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 | Course Hero

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 ? ? 2 ? ( a ? 1) ? + ( b ? a ) =0
and we have eigenvalues ? =
?
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
?=
( 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.

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