(T)otal -- Change total time
(S)tart time -- Change start time of simulation
t(R)ansient -- Only store after this much time
(D)t -- Integration timestep
n(C)line ctrl -- nullcline grid
s(I)ng pt ctl -- fixed point accuracy
n(O)ut -- frequency of output
(B)ounds -- global bounds for all variables
(M)ethod -- integration method
d(E)lay -- delay equations params
(C)olor code -- color code trajectories
(P)oincare map -- c'mon you should know!
r(U)elle plot -- make delayed embedding plots
stoc(H)astic -- all sorts of
statistics, curve fitting, miscellany!
loo(K)up -- change tables
bndry(V)al -- params for BVP solver
(A)djoint -- weakly coupled oscillator stuff. |
y2'' = f2(y1,...,yn)
...
yn'' = fn(y1,...,yn)
This is the initial time
T0. For autonomous systems it is usually irrelevant.
The program will
integrate silently for this amount of time before plotting output. It
is used to get rid of transients.
This is the step size used by the
fixed step integrators and the output step for the adaptivve
algorithms. It is positive or negative depending on the direction of
integration. That is, to integrate backwards, make it negative.
will prompt you for the
grid size for computing nullclines.
This prompts you for errors and epsilons for eigenvalues and equilibria calculations as well as the maximum iterates.
This sets the number of integration steps to perform between output to storage for fixed step integrators.
Thus, if you output every 10 steps with a Dt of .05, XPP will yield output at times that are 10*.05=.5 timesteps apart. The advantage of this is that lengthier records of data can be made without losing accuracy of the integrator.
This sets a global bound on the integrator. If any variable exceeds this value in magnitude, a message appears and the integration stops.
allows you to choose the
methods of integration. There are many integrators available for use:
This sets the upper bound for
the maximum delay as well as some other parameters used in computing
the stability of fixed points for delay equations. The first pair are
guesses for the eigenvalue with maximal real part and the grid
is for the integration around a contour to determine how many
eigenvalues there are with positive real part using the argument
principle. If in your delay
equations, the delay exceeds the maximal delay , a message will appear and the
integration will stop. Each time the maximal delay
is changed all previous delay
data is destroyed and you must begin your integration anew. Thus, it
should be the first thing you set when solving a delay equation.
Since the storage depends on the size of Dt when you change this, then
the delay storage will also be destroyed. Delay equations require
data for t <t0 so that you should edit the Delay ICs to
achieve this. Use caution when integrating with adaptive step-size
integrators.
If you have a color system,
XPP can code the output according to the magnitude of the velocity or
the magnitude of a different quantity. A choice pops up for these two
options or for turning off the color. This overides any color on any
other curves in your picture. Once you choose to color code, you are
asked to either choose max and min values or have XPP do it for you
via optimize. The latter will compute the max and min and set the
scales accordingly.
This sets up
parameters for Poincare sections. A choice of four items will appear:
the Max/Min option, the Poincare section option, period option, and
the option to turn off all maps. Only sections orthogonal to
the coordinates are allowed, however, by defing the appropriate
auxiliary variables, you can make sections for any quantity.
The Max/Min finds local extrema of variable and the
Period option records the time between events in the Time
column as well as the local section info.
A window will pop up and you should
type in the parameters. They are the variable to check and the
section and the direction. That is, a point will be recorded if the
variable crosses the section such that it is either positive going to
negative or vice versa according to the direction parameter. If the
direction is set to zero, the, the point will be recorded from either
direction. The flag Stop on Section instructs XPP to halt when
the section is crossed. Note that automatic interpolation is done.
If the section variable is Time, T and the section is say
T1,then each time T=0 modulo T1 the point is recorded.
This is useful for periodically driven systems. If you have opted for
the Max/Min option, then the section is irrelevant and the
point will be recorded when a local maximum (if the direction is 1)
minimum (direction=-1) or both (direction=0) of the variable is
encountered.
