EE6415: Nonlinear Systems Analysis Tutorial 1 solution

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1. Consider the following system :
x˙ = y
y˙ = −x − βy

(a) Plot the Phase portrait for this system for
i. β = 2
ii. β = 0
iii. β = −2

(b) Provide inferences based on the difference in the graphs in terms of stability of equilibrium
points.

2. Plot the phase portrait and comment on the stability of each of the equilibrium points and check
whether there are any limit cycles in the systems described below.

(a)
x˙ = x(5 − x − 2y)
y˙ = y(4 − x − y) x, y ≥ 0
1
(b)
x˙ = −x
y˙ = y
2
(c)
x˙ = y
y˙ = x − x
3
(d)
x˙ = sin(y)
y˙ = sin(x)

3. In the following system, plot phase portraits for µ = −1, 0 and 1. Explain the differences in the
phase portraits in terms of number and stability of equilibrium points, and existence of limit cycles.
x˙ = µx − y + xy2
y˙ = x + µy + y
3

4. In the following system, plot phase portraits for µ = −2.5, −2 and −1.5. Explain the differences in
the phase portraits in terms of number and stability of equilibrium points, and existence of limit
cycles.
x˙ = µx + y + sin(x)
y˙ = x − y

5. A particle moves along a line joining two stationary masses, m1 and m2, and which are separated
by a fixed distance a . Let x denote the distance of the particle from m1.
(a) Find a relationship between ¨x and x, using Newton’s Law of Gravitation.
(b) For m1 = 1, m2 = 10, a = 10, plot the phase portrait for the system, and identify the nature
of the particles equilibrium.

6. In the following system, plot phase portraits for µ = −2, 0 and 2. Explain the differences in the
phase portraits in terms of number and stability of equilibrium points, and existence of limit cycles.
x˙ = µ − x
2
y˙ = −y
2

7. Consider the equations of a normalized pendulum, given by
q¨+ sin(q) + ