I(r ) Beam cross section r