UNIVERSITY EXAMINATIONS
JANUARY/FEBRUARY 2022
APM2614
Applied dynamical systems
Welcome to the
APM2614 exam.
This paper consists of 4 pages.
Instructions:
Duration
:
2 hours
Examiners
First
:
Mr Nyathi F
Second
:
Prof Khumalo M & Dr Moremedi MG
Answer all questions
The use of non-programmable calculator is permissible.
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Open Rubric
2
APM2614
Jan/Feb 2022
Question 1
Consider the dynamical systems given below:
ẋ = 2x + 3y
ẏ = 2x + y
(a) Calculate the fixed point of the system
(2)
(b) Determine the nature of the fixed point
(4)
(c) Solve the system subject to the initial conditions: x(0) = 7; y(0) = −3
(16)
[22]
Question 2
Consider the discrete dynamical systems given below:
x(n + 1) = −5x(n) + y(n)
y(n + 1) = −x(n) − 3y(n)
(a) Find the fixed point of the system
(3)
(b) Calculate the eigenvalues of the system and state the nature of the fixed point
(4)
(c) Find the general solution of the system
(8)
[15]
Question 3
Consider the system below:
"
# "
#" # "
#
x˙1
0 1 x1
−1 1
=
+
u
x˙2
1 0 x2
0 0
with output
" #
h
i x
1
y = 1 −1
x2
(a) Draw a dynamical diagram for the system
(9)
(b) Is the system
(i) Completely controllable?
(2)
[TURN OVER]
3
(ii) Completely observable?
APM2614
Jan/Feb 2022
(2)
[13]
Question 4
Consider the system below:
x˙1 = 3x1 − 7x2
x˙2 = αx1 + 4x2
with output
y = αx1 + 3x2
where α is a costant.
Find the values of α for which the system is completely observable.
[7]
Question 5
Given the planner system below:
ẋ = y − x2 + 1
ẏ = x2 − xy
(a) Find the Jacobian of the system
(2)
(b) Calculate the critical points of the system
(5)
(c) Classify the critical points as stable, unstable, stable spiral, unstable spiral or a saddle point where
possible.
(6)
[13]
Question 6
Consider the second order differential equation:
1
ẏ = −x − x2 + y − 3y 3
2
classify the critical points as stable, unstable or saddle
[10]
[TURN OVER]
4
APM2614
Jan/Feb 2022
Question 7
Given the dynamical system:
ẋ = (x + λ)(x2 − λ)
where λ is a parameter.
(a) Determine the fixed point(s) of the system, specifying the range of values of λ for which the solutions
are valid.
(5)
(b) With the aid of a suitable function determine whether or not the fixed point(s) are asymptotically
stable.
(10)
(c) Sketch the bifurcation diagram.
(5)
[20]
Total: [100]
c
UNISA 2022