Description
1.
x˙ = y + y
2
y˙ = −x +
1
5
y − xy +
6
5
y
2
2.
r˙ = r(1 − r
2
)
˙θ = 1 − cos(θ)
3. Find the equilibrium point(s), Explain how the system behaves locally around the equilibrium
point(s).
x˙ = sin(y)
y˙ = x − x
3
4. (Leftists, rightists, centrists) Vasquez and Redner (2004, p. 8489) mention a highly simplified model
of political opinion dynamics consisting of a population of leftists, rightists, and centrists. Let x, y, z
represent the fraction of the population of leftists, rightists and centrists respectively. The leftists
and rightists never talk to each other; they are too far apart politically to even begin a dialogue.
But they do talk to the centrists, − this is how opinion change occurs.
The population dynamics
can be modelled as given below:
x˙ = rxz
y˙ = ryz
z˙ = −rxz − ryz
where r ∈ R\{0}.
Linearize around the fixed point(s) and explain how the population behaves for r > 0 and for r < 0.
5. A simple model of a satellite of unit mass moving in a plane can be described by the following
equations of motion in polar coordinates:
r¨(t) = r(t)
˙θ
2
(t) −
β
r
2(t)
+ u1(t)
¨θ(t) = −
2 ˙r(t)
˙θ(t)
r(t)
+
u2(t)
r(t)
Linearize the system around u
∗ =
u
∗
1
u
∗
2
=
0
0
and the trajectory
r
∗
r˙
∗
θ
∗
˙θ
∗
=
r0
0
ω0t + θ0
ω0
where
ω0 =
q β
r
3
0
6. Consider the nonlinear system
x˙ = y + x(x
2 + y
2 − 1) sin( 1
(x
2 + y
2 − 1))
y˙ = −x + y(x
2 + y
2 − 1) sin( 1
(x
2 + y
2 − 1))
Without solving the above equations explicitly, show that the system has infinite number of limit
cycles.
7. The system
x˙ 1 = −x1 −
x2
ln p
x
2
1 + x
2
2
x˙ 2 = −x2 +
x1
ln p
x
2
1 + x
2
2
has an equilibrium point at the origin
(a) Linearize the system about the origin, and show that the origin is a stable node.
(b) Plot the phase portrait of the system about the origin, and show that the origin is a stable
focus
(c) Explain the discrepancy between the two results.
8. For the following systems, show that there exists a limit cycle
(a) y¨ + y = ϵy˙(1 − y
2 − y˙
2
)
(b) x˙ 1 = x2 , ˙x2 = −x1 + x2(2 − 3x
2
1 − 2x
2
2
)
(c) x˙ 1 = x2 , ˙x2 = −x1 + x2 − 2(x1 + 2×2)x
2
9. The following model is used to analyze the interaction between inhibitory and excitatory neurons
in a biological system. In its simplest form, x1 is the output of the excitatory neuron, and x2 is the
output of the inhibitory neurons.
x˙ 1 = −
1
τ
x1 + tanh(λx1) − tanh(λx2)
x˙ 2 = −
1
τ
x2 + tanh(λx1) + tanh(λx2)
Show that, when λτ > 1, the system has a periodic orbit