Worksheet #9 – PHY102 (Spring 2010)More on Do loops: Intentional chaosTools you will needThis week you will need to use ListPlot, Animate, and Do or For. Youcan review these in your notebook from Worksheet 6, where they were intro-duced; or you can look them up in the online help. You will also want to useTable or NestList, so have a look at those in the online help and make upsome examples for yourself to get familiar with them before starting in onthe assigned problem.The new physics – ChaosChaos, though it had been discussed extensively for a couple of centuries(e.g. Boltzmann and Maxwell discussed “molecular chaos”), has really comeinto its own since the widespread use of computers. That is because the so-lutions to chaotic systems—even simple ones—do not lend themselves to thekind of mathematical closed-form solutions that are amenable to traditionalanalytic mathematical methods.An early surprise was that even quite simple looking systems can displaychaotic behavior, whereas it was originally thought that chaos only occured insystems with billions of molecules. In this worksheet you will study perhapsthe simplest system which shows chaos: namely, the purely mathematicalnonlinear “mapping”xn+1= λ xn(1 − xn) (1)This mapping model can be used, for example, to describe how a populationdensity, xnchanges from one generation (n) to the next (n+1). Actually, it isnot a very realistic model; but it does illustrate many of the features of morecomplex systems. The parameter λ can be considered to be the “birth rate”,i.e., the number of offspring from the last generation. The way it works isthat if we know the population density at some time and call that densityx1, then the population density of the next generation is x2= λ x1(1 − x1).This procedure is continued using Eq. 1 to find the population density forlater generations. Intuitively, chaos means a lack of order. Mathematically,it is defined by how stable the behavior of a set of equations is to small per-turbations in the initial conditions. In the context of equation 1, this meanshow stable are the set of iterates (x1, x2, x3, . . .) when you make a very small1change x1→ xδ1= x1+ δx. When this change is made, we get a new set ofiterates (xδ1, xδ2, xδ3, . . .).If a set of equations is in a chaotic regime then the trajectories defined bythese series of points diverge exponentially. In the context of our example,|xδn− xn| ∼ eνn, (2)where in a chaotic system, the Lyapunov exponent ν is positive.Problem 1(i) Write a Mathematica code to iterate the mapping in Eq. (1). (Youcan use Do, For, or NestList for it. Another useful function is Range,which can be used to specify the dimension of an array—even if youdon’t want to use the values that Range puts into that array.) Plot thesteady-state behavior of the mapping as a function of the parameter λfor 1 < λ < 4. Do this for several different values of the starting pointx1in the range 0 < x1< 0.5 .(ii) For some particular value of λ in the regime that looks chaotic in yourgraph, obtain an estimate of the Lyapunov exponent using Eq. 2.Hint: choose x1and xδ1= x1+ δx, where δx is small. Plot the seriesxδn− xnas a function of n and look for exponential growth on theaverage. Since you are looking for xδn− xn≈ const eνn, you might wantto also plot the sequence log(|xδn−xn|) or even (1n) log(|xδn−xn|). If allis well, your estimate of the Lyapunov exponent ν will be independentof the choices of x1and
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