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Georgia Tech | PHYSICS 4267
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Georgia Tech PHYSICS 4267
INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS - Georgia Tech PHYS 4426/6268 Spring semester 2007 Predrag Cvitanovic\' Problem set 8: Period doubling, Lyapunov due Tue Mar 15 2007 - Exercise 26.1 [ChaosBook.org] Per -
Georgia Tech PHYSICS 4267
georgia tech PHYS 4267/7224 introduction to nonlinear dynamics and chaos instructor: P Cvitanovi c spring semester 2007 take-home nal exam due no later than 10:50am, Thursday, May 3 [delivered either to Jonathan or Predrag, 5th oor Howey] 1 Flutte -
Georgia Tech PHYSICS 4267
INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS -Georgia Tech PHYS 4426/6268 Predrag Cvitanovic\' Problem set 5: Baker map -Spring semester 2007 due Tue Feb 13 2007 Problem 5.0 Program baker map (5.2) in programming language of your choice (Matlab, C, -
Georgia Tech PHYSICS 4267
INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS -Georgia Tech PHYS 4426/6268 Predrag Cvitanovic\' Problem set 6: Lozi map -Spring semester 2007 due Tue Feb 20 2007 Feb 18 2007 NOTE added comments, in response to several emails now reaching lenght of a n -
Georgia Tech PHYSICS 4267
182 References Exercises Exercise 11.1 Binary symbolic dynamics. Verify that the shortest prime binary cycles of the unimodal repeller of figure 11.8 are 0, 1, 01, 001, 011, . Compare with table 11.1. Try to sketch them in the graph of the unimod -
Georgia Tech PHYSICS 4267
Exercise 8.3 A limit cycle with analytic stability exponent. There are only two examples of nonlinear flows for which the stability eigenvalues can be evaluated analytically. Both are cheats. One example is the 2-d flow q = p + q(1 - q 2 - p2 ) , p -
Georgia Tech PHYSICS 4267
Chapter 24 Turbulence? I am an old man now, and when I die and go to Heaven there are two matters on which I hope enlightenment. One is quantum electro-dynamics and the other is turbulence of uids. About the former, I am rather optimistic. Sir Horac -
Georgia Tech PHYSICS 4267
EXERCISES Exercise 11.6 185 Unimodal map symbolic dynamics. Show that the tent map point (S + ) with future itinerary S + is given by converting the sequence of sn s into a binary number by the algorithm (11.11). This follows by inspection from the -
Georgia Tech PHYSICS 4267
Chapter 11 Qualitative dynamics, for pedestrians The classication of the constituents of a chaos, nothing less is here essayed. Herman Melville, Moby Dick, chapter 32 In this chapter we begin to learn how to use qualitative properties of a ow in or -
Georgia Tech PHYSICS 4267
Chapter 26 Universality in transitions to chaos The developments that we shall describe next are one of those pleasing demonstrations of the unity of physics. The key discovery was made by a physicist not trained to work on problems of turbulence. I -
Georgia Tech PHYSICS 4267
4 CHAPTER 1. OVERTURE Figure 1.1: A physicists bare bones game of pinball. 1.3 The future as in a mirror All you need to know about chaos is contained in the introduction of [ChaosBook]. However, in order to understand the introduction you will r -
Georgia Tech PHYSICS 4267
EXERCISES Exercise 9.4 Escape rate of the tent map. 147 (a) Calculate by numerical experimentation the log of the fraction of trajectories remaining trapped in the interval [0, 1] for the tent map f (x) = a(1 2|x 0.5|) for several values of a. 3 -
Georgia Tech PHYSICS 4267
522 CHAPTER 26. UNIVERSALITY IN TRANSITIONS TO CHAOS (a) (b) Chapter 26 Figure 26.1: Universality in transitions to chaos The developments that we shall describe next are one of those pleasing demonstrations of the unity of physics. The key dis -
Georgia Tech PHYSICS 4267
42 References Exercise 2.6 Runge-Kutta integration. Implement the fourth-order Runge-Kutta integration formula (see, for example, ref. [2.7]) for x = v(x): xn+1 = xn + k1 k1 k2 k3 k4 + + + + O( 5 ) 6 3 3 6 = v(xn ) , k2 = v(xn + k1 /2) k4 = v(x -
Georgia Tech PHYSICS 4267
Exercises Exercise 26.1 Period doubling in your pocket: Take a programmable pocket calculator or Matlab or whatever makes you feel good and program the function f (x) = - x2 . The game consists in staring at the display, and looking for regularities -
