var x1;
var x2;
var x3;

subject to

cons1:     - 2*x1 + x2 + 0.835634534*x1*(1-x1) = 0;
cons2:  x1 - 2*x2 + x3 + 0.835634534*x2*(1-x2) = 0;
cons3:  x2 - 2*x3      + 0.835634534*x3*(1-x3) = 0;

solve;
display x1, x2, x3;

# TITLE : 3-dimensional reaction-diffusion problem

# ROOT COUNTS :

# total degree : 8
# mixed volume : 7

# REFERENCES :

# Communicated to me by Arieh Iserles, at the conference held
# in Park City, Utah, July 1995.

# The general formulation is as follows:

#   alpha > 0, x_0 = x_{n+1} = 0

#   f_k = x_{k-1} - 2*x_k + x_{k+1} + alpha*x_k*(1-x_k) = 0, k = 1,2,..,n.

# It stems from a reaction diffusion problem.
# For general dimension n, there are 2^n solutions, with the number of
# real solutions increasing when the parameter alpha increases.
# Though, there is only one real solution with all its components positive.
# This parameter alpha is here the only real random constant in the system.

# Note that the mixed volume does not count the trivial zero solution.

# THE SOLUTIONS :

# 7 3
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  9.90452246822526E-03   9.20668856728156E-01
#  x2 : -6.96695396506914E-01   1.08723496633471E+00
#  x3 : -1.40329531548205E+00  -9.20668856728156E-01
# == err :  2.381E-15 = rco :  2.697E-01 = res :  3.140E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.51954966926035E+00   1.43763339100755E+00
#  x2 : -1.56688585394284E+00  -1.97704910020807E+00
#  x3 : -1.51954966926035E+00   1.43763339100755E+00
# == err :  6.957E-15 = rco :  2.882E-01 = res :  6.280E-16 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.40329531548205E+00  -9.20668856728156E-01
#  x2 : -6.96695396506914E-01   1.08723496633471E+00
#  x3 :  9.90452246822522E-03   9.20668856728156E-01
# == err :  2.410E-15 = rco :  3.290E-01 = res :  4.965E-16 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.40329531548205E+00   9.20668856728156E-01
#  x2 : -6.96695396506914E-01  -1.08723496633471E+00
#  x3 :  9.90452246822524E-03  -9.20668856728156E-01
# == err :  2.366E-15 = rco :  3.290E-01 = res :  3.140E-16 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  2.52317752493039E-01   9.48037483736659E-62
#  x2 :  3.46990121858033E-01   1.31266728517383E-61
#  x3 :  2.52317752493039E-01   9.48037483736659E-62
# == err :  1.413E-15 = rco :  6.653E-02 = res :  1.110E-16 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.51954966926035E+00  -1.43763339100755E+00
#  x2 : -1.56688585394284E+00   1.97704910020807E+00
#  x3 : -1.51954966926035E+00  -1.43763339100755E+00
# == err :  6.957E-15 = rco :  2.882E-01 = res :  6.280E-16 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  9.90452246822521E-03  -9.20668856728156E-01
#  x2 : -6.96695396506914E-01  -1.08723496633471E+00
#  x3 : -1.40329531548205E+00   9.20668856728156E-01
# == err :  2.390E-15 = rco :  2.697E-01 = res :  3.514E-16 ==