All ODE files are just text files which consist of a series of definitions and directives to the XPP parser. The order in which the definitions are given is usually unimportant with one major exception. So-called fixed or temporary variables are evaluated in the order in which they are defined. Thus, you should never use a named fix variable before it is defined as this will lead to some rather bizarre results when you attempt to solve an ODE. ODE files are all line oriented with a limit of 1024 characters per line. You can use the standard line continuation symbol \ . Here are all the possible ODE file directives:

#: comment line. This is ignored by XPP

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'': A comment that is extracted and can be displayed in a separate window.

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'' {a=1,b=2,...}: A comment with some ``action'' associated with it. When these comments are displayed in the comment window, an * appears next to them. When clicked by the user, various initial conditions, parameters, and XPP internal options can be set.

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!name=formula This defines a derived parameter, name whose values could depend on other parameters. These do not appear in the parameter list since they are presumably tied to the other parameters. Ecah time you change a parameter, they also are updated.

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options filename: Insert a file name with common options. This is include only for backward compatibility and is generally obsolete.

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name(t+1)=formula or

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dname/dt=formula or

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name'=formula: Define a differential/difference equation dependent variable, name and the formula defining its right-handside. For example

x'=-x+sin(t)
z(t+1)=z*(4-z)
dw/dt=1-cos(w)+(1+cos(w))*a

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name(t)=formula or

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volterra name = formula: Define a Volterra integral equation for the variable name. For example

y(t)=int{exp(-t)#x}
y'=-y + a int{exp(-(t-t'))*max(y-1/(1+(t-t')^2),0)}
y(t)=int[.5]{1#y}

The # symbol means a convolution and [a] multiplies the integrand by

(t-t')^-a

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markov name nstates

{0}	{P0,1}	...	{P0,N-1}
{P1,0}	{0}	...	{P1,N-1}
...	...	...	...
{PN-1,0}	{PN-1,1}	...	{PN-1,N-1}

This defines a Markov variable, name which has nstates states. Then following is the transition table as a series of formulas that are delimited by the curly brackets { } . The diagonal entries are ignored but should still contain a number or formula. For example:

markov z 2
{0} {alpha(v)}
{beta(v)} {0}

defines a two-state Markov process with transition from state 0 to 1 determined by the rate alpha(v) and the transition from 1 to 0 determoined by beta(v)

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aux name=formula defines a named quantity, name which appears in the data browser and is available for plotting. Note that auxiliary variables are not known internally to XPP so that you can't use them in formulas

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name= formula defines an internal or quantity which can be used in other formulas. This fixed variable serves the same function as temporary quantities in C code.

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par name1=value1, name2=value2, ... defines named parameters with default values. These can later be altered and varied in XPP. Never put spaces between the name, value, and equal sign. For example:

par a=1,b=2,c=2,big_g=20

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number name1=value1, name2=value2, ... defines named parameters with default values. These are fixed and cannot be changed in XPP. For example:

number faraday=96485,rgas=8.3147,tabs0=273.15

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name(x,y,...)=f(x,y,...) defines a function of up to 9 variables. This function can be used in any right-hand side or other functions. For example:

f(x)=if(x>1)then(1/ln(x/(x-1)))else(0)
g(v,w)=2v^2(1-w)-w/10

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table name filename defines a lookup table called name defined the values found in the file, filename. This behaves as a one variable function which uses linear interpolation on the values of the tabulated file. The file has the structure:

npts
xlo
xhi
y(xlo)
...
y(xhi)

The domain of the function is [x_lo,x_hi]

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table % name npts xlo xhi f(t) defines a precomputed table called name using npts and evaluating f(t) at t=x_lo,...,x_hi For example:

table h % 101 0 6.283 sin(t)+.6+.4*cos(t)+.2*sin(2*t)

Tables are automatically re-evaluated whenever you change parameters unless you turn off the AUTOEVALUATE flag from within XPP or in the ODE file with

@ autoeval=0

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wiener name1, name2, ... defines a series of normally distributed random variables with mean 0 and standard deviation sqrt(dt) where dt is the internal time step. The advantage of using wiener is that when you change the time-step, the magnitude of these variables is automatically adjusted.

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global sign condition {name1=form1;...} defines a global flag allowing XPP to implement delta functions. If sign=1 and condition changes from negative to positive or if sign=-1 and condition changes from positive to negative, or if sign=0 and condition vanishes identically, then each of the variables, name is immediately changed to the value of formula. For example:

global 1 x-1 {x=0;y=y+a*sin(y)}

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init name1=val1,name2=val2,... sets the initial data of name to the number val.

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name(0)= expression sets the initial value of name to the expression. If name is a delay-differential equation variable, then if expression is a function of time, t, then name(t) is set to the function for -M < t < 0 where M is the maximum delay. For example:

x'=-delay(x,2)
x(0)=sin(t)

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bdry expression defines a boundary condition at the start or the end of an interval. If the variable name is primed, then it refers to the end of the interval, otherwise to the start of the interval. For example:

bdry u'-v-1

forces the boundary condition u(L)-v(0)-1=0 where [0,L] is the interval for the ODE.

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0= expression defines an algebraic condition for differential-algebric equations.

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solve name=expression tells XPP to solve the algebraic conditions for the variable name with initial guess expression. For example:

x'=-y
init x=0
0=y+exp(y)-x
solv y=-.56715

Solves the DAE x'=-y where y+e^y=x Note that there is no closed form inverse of y+e^y.

