SequenceTools Package

BifurcationPlot[f, x0, {a, amin, amax}] produces a bifurcation diagram for a function f[x] which is defined in terms of the parameter a. (A good example function to start with is f[x_]:= a*x*E^-x, with amin=2 and amax=20, and x0 between them). For each of many sampled values of the parameter a between amin and amax, the plot shows the long-term orbit of x0 under repeated iteration of f. By default, above each a-value on the horizontal axis appears the values Nest[f,x0,100] through Nest[f,x0,200]. This can be adjusted with the option Iterations, whose default setting is Iterations->{100,200}. By default, the interval from amin to amax is subdivided into 99 subintervals of equal width, and the 100 endpoints of the subintervals are sampled. This can be adjusted in either of two ways: the first is to use the option Subintervals, whose default setting is 99. The second is to use the syntax BifurcationPlot[f, x0, {a, amin, amax, da}], where da is the width of a single subinterval (the distance between successive sample points).For the true control freaks, the option Subintervals may also be set to a list of values that are to be sampled. BifurcationPlot will accept any options accepted by ListPlot. A different use of BifurcationPlot is also possible. What if the parameter a is fixed at the value a0 and the intitial value x0 is permitted to vary? Simply adjust the syntax to reflect this change: BifurcationPlot[f, {x0min, x0max}, {a, a0}] does the job. In this case the horizontal axis represents the varying values of the initial point x0. The options Iterations and Subintervals work as above. One can also proceed in this investigation for a function f defined without a parameter (such as f[x]=2*x*E^-x). In this case, use the even simpler syntax BifurcationPlot[f, {x0min, x0max}] to get the plot.

Cobweb[f, x0, n] gives a visual representation of the sequence {x0, f[x0], f[f[x0]], ...} where each term is obtained from the previous term by applying the function f, and where the last term is obtained by applying f precisely n times. The command requires that f be a real-valued function of a single variable, x0 be a real number, and n be a nonnegative integer. The sequence itself can be generated with the command NestList[f, x0, n]. The Cobweb command produces a plot of f togther with the line y=x, and the `cobweb' comprised of the horizontal and vertical segments between these graphs. Also displayed are red dots on the x axis indicating the values of the sequence produced. Cobweb will accept any option accepted by the Plot command; in particular, the setting AspectRatio->Automatic can be used to give both axes the same scale. Also, the PlotRange option can be used to adjust the range of values on each axis.

CobwebMovie[f, x0, n] gives a visual representation of the sequence {x0, f[x0], f[f[x0]], ...} where each element is obtained from the previous element by applying the function f, and where the last member is obtained by applying f precisely n times. The command requires that f be a real-valued function of a single variable, x0 be a real number, and n be a nonnegative integer. The sequence itself can be generated with the command NestList[f, x0, n]. The CobwebMovie command produces a movie, that is a sequence of graphics frames. Each frame shows a plot of f togther with the line y = x, and the emerging `cobweb' comprised of the horizontal and vertical segments between these graphs. Also displayed are dots on the x axis indicating the values of the emerging sequence. CobwebMovie will accept any option accepted by the Plot command; in particular, the setting AspectRatio->Automatic can be used to give both axes the same scale, and the PlotRange option can be used to adjust the range of values on each axis. There is one frame in the movie for each of the n+1 members of the sequence.

DeltaTable[y, n] produces a table of first through n-th differences for the values of the sequence y. To use the command, the first argument y must be a list of the first several values of a sequence. If the values are empirical data, be sure that the corresponding x-values are evenly spaced (i.e. the first differencs of the x data should be constant).

SequencePlot[s[n], {n, nmin, nmax}] produces a plot of the sequence s, as n ranges from nmin to nmax. SequencePlot[s[n], {n, nmin, nmax, step}] produces a plot of the sequence s, as n ranges from nmin to nmax in increments of step. SequencePlot will accept any option that ListPlot does. In particular, you can connect the dots by adding the option PlotJoined->True. SequencePlot will also accept a list of sequences as its first argument.

SequenceTable[s, {n, nmin, nmax}] produces a table of values for the sequence s as n ranges from nmin to nmax. SequenceTable[s, {n, nmin, nmax, step}] produces a table of values where n ranges from nmin to nmax in increments of step. SequenceTable will accept any options that TableForm does. SequenceTable will also accept a list of sequences as its first argument.

LogisticFit[xdata,ydata] gives the coefficients for the `best fitting' logistic growth sequence for the list of values ydata. The best fit sequence is given by the difference equation p[n]=a*p[n-1]-b*p[n-1]^2, with initial value corresponding to the first value in ydata. LogisticFit[data], where data is a list of two-tuples {{x1,y1},{x2,y2},...}, will give the same result. LogisticFit assumes the members of xdata are evenly spaced, and will give poor results when this is not the case. The output also includes a scatterplot of the original data, with the best fitting sequence superimposed in blue.