Worksheet #9 – PHY102 (Spring 2011)More on Do loops: Intentional chaosTools you will needThis week you will need to use ListPlot, Animate, and Do or For. You can review these in yournotebook from Worksheet 6, where they were introduced; or you can look them up in the onlinehelp. You will also want to use Table or NestList, so have a look at those in the online help andmake up some examples for yourself to get familiar with them before starting in on the assignedproblem.The new physics – ChaosChaos, though it had been discussed extensively for a couple of centuries (e.g. Boltzmann andMaxwell discussed “molecular chaos”), has really come into its own since the widespread useof computers. That is because the solutions to chaotic systems—even simple ones—do not lendthemselves to the kind of mathematical closed-form solutions that are can be handled by traditionalanalytic mathematical methods.An early surprise was that even quite simple looking systems can display chaotic behavior, whereasit was originally thought that chaos only occured in systems with billions of molecules. In thisworksheet, you will study perhaps the simplest system which shows chaos: namely, the purelymathematical nonlinear “mapping”xn+1= λ xn(1 − xn) (1)This mapping model can be used, for example, to describe how a population density, xnchangesfrom one generation (n) to the next (n + 1). Actually, it is not a very realistic model; but it doesillustrate many of the features of more complex systems. The parameter λ can be considered tobe 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 density x1, then the populationdensity of the next generation is x2= λ x1(1 − x1). This procedure is continued using Eq. 1 tofind the population density for later generations. Intuitively, chaos means a lack of order . Mathe-matically, it is defined by how stable the behavior of a set of equations is to small perturbations inthe initial conditions. In the context of equation 1, this means how stable is the series of iterates(x1, x2, x3, . . . ) when you make a very small change x1→ xδ1= x1+ δx. When this change ismade, we get a new set of iterates (xδ1, xδ2, xδ3, . . . ).If a set of equations is in a chaotic regime then the trajectories defined by these series of pointsdiverge exponentially. In the context of our example,|xδn− xn| ∼ eνn, (2)1where in a chaotic system, the Lyapunov exponent ν is positive.Problem 1(i) Write a Mathematica code to iterate the mapping in Eq. (1). (You can use Do, For, orNestList for it. Another useful function is Range, which can be used to specify the dimen-sion of an array—even if you don’t want to use the values that Range puts into that array.)Plot the steady-state behavior of the mapping as a function of the parameter λ for 1 < λ < 4.Do this for several different values of the starting point x1in the range 0 < x1< 0.5 .(ii) For some particular value of λ in the regime that looks chaotic in your graph, make an esti-mate of the Lyapunov exponent using Eq. 2.Hint: choose x1and xδ1= x1+ δx, where δx is small. Plot the series xδn− xnas a functionof n and look for exponential growth on the average. Since you are looking for xδn− xn≈const × eνn, you might want to also plot the sequence log(|xδn−xn|) or even (1n) log(|xδn−xn|).If all is well, your estimate of the Lyapunov exponent ν will be independent of the choices ofx1and
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