This brings up a series of items that allow you to compute many trajectories and find their mean and variance. It is most useful when used with systems that are either Markovian or have noise added to the right-hand sides. The items are:
Use this to reseed the random number generator. If you use the same seed then the results will not change from run to run.
This will put up the same dialog box as the ``Integrate'' ``Range'' choice. Two new data sets will be created that will compute the mean and the variance of the point by point values of the trajectories over the number of trial runs you choose. If the system is completely deterministic and the parameters and initial conditions are identical for each run, then this is superfluous. Otherwise, the mean and variance are computed. You can then access these new arrays as described below. If you fire up the sample Markov problem, choose the ``Compute'' option, and set keep the initial data constant over say 20 runs, then you can look at the mean trajectory and its variance for each of your variables.
This puts the results of the most recent run into the data browser and enables plotting of them.
This puts the results of the mean value of the most recently computed set of trials.
This does the same for the variance.
This computes a histogram for a chosen variable and additional conditions and replaces the ``t'' column and
the first variable column with the bin values and the number per bin respectively. You will be prompted for the number of bins, a maximum and minimum value and the variable on which to perform the histogram. Finally, you will be asked for additional conditions that involve the other stored variables (not the fixed ones though.) For example, suppose you have run an ODE/Markov system and you want the distribution of a continuous variable when the Markov variable is in state 1. Then the additional condition would be z==1 where z is the Markov variable. Multiple conditions are made by using the & and | expressions. Note that == is the logical equal and is not the same as the algebraic one.
brings back the most recently computed histogram.
This prompts you for
a data column and the number of modes you want. It then
computes a Fourier transform (not an FFT, since I don't want
to worry about zero padding and other matters) for the number
of modes you have chosen. The results are in the Browser.
The first column (labeled ``T'') is the mode. The second, the
cosine component and the third, the sine component.
This is just like Fourier
except the magnitude and phase of the FFT are kept.
This is a routine based on
Marquardt-Levenberg algorithm for nonlinear least squares fitting. A description of the method can be found in Numerical Recipes in C. In this implementation, one can choose parameters and initial data to vary in an attempt to minimize the least-squares difference between solutions to a dynamical system and data. The data must be in a file in which the first column contains the independent values in increasing order. The remaining columns contain data which are to be fitted to solutions to a DE. Not all the columns need be used. When you choose this option, a window pops up with 10 entries describing the fit parameters. The items are:
On return, the program will put the best set of parameters that it has
found. It currently is quite verbose and prints a lot of stuff to the
console. This is mainly info about the current values of the
parameters and the least square.
This allows you to retard any of the axes by an integral number of steps. This is useful for chaotic orbits and delayed systems. Choosing a number for any of the axes will result in the variable associated with that axis being delayed by the number of steps inputted. Thus if you plot X vs X then of course you will get a diagonal line, but if you make the Y-axis delayed by say 50 and the output is every .1 timesteps, then the plot will be X(t-5) vs X(t). This does not appear during integration of the equations and is available only after a computation. You set it up and then click Restore from the main menu.
This allows you to change the definitions of tabulated functions by reading in a different file. Thus, if you have many experimental sets of data, you can read them in one by one and integrate the equations. You are prompted for the name of a tabulated function. Then you give the filename to read in. You will continue to be prompted and can type a few carriage returns to get out. If the table was defined as a function instead of a file, then you will be prompted for the number of points, the limits of the range (Xhi,Xlo) ad finally, the formula of for the function defining the table. Note that it must be a function of t. Note that if the
function contains parameters and these are changed, it will be automatically recomputed.
prompts you for the maximum iterates, the error tolerance, and the deviation for the numerical Jacobian for the shooting method for solving BVPs.
This allows you to compute
the adjoint and do averaging for weakly coupled oscillators. To use
this option, you must successfully compute a periodic orbit and set
the total integration time to one full orbit. If you are only
interested in the adjoint of a single component of an oscillation, the
algorithm works best if you start at a maximum of that component.
The menu that pops up is:
Anytime you integrate, etc, the data will be placed back into the storage area.