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## From Linear

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**IAPT Workshop**2nd August ISP, CUSAT From Linear Nonlinear Physics To V P N Nampoori nampoori@gmail.com**Simple pendulum**Oscillate Rotate**Simple pendulum**Oscillate Rotate Double pendulum**Taking the amplitude**small θ**The hallmark of linear equations.**We can predict state of the simple pendulum at any future time Linearlty helps us to predict future.**Alternate representation of damped motion of simple pendulum**Phase space plot**Second order nonlinear differential equation**Superposition principle is not valid Prediction is not possible**One-dimensional maps**One-dimensional maps, definition: - a set V (e.g. real numbers between 0 and 1) - a map of the kind f:VV Linear maps: - a and b are constants - linear maps are invertible with no ambiguity Non-linear maps: The logistic map**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**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**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) , …**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?**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**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 ?**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?**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?**Plot of fixed points vs**Bifurcation diagram**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**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.**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)**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) , …**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**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?**Plot of fixed points vs**Tension builds up Bifurcation diagram Peace time WAR!!!**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**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**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)**Photograph of melting ice landscape –**Face of Jesus Christ - evolution leading to fixed state**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 - 00 2 E/t2 = 0 or 2E - 1/c22 E/t2 = 0 In a medium, 2E - 00 2 E/t2 - 02P/t2 = 0 writing P = 0 E, 2E - 1/v22 E/t2 = 0 where, v = (0 )-1/2 Defining c/v = n, 2E - n2/c22 E/t2 = 0 we get a second order linear diffl eqn describing what is called, Linear Optics**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**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/c22 E/t2 = 02PNL/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****2 Optical second harmonic generation 1 3 2 Sum (difference) frequency generation OPC by DFWM**Consequence of 3rd order optical nonlinearity**intensity dependent complex refractive index**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.**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**It**I Optical Limiters : Materials which are opaque above a threshold laser intensity Materials become opaque at higher intensity Is**Materials become opaque at higher pump intensity – optical**limiter Materials become transparent at higher pump intensity- saturable absorber**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 .**Intensity dependent refractive index**Laser beam :Gaussian beam I(r ) = I0exp(-2r2/w2) I(r ) Beam cross section r