For: y1'= ay1 + ay2 and y2'= ay1 - y2
find the values of a, such that the critical point at y=0 is stable and attractive.
So, the question seems simple but I get stuck on the working. Any help would be great:
I know that the critical point would be stable when the sum of the eigen values of the system are equal to zero and the product of them is greater than zero.
So, I thought it would be a simple case of calculating the eigen values and then solving for a.
I set up the matrix:
y1' a a y2
= a -1
y2' y2
I then calculated the eigen values (or tried to):
a-λ a
= 0
a -1-λ
when solving this I get:
λ^2 +(-a+1)λ - a^2 -a =0
After that I tried to solve for lambda using the quadratic formula, but the result is a mess and doesn't help.
Any help on how to solve the problem would be greatly appreciated.
find the values of a, such that the critical point at y=0 is stable and attractive.
So, the question seems simple but I get stuck on the working. Any help would be great:
I know that the critical point would be stable when the sum of the eigen values of the system are equal to zero and the product of them is greater than zero.
So, I thought it would be a simple case of calculating the eigen values and then solving for a.
I set up the matrix:
y1' a a y2
= a -1
y2' y2
I then calculated the eigen values (or tried to):
a-λ a
= 0
a -1-λ
when solving this I get:
λ^2 +(-a+1)λ - a^2 -a =0
After that I tried to solve for lambda using the quadratic formula, but the result is a mess and doesn't help.
Any help on how to solve the problem would be greatly appreciated.