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# Numerics

This menu controls the numerical aspects of the program.

### Press nUmerics from main menu

### (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.

- (T)otal
  This is the amount of time to integrate. If it is negative then it will be made positive and no data will be stored. Thus you can integrate for very long periods of time without being told that the storage is full.

- (S)tart time
  This is the initial time T0. For autonomous systems it is usually irrelevant.

- t(R)ansient
  The program will integrate silently for this amount of time before plotting output. It is used to get rid of transients.

- (D)t
  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.

- n(C)line ctrl
  will prompt you for the grid size for computing nullclines.

- s(I)ng pt ctrl
  This prompts you for errors and epsilons for eigenvalues and equilibria calculations as well as the maximum iterates.

- n(O)utput
  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.

- (B)ounds
  This sets a global bound on the integrator. If any variable exceeds this value in magnitude, a message appears and the integration stops.

- (M)ethod
  allows you to choose the methods of integration. There are many integrators available for use:
  - (D)iscrete Use this for discrete dynamical systems or maps.
  - (E)uler is the usual Euler one-step method.
  - (M)od. Euler is the two-step Euler or Heun method.
  - (R)unge-Kutta is the fourth order Runge-Kutta method described in every text book
  - (A)dams is an Adams-Bashforth fourth order predictor-corrector method. It requires fewer right-hand side eva,uations than RK and achieves the same accurace. However, it is notoriously unstable.
  - (G)ear is a stiff integrator based on the algorithm in Gear's classic text. The step size is adaptive. You need to specify a tolerance (the smaller, the more accurate), a minimum step size, and a maximum step size.
  - (V)olterra is a backward integrator used for solve Volterra integral equations. It is first order accurate. XPP prompts for a tolerance, maximum iterates, and a ``memory size.'' The memory size sets the storage used by the solver. If the kernel decays rapidly, you can make this small. Tolerance and maximum iterates are used to advance to the next step. You will also be asked if you want the convolution kernels to be re-evaluated after any parameter is changed in either the range integration or through a manual change in parameters. If this flag is 1 then the kernels will be re-computed. The default is to not recompute them.
  - (B)ackEul is a simple backward Euler integrator. Smaller tolerances imply tighter requirements for the Newton's method and maximum iterates specifies how many Newton attempts will be made.
  - (Q)ualst.RK4 is an adaptive stepsize Runge-Kutta integrator recommended for nonstiff systems. Choose small tolerances for increased accuracy and a minimum/maximum allowable step size.
  - (S)tiff is another adaptive stiff integrator similar to Gear. You are prompted for error tolerance and maximum/minimum allowable steps.
  - (C)Vode a C implementation of LSODE based on Gear. It is the recommended stiff integrator. Relative and absolute tolerances are requested. You will be asked if you want to use a banded solver. For systems like discretized PDEs, using banded solvers can increase the speed by factors of 10 to 100. The upper and lower bands will be requested.
  - DoPri(5) Dormand-Prince adaptive step size integrator for nonstiff systems. Absolute and relative tolerances are requested.
  - DoPri(8)3 is a higher order DP integrator. Absolute and relative tolerances are requested.
  - Rosen(2)3 is a Rosenbrock adaptive integrator for stiff systems. Tolerances and bandedness are requested,
  - s(Y)mplect is a symplectic integrator for conservative systems of the form:

    y1'' = f1(y1,...,yn)

    y2'' = f2(y1,...,yn)

    ...

    yn'' = fn(y1,...,yn)

  - d(E)lay
    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.

  - (C)olor code
    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.

  - (P)oincare map
    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.

  - stoc(H)astic
    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:

    - New seed
      Use this to reseed the random number generator. If you use the same seed then the results will not change from run to run.

    - Compute
      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.

    - Data
      This puts the results of the most recent run into the data browser and enables plotting of them.

    - Mean
      This puts the results of the mean value of the most recently computed set of trials.

    - Variance
      This does the same for the variance.

    - Histogram
      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.

    - Old Hist
      brings back the most recently computed histogram.

    - Fourier
      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.

    - Power
      This is just like Fourier except the magnitude and phase of the FFT are kept.
    - f(I)t curve
      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:
      - File This is the name of the data file. The first column must contain the times at which the data was taken.
      - NCols This contains the total number of columns in the data file. This includes all columns in the file, even those that you will not use.
      - Fitvar This is a list separated by commas or spaces of variables that you want to fit to the data. These must not be Markov variables or Auxiliary variables. They are restricted to the items that you define as ``Variables'' in the ODE file. Due to laziness, you can only have as many variables as you can type in 25 or fewer characters.
      - To Col This should contain a list of column numbers in the data file associated with each of the variables you want to fit. Thus, for example, if you want to fit ``x'' to column 5 and ``y'' to column 2, you would type ``x y'' in the ``Fitvar'' entry and ``5 2'' in the ``To Col'' entry. The number of columns in this must equal the number of variables to be fit.
      - Params These items (there are 2 of them in case you have lots of parameters you need to vary) contain the names of parameters and variables. If the name is a variable, then the initial data for that variable will be adjusted. If it is a parameter, then the parameter will be adjusted. On the initial call, the current initial data and parameter values are used. The lists of parameters and initial data can be separated by spaces or commas.
      - Tolerance This is just a small number that tells the algorithm when the least square error is not changing enough to be significant. That is if either the difference is less than ``TOL'' or the ratio of the difference with the least square is less than ``TOL'' then the program will halt sucessfully. One should not make this too small as such differences are insignificant and a waste of CPU time. The default of .001 seems to work well.
      - Npts This is the number of points in the data file that you want to fit to.
      - Epsilon This is used for numerical differentiation. 1e-5 is a good value since we really don't need precise derivatives.
      - Max iter This is the maximum number of iterates you should use before giving up.
      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.
    - Stat brings up two choices: Mean/Dev which computes the mean and standard deviation of a given column of numbers; and autoCor which computes the autocorrelation within a column of data. You should give a number for the bins and for the max and min. It is a symmetric function.
    - Liapunov computes the maximal Liapunov exponent for the current simulation. If you choose to compute it over a range of parameters, the parameter and the maximal exponent are stored in the first two columns of the browser.

  - R(U)elle plot
    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.

  - loo(K)up
    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.

  - bndry(V)al
    prompts you for the maximum iterates, the error tolerance, and the deviation for the numerical Jacobian for the shooting method for solving BVPs.

  - (A)djoint
    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:
    - (N)ew adj This makes the adjoint from the computed periodic data. Success or failure will be noted. Separate storage is maintained for the adjoint. The program automatically puts the data from the adjoint into the browser so it can be viewed and plotted or saved. If Out of bounds increase the bounds.
    - (A)djoint This will place the adjoint data in the browser,
    - (O)rbit This places the periodic orbit in the browser.
    - (M)ake H This will prompt you for the coupling function and the result will be averaged and placed in 2 columns of storage, the first is the time, then the H function. If there are enough columns, the odd and even parts of the averaged function will be retained. As with the adjoint, this list is automatically placed in the browser. The user will be prompted for the coupling functions and should type in the formulas. The coupling is between two identical units. Say you want to couple two oscillators via a variable called V Then, you V refers to the oscillator getting the input and V' refers to the oscillator providing it. So, e.g, for diffusive coupling you would type in `V'-V`
    - (H) function This places the computed H function in the browser.
    Anytime you integrate, etc, the data will be placed back into the storage area.