I would appreciate any hint to find (if exists, I do not know) an analytical solution of the following nonlinear system:
x_j = (a_j + SUM_i b_i x_i) / (c_j + SUM_i d_i x_i) for j = 1..m, i (index of sums) = 1..m, ij and a,b,c and d constant coefficients.
For m=2 solution is simple as it becomes a second degree equation but things go tougher as m increases.
The possible maximum number of solutions is 2^m if the solution set is finite, using bezout theorem. Now, getting exact values of the solutions is harder. U can convert the system into a polynomial system by moving the denominator to the other side then use methods in solving polynomial systems - an example is using the concept of resultants.
One possible case for infinite solution is when a_1=a_2=c_1=c_2 and b_1=b_2=d_1=d_2, since at least two eqations will have nonconstant common factors.
If you are using this to solve equilibrium points of a biological system then the solutions you will get should be further restricted to nonnegative real-valued numbers
yuo can read this: http://link.springer.com/chapter/10.1007%2F3-540-27357-3_3
Thank you, Jomar. Yes, I need it for equilibrium points of a biological system.
You are welcome.
we have actually analyzed this case:
\frac{d[X_i]}{dt}=\frac{\beta_i[X_i]^{c_i}}{\overline{{K}_i}^{c_i}+[X_i]^{c_i}+{\displaystyle\sum^{n}_{j=1,j\neq{i}}{\gamma_{ij}[X_j]^{c_{ij}}}}}+\alpha_is_i-\rho_i[X_i]
$i=1,2,...,n$
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics