The most important Mathematica functions for studying one dimensional maps are Nest, and

Transcription

1 The Standad Map Based on a notebook by T. Timbelake (Bey College, GA). See also the aticle T.Timbelake, Am. J. Phys. 7 (8) Aug 4, available at Mathematica basics The most impotant Mathematica functions fo studying one dimensional maps ae Nest, and NestList. Nest calculates the esult of iteating a function a cetain numbe of times, using a cetain initial value fo the vaiable. NestList calculates each esult in the iteation pocess and foms all of these esults into a list. The examples below illustate how these functions can be used with the Standad Map. Fist, we must define the function we wish to iteate. In[]:= K =.5; f@8_, _<D := N@8Mod@ + - K Sin@D, p, -pd, Mod@ - K Sin@D, p, -pd<d; Now suppose we wanted to find the esult of iteating this function fou times using an initial point {.,.7}. In othe wods, we want to find f(f(f(f({.,.7})))). In Mathematica we can simply use the Nest function to geneate this esult. Nest@f, 8.,.7<, 4D 8-.44,.6< If we wish to see a list of all of the fist fou iteates of f, using an initial point {.,.7}, we use NestList instead. NestList@f, 8.,.7<, 4D 88.,.7<, ,.567<, 8.8, <, , <, 8-.44,.6<< Note that this list contains five elements, the fist element being ou initial point (o the th of the map). Sufaces of Section We can geneate Poincae sufaces of section by applying NestList to seveal diffeent initial points in phase space. In the example below each initial point is used to geneate 4 othe points though iteation of the map with K=.8. The initial points ae chosen to have = o =p/, with diffeent values fo distibuted evenly fom to.

2 StandadMap_modpi.nb In[]:= K =.8; n = 4; = ; = p ê ; max = p; n = ; MapData = Flatten@Table@Join@NestList@f, 8, i max ê n<, nd, NestList@f, 8, i max ê n<, ndd, 8i,, n<d, D; ListPlot@MapData, PlotStyle Ø PointSize@.D, PlotRange Ø p 88-, <, 8-, <<, Fame Ø Tue, RotateLabel Ø False, FameLabel Ø 8"", "", "", ""<, Axes Ø FalseD Out[4]= Aea-Pesevation (Evolution of a Box of Initial Conditions) To illustate the aea-peseving natue of the Standad Map we stat with a lage numbe of initial points scatteed andomly thoughout some egion. By applying the map to all of these points we find that this egion is mapped to a new egion that has a diffeent shape but the same aea. Repeated application of this pocedue futhe distots the shape of the egion, but the aea emains constant. In the example below we stat with 5 points andomly distibuted in the ectangula egion.4<<.6 and <<.. The map is iteated 4 times (nite=4) fo these points and the esulting set of points is plotted. In[5]:= K =.; min = -. p; max =. p; min =. p; max =.4 p; n = 5; nite = ; MapData@nite_D := Table@Nest@f, 8Random@Real, 8min, max<d, Random@Real, 8min, max<d<, nited, 8i,, n<d; aeaplot@nite_d := ListPlot@MapData@niteD, PlotStyle Ø PointSize@.D, PlotRange Ø p 88-, <, 8-, <<, Fame Ø Tue, RotateLabel Ø False, FameLabel Ø 8"", "", "", ""<D

3 StandadMap_modpi.nb In[9]:= Out[9]= Animation Manipulate@aeaplot@nD, 8n,,, <, AppeaanceElements Ø "ResetButton"D n Single Tajectoy Plotting a single tajectoy is vey simple. The pocedue is identical to that used to ceate the suface of section, but with only a single initial point used.

4 4 StandadMap_modpi.nb In[4]:= K =.98; = p ê ; =.; n = ; MapData = NestList@f, 8, <, nd; ListPlot@MapData, PlotStyle Ø PointSize@.D, PlotRange Ø p 88-, <, 8-, <<, Fame Ø Tue, RotateLabel Ø False, FameLabel Ø 8"", "", "", ""<D Out[4]= Exponential Divegence and Lyapunov Exponents To illustate exponential divegence of neighboing tajectoies we can stat with two diffeent, but vey close, initial points. The distance between the iteates of these points should incease exponentially with time if the initial points ae in a chaotic egion of the phase space. Theefoe, a plot of the natual log of this distance should incease linealy with the numbe of iteations. In the example below we ceate such a plot using initial points {p,} (which is a fixed point) and a slightly displaced point. The gaph shows the log of the distance between the tajectoies vesus the numbe of iteations of the map, up to iteations. Note that the function is no longe defined Modulo, because this leads to incoect measues of the distance between the tajectoies. K =.5; = p; = ; = p +.; = ; n = ; f@8_, _<D := N@8 + - K Sin@D, - K Sin@D<D; L = NestList@f, 8, <, nd; DList = Table@Log@St@H - L@@i, DDL^ + H - L@@i, DDL^DD, 8i, n + <D; ListPlot@DList, Fame Ø Tue, Axes Ø False, FameLabel Ø 8"n", "LnHd n L", "", ""<, PlotStyle Ø PointSize@.DD -5 LnHdnL n This esult can be compaed to the positive Lyapunov exponent fo this fixed point, which should be eual to the slope of the line in the pevious plot. The Lyapunov exponent of the fixed point is found by taking the natual log of the eigenvalues of the Jacobian matix fo the fixed point (see pevious section). Using the Jacobian given in the pevious section we find:

5 StandadMap_modpi.nb 5 eual to the slope of the line in the pevious plot. The Lyapunov exponent of the fixed point is found by taking the natual log of the eigenvalues of the Jacobian matix fo the fixed point (see pevious section). Using the Jacobian given in the pevious section we find: K =.5; Jac = 88 + K, <, 8K, <<; Log@Eigenvalues@JacDD , < The slope in the above plot is oughly given by HDList@@5DD - DList@@DDL ê So we see that the ageement between the two is good.