Georgia Tech PHYSICS 4267
Chaos: Classical and Quantum Part I: Deterministic Chaos Predrag Cvitanovi Roberto Artuso Ronnie Mainieri Gregor c Tanner Gbor Vattay Niall Whelan Andreas Wirzba a ChaosBook.org/version11.4, Aug 5 2005 printed August 4, 2005 ChaosBook.org com -
Georgia Tech PHYSICS 4267
18 flows - 8may2005 A typical numerically integrated long-time trajectory Rssler flow _ x = `y ` z _ y = x + ay _ z = b + z(x ` c) ; a = b = 0:2 ; c = 5:7 : Z(t) 30 25 20 15 10 5 0 5 0 Y(t) -5 -10 -10 0 -5 15 10 5 X(t) version -
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Georgia Tech PHYSICS 4267
Predrag Cvitanovic\' Chaos and what to do about it Classics Illustrated version H. Poincar, describing in `Les mthodes nouvelles de la mchanique mleste\' his discovery of homoclinic tangles: The complexity of this figure will be striking, and I shal -
Georgia Tech PHYSICS 4267
EXERCISES 479 Exercises Exercise 27.1 WKB ansatz. Try to show that no other ansatz other than (28.1) gives a meaningful definition of the momentum in the 0 limit. Exercise 27.2 1 2 - Fresnel integral. x2 Derive the Fresnel integral dx e- 2ia -
Georgia Tech PHYSICS 4267
confession!St. Augustine St. Augustine coarse-graining Chapter 9 Transporting densities Paulina: I\'ll draw the curtain: My lord\'s almost so far transported that He\'ll think anon it lives. W. Shakespeare: The Winter\'s Tale what does \"anon it lives\" -
Georgia Tech PHYSICS 4267
Instructor: Predrag Cvitanovic Spring semester 2006 - Jan 09, 2006 - May 06, 2006 Registration: Nov 01, 2005 - Jan 13, 2006 Undergraduate Research Assistantship - 25599 - PHYS 4698 Undergraduate Research - 25600 - PHYS 4699 -
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Georgia Tech PHYSICS 4267
~ r , m on LEa~arn Looks k t Ue b ~rc=*(o\"i it h a c a q-SQS Cp , s, ~n Cl~we~rr, t j k ~ c,q be. s - hy din 9 -
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Georgia Tech PHYSICS 4267
2 > x( t ) + x( t ) + x( t ) = 0 ; x( 0 ) = 1 ; x( 0 ) = 0 t t t t > = .3 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.5 1 1.5 t~ 2 2.5 3 Exact solution (red) Outer solution (yellow) Inner solution (green) Page 1 2 > x( -
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Georgia Tech PHYSICS 4267
Phys. 4267/6268 Final Exam Due 04/30/09 (12:00 noon) Problem 1 Consider a two dimensional system: x = x(3 - 2x - y), y = y(2 - x - y). 1. Find the fixed points and determine their stability. 2. Draw the nullclines and sketch the phase portrait. -
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Georgia Tech PHYSICS 4267
PHYS 4267 Solution to Problem #9 Spring 2003 PAGE 1 of 3 PAGE 2 of 3 PAGE 3 of 3 -
Georgia Tech PHYSICS 4267
Phys 4267 Solution Set #5 Spring 2003 Stabilizing the Inverted Pendulum (a) To find the critical value of T , I guess a value for T , evaluate the expressions for c and d, and then use these to evaluate the formula X = 2 cos(dT /2) cosh(cT /2) + c -
Georgia Tech PHYSICS 4267
Phys 4267 Solution to Problem #15 Spring 2003 Stability Analysis of the Rossler Equations (a) Start with the equation X = -Y - Z Y = X + aY Z = b + Z(X - c) To find the fixed points, set the time derivatives to zero: 0 = -Y0 - Z0 0 = X0 + aY0 -
Georgia Tech PHYSICS 4267
Phys 4267 Solution to Problem #10 Spontaneous Symmetry Breaking Spring 2003 x + x + x3 = cos t (a) This is a first order non-autonomous equation, so the phase space dimension is 1 + 1 = 2. (b) Let x = A + B cos where for convenience I\'ve introdu -
Georgia Tech PHYSICS 4267
Phys 4267 Solution to Problem #18 Spring 2003 Perturbing the Transcritical Bifurcation x = x - x2 - (a) The fixed points are determined by setting x = 0. For the fixed points are x = 0andx = Stability is determined by checking the Jacobian mat -
Georgia Tech PHYSICS 4267
Phys 4267 Solution to Problem #30 Spring 2003 Lyapunov Exponents for the Logistic Map (a) Logistic Map R=3.2 1 1 Logistic Map R=3.9 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 Xn 0.4 X 0 100 200 300 400 500 600 700 800 900 1000 0.5 0.3 0.2 0.1 0
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