Solve systems of algebraic equations with parameters obtained by discretizing non-linear partial differential equations. Study how solutioins depend on parameters (local and global bifurcation problems).

Here is one test case. Consider the following reaction-diffusion equation.

Here, a and b are parameters. Set b=1 without loss of generality.
If 0 < a is small, there exists only trivial solution $u=0$ (extinction). If a >> 0, there exists non-trivial solution (stable population). Finding the border of the set is the first step of the bifurcation problem. Bifircation value is about 9.85.

Time evolution of u(t,x) for a=1, a=11, a=20. Here, u(0,x)=if (x < 0.5) then 2x else 2-2x. u(t,x) converges to a solution of the equation for the stable state when t goes to infinity.
Let us consider a difference scheme to solve this differential equation for stable state numerically. Devide [0,1] into N+1 segments.

We call the system of algebraic equations parter-N . The unknowns are u_1, ... , u_(N).
u_k=0 is a trivial solution. A bifurcation question is to find a non-trivial solution of the parter-N.
As to this questioin, Parter gave the following answer.
Theorem (Parter, 1965):
Put a_0 = 4 (N+1)^2 sin^2(\pi/(2 (N+1))).
If a < a_0 b, then there exists only trivial real solution. If a > a_0 b, then there exist two non-trivial real solutions.

Here are questions.
(子曰. 工欲善其事, 必先利其器.)
  1. Find a shape basis of parter-N. How far can we compute? Find rational univariate expression.
  2. Compute a primary (prime) ideal decomposition of parter-N.
  3. Can we rediscover Parter's result by our software systems?
  4. What is the function u_k(a,b)? Describe the global behavior of the function u_k(a,b).
parter-10   (d is b)

[[(-1/11*u_1^2+1/11*u_1)*a+(-2*u_1+u_2)*d,
(-1/11*u_2^2+1/11*u_2)*a+(u_1+u_3-2*u_2)*d,
(-1/11*u_3^2+1/11*u_3)*a+(u_4-2*u_3+u_2)*d,
(-1/11*u_4^2+1/11*u_4)*a+(u_5-2*u_4+u_3)*d,
(-1/11*u_5^2+1/11*u_5)*a+(u_6-2*u_5+u_4)*d,
(-1/11*u_6^2+1/11*u_6)*a+(u_7-2*u_6+u_5)*d,
(-1/11*u_7^2+1/11*u_7)*a+(u_8-2*u_7+u_6)*d,
(-1/11*u_8^2+1/11*u_8)*a+(u_9-2*u_8+u_7)*d,
(-1/11*u_9^2+1/11*u_9)*a+(-2*u_9+u_10+u_8)*d,
(-1/11*u_10^2+1/11*u_10)*a+(u_9-2*u_10)*d],
 [u_1,u_2,u_3,u_4,u_5,u_6,u_7,u_8,u_9,u_10]]

parter-20

[[(-1/21*u_1^2+1/21*u_1)*a+(-2*u_1+u_2)*d,
(-1/21*u_2^2+1/21*u_2)*a+(u_1+u_3-2*u_2)*d,
(-1/21*u_3^2+1/21*u_3)*a+(u_4-2*u_3+u_2)*d,
(-1/21*u_4^2+1/21*u_4)*a+(u_5-2*u_4+u_3)*d,
(-1/21*u_5^2+1/21*u_5)*a+(u_6-2*u_5+u_4)*d,
(-1/21*u_6^2+1/21*u_6)*a+(u_7-2*u_6+u_5)*d,
(-1/21*u_7^2+1/21*u_7)*a+(u_8-2*u_7+u_6)*d,
(-1/21*u_8^2+1/21*u_8)*a+(u_9-2*u_8+u_7)*d,
(-1/21*u_9^2+1/21*u_9)*a+(-2*u_9+u_10+u_8)*d,
(-1/21*u_10^2+1/21*u_10)*a+(u_9-2*u_10+u_11)*d,
(-1/21*u_11^2+1/21*u_11)*a+(u_10-2*u_11+u_12)*d,
(-1/21*u_12^2+1/21*u_12)*a+(u_11+u_13-2*u_12)*d,
(-1/21*u_13^2+1/21*u_13)*a+(u_14-2*u_13+u_12)*d,
(-1/21*u_14^2+1/21*u_14)*a+(u_15-2*u_14+u_13)*d,
(-1/21*u_15^2+1/21*u_15)*a+(u_16-2*u_15+u_14)*d,
(-1/21*u_16^2+1/21*u_16)*a+(u_17-2*u_16+u_15)*d,
(-1/21*u_17^2+1/21*u_17)*a+(u_18-2*u_17+u_16)*d,
(-1/21*u_18^2+1/21*u_18)*a+(u_19-2*u_18+u_17)*d,
(-1/21*u_19^2+1/21*u_19)*a+(-2*u_19+u_20+u_18)*d,
(-1/21*u_20^2+1/21*u_20)*a+(u_19-2*u_20)*d],
[u_1,u_2,u_3,u_4,u_5,u_6,u_7,u_8,u_9,u_10,
 u_11,u_12,u_13,u_14,u_15,u_16,u_17,u_18,u_19,u_20]]
Remark: It is known that the function u_k(a,b) can be expressed in terms hypergeometric functions of several variables. As to details, see here