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
- '': A comment that is extracted and can be displayed in a
separate window.
- '' {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.
- !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.
- options filename: Insert a file name with common
options. This is include only for backward compatibility
and is generally obsolete.
- name(t+1)=formula or
- dname/dt=formula or
- 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
- name(t)=formula or
- 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}
- markov name nstates
{0} |
{P_{0,1}} |
... |
{P_{0,N-1}} |
{P_{1,0}} |
{0} |
... |
{P_{1,N-1}} |
... |
... |
... |
... |
{P_{N-1,0}} |
{P_{N-1,1}} |
... |
{P_{N-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)
- 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
- 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.
- 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
- 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
- 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
- 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}]
- 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
- 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.
- 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)}
- init name1=val1,name2=val2,... sets the initial
data of name to the number val.
- 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)
- 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.
- 0= expression defines an algebraic condition for
differential-algebric equations.
- 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.
- 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.
- 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}
- @ 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.
- anything[j1..j2] expr[j] is expanded by XPP into j_{2}-j_{1}+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
- %[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
- export
{x1,x2,...,xn} {y1,...,yn}
sends and receives variables and parameters to and from dynamically loaded libraries.
Click here for details.
- done tells XPP the file is complete.
These are mainly self-explanatory. The nonobvious ones are: