MATH 275 San Mateo SEC3 Phase Diagram of A 2D Differential Equation System HW In the lab you are to address question one only. The task is to draw the a-b

MATH 275 San Mateo SEC3 Phase Diagram of A 2D Differential Equation System HW In the lab you are to address question one only. The task is to draw the a-b plane in a fashion similar to the trace determinant plane discussed in Section 3.7DO NOT work on question 2 which involves bifurcations! Project #3 Lab 3.1
Lab 3.1 Assignment details Summer 2020
In the lab you are to address question one only. The task is to draw
the a-b plane in a fashion similar to the trace determinant plane
discussed in Section 3.7
DO NOT work on question 2 which involves bifurcations!
Due date:- 8/1/20 by 6:00 PM
We have the system
dx
= a x+by
dt
dy
=− x − y
dt
where a, b ∈ .
The idea is to draw the a − b plane
We have the system
 dx 
 dt   a b   x 
 =
  y
−
−
dy
1
1
  
 
 dt 
with T =a − 1 and D =− a + b ⇒ λ 2 − ( a − 1) λ + ( b − a ) =0
and we have eigenvalues λ =
λ
2
( a − 1) − 4 ( b − a )
a −1±
2
a − 1 ± a 2 − 2a + 1 − 4b + 4a a − 1 ± a 2 + 2a + 1 − 4b
=
2
2
a −1±
( a + 1)
2
2
− 4b
2
. Set ( a + 1) − 4b = 0 ⇒ b =
( a + 1)
4
2
We have the eigenvalues
λ=
( a + 1)
a −1±
2
− 4b
2
The first thing to do is to plot the parabola
b=
( a + 1)
2
4
Go from there to complete the a − b phase plane diagram.

Purchase answer to see full
attachment