For: y1'= ay1 + ay2 and y2'= ay1 - y2: find values of a such that the critical.....

mooker

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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.
 
Check definitions once more

These condition are possible? Sum=0 ;product greater than zero.
 
These condition are possible? Sum=0 ;product greater than zero.

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These condition are possible? Sum=0 ;product greater than zero.
You were right in your sceptism! It turns out that my book had a misprint. It should have read 'q=0'.

Therefore I solved as follows:

Used the quadratic formula to solve λ1,2 = 0.5((a-1) +- (5a^2 +2a +1)^0.5)

From here it can be seen that if a is = 0 or -1, the conditions of the sum of the two eigen values being negative and having their product =0 is met.
1,2
 
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