Worksheet #9 - PHY102 (Spr. 2008)Due Friday March 21st 6pmMore on “Do loops”, Illustrating ChaosTools that you needDo (you can also use Table or NestList)You will also need to learn how to plot lists of numbers using:ListPlotThe new physics - ChaosChaos, though discussed extensively for a couple of centuries (e.g. Boltz-mann and Maxwell discussed “molecular chaos”), has really come into its ownsince the widespread use of computers. An early surprise is that even quitesimple looking systems can have chaos, whereas it was originally thought thatchaos only occured in systems with billions of molecules. In this worksheetyou will study perhaps the simplest system which shows chaos, namely the“mapping”xn+1= λxn(1 − xn) (1)This mapping models, for example, how a population density, xn+1, changesas a function of the number of generations, n. Actually, it is not a veryrealistic model but it does illustrate many of the features of more complexsystems. The parameter λ can be considered to be the “birth rate”, ie. thenumber of offspring from the last generation. Anyway the way it works isthat if we know the population density at some time and call that densityx0, then the population density of the next generation is x1= λx0(1 − x0).This procedure is continued using Eq. 1 to find the population density forlater generations. Intuitively, chaos means lack of order. Mathematically,it is defined by how stable the behavior of a set of equations is to smallperturbations in the initial conditions. In the context of equation 1 thismeans, how stable are the set of iterates (x0, x1, x2, x3...) when you make thesmall change x0− > xδ0= x0+ δx0. If this change is made, we get a ne wset of iterates (xδ0, xδ1, xδ2, xδ3...). If a set of equations is in a chaotic regimethen the divergence of trajectories is exponential with a positive “Lyapunov”exponent. In the context of our example,|xδn− xn| ≈ eνn, (2)1where, in a truly chaotic system, the Lyapunov exponent ν is positive. Inthat case small changes in the initial conditions lead to very large changes inthe final behavior. This is the “butterfly effect” which makes it very difficultto compute the precise dynamical behavior of complex dynamical systems,such as the weather or the stock market.Problem(i) Write a Mathematica code to iterate the mapping (Eq. 1). Plot thebehavior of xnas a function n for λ = 2.5, λ = 3.5, λ = 3.95. Discussthe behavior you observe.(ii) By extracting the steady state behavior at 100 values in the range1 < λ < 4, plot the steady state behavior as a function of λ on onegraph. This is achieved by plotting xnvs λ at these 100 values.(iii) Consider to close starting values of x1, namely x1= 0 .51 and x1=0.510000000000001. Using these starting values find the way in whichthe trajectories diverge as n increases. From this data estimate theLyapunov exponent using Eq.
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