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Section 4 exercises: Linear systems in 2 dimensions

Complete these as you are going through part 4. Try and do questions from this regularly. These questions are taken from Nonlinear Dynamics and Chaos by Strogatz.
1
.
For the following differential equations, calculated the eigenvalues and eigenvectors of the eigenvalue equations, plot the phase portraits and classify the systems. (Answers)
​
1
.
x
=y,
y
=-2x-3y
2
.
x
=5x+10y,
y
=-x-y
3
.
x
=3x-4y,
y
=x-y
4
.
x
=-3x+2y,
y
=x-2y
5
.
x
=5x+2y,
y
=-17x-5y
6
.
x
=-3x+4y,
y
=-2x+3y
7
.
x
=4x-3y,
y
=8x-6y
8
.
x
=y,
y
=-x-2y
​
​
2
.
What will Thando and Marsha’s future be in the following situations?
1
.
T
=M,
M
=-T+M
2
.
T
=aM,
M
=bT(foranyrealaandb)
3
.
T
=aT+bM,
M
=-bT-aM
4
.
T
=aT+bM,
M
=bT+aM.Whatifa=b?
5
.
T
=0,
M
=aT+bM

​​JonathanShock,UniversityofCapeTown2021​​jon.shock@gmail.com

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