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special name=FUN(...) defines a one-dimensional array name that represents a special summation involving an array of variables and a table. FUN can be one of the following:

• conv(type,n,m,w,u):
  
  name(j)= sum(w(m+i)shift(u,j+i),i=-m..m), j=0,...,n-1

  
  

  where w is a table of length 2m+1 and type={even,0,periodic} determines how u(j+i) is defined for j+i <0 and j+1 > n-1

• fconv(type,n,m,w,u,v,f):
  
  name(j)= sum(w(m+i)f[shift(u,j+i),shift(v,j)],i=-m..m), j=0,...,n-1

  
  

• sparse(n,m,w,l,u) evaluates to
  
  name(j)= sum(w[j m + i]shift(u,c[j m + i]),i=0..m-1), j=0,...,n-1

  where w is a table of values of length m n and c is a table of indices of length m n

• fsparse(n,m,w,l,u,v,f) evaluates to
  name(j)= sum(w[j m + i]f(shift(u,c[j m + i]),shift(v,j)),i=0..m-1), j=0,...,n-1

• mmult(m,n,w,u) evaluates to:
  
  name(j)=sum(w(i + m j)*u(i),i=0..m-1), j=0,...,n-1

  In other words, it returns the product of an n × m matrix with a m dimensional vector.

• fmmult(m,n,w,u,v,f) evaluates to:
  
  name(j)=sum(w(i + m j)*f(u(i),v(j)),i=0..m-1), j=0,...,n-1

These can then be used in right-hand sides of ODEs. For example

table w % 21 -10 10 exp(-abs(t))
special k=conv(even,101,21,w,u0)
u[0..100]'=-u[j]+f(k([j]))

Here k evaluates as the discrete convolution of exp(-|x|) with the array u.

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set name {x1=z1,x2=z2,...,} defines a named set of declarations including parameter values, initial data, and options, which can be invoked while running XPP with the File Get par set command. For example:

set hopf {a=.3,x=1.2,b=9}

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@ opt1=val1,opt2=val2,... sets various internal XPP options. For example:

@ maxstor=100000,total=1000,dt=.02

See OPTIONS for a complete list of options.

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anything[j1..j2] expr[j] is expanded by XPP into j₂-j₁+1 statements e.g.

x[1..4]'=-x[j]+[j]

is expanded into:

x1'=-x1+1
x2'=-x2+2
x3'=-x3+3
x4'=-x4+4

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%[j1..j2] expands all following statements until another % is encountered as above. For example:

%[1..4]
x[j]'=-y[j]-x[j-1]
y[j]'=x[j]
%

is expanded into:

x1'=-y1-x0
y1'=x1
x2'=-y2-x1
y2'=x2
x3'=-y3-x2
y3'=x3
x4'=-y4-x3
y4'=x4

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export {x1,x2,...,xn} {y1,...,yn} sends and receives variables and parameters to and from dynamically loaded libraries. Click here for details.

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done tells XPP the file is complete.

sin cos tan atan atan2 sinh cosh tanh
exp delay ln log log10 t pi if then else
asin acos heav sign ceil flr ran abs del_shft
max min normal besselj bessely erf erfc hom_bcs
arg1 ... arg9  @ $ + - / * ^ ** shift
| > < == >= <= != not \# int sum of i'

These are mainly self-explanatory. The nonobvious ones are:

• heav(arg1) the step function, zero if arg1<0 and 1 otherwise.
• sign(arg) which is the sign of the argument (zero has sign 0)
• ran(arg) produces a uniformly distributed random number between 0 and arg.
• besselj, bessely take two arguments, (n,x) and return J_n(x) and Y_n(x) respectively, the Bessel functions.
• erf(x), erfc(x) are the error function and the complementary function.
• normal(arg1,arg2) produces a normally distributed random number with mean arg1 and variance arg2.
• max(arg1,arg2) produces the maximum of the two arguments and min is the minimum of them.
• if(<exp1>)then(<exp2>)else(<exp3>) evaluates <exp1> If it is nonzero it evaluates to <exp2> otherwise it is <exp3>. E.g. if(x>1)then(ln(x))else(x-1) will lead to ln(2) if x=2 and -1 if x=0.
• delay(<var>,<exp>) returns variable <var> delayed by the result of evaluating <exp>. In order to use the delay you must inform the program of the maximal possible delay so it can allocate storage. (see the numerics menu)
• ceil(arg),flr(arg) are the integer parts of<arg> returning the smallest integer greater than and the largest integer less than <arg>.
• t is the current time in the integration of the differential equation.
• pi is 3.1415926...
• arg1, ..., arg9 are the formal arguments for functions
• int, # concern Volterra equations.
• shift(<var>,<exp>) This operator evaluates the expression <exp> converts it to an integer and then uses this to indirectly address a variable whose address is that of <var> plus the integer value of the expression. This is a way to imitate arrays in XPP. For example if you defined the sequence of 5 variables, u0,u1,u2,u3,u4 one right after another, then shift(u0,2) would return the value of u2.
• del_shft(<var>,<shft>,<delay>). This operator combines the delay and the shift operators and returns the value of the variable <var> shifted by <shft> at the delayed time given by <delay>. It is of limited utility as far as I know, but I needed it for a problem, so here it is.
• sum(<ex1>,<ex2>)of(<ex3>) is a way of summing up things. The expressions <ex1>,<ex1> are evaluated and their integer parts are used as the lower and upper limits of the sum. The index of the sum is i' so that you cannot have double sums since there is only one index. <ex3> is the expression to be summed and will generally involve i'. For example sum(1,10)of(i') will be evaluated to 55. Another example combines the sum with the shift operator. sum(0,4)of(shift(u0,i')) will sum up u0 and the next four variables that were defined after it.
• hom_bcs(<int>) is used only to tell XPP about heteroclinic and homoclinic connections at boundary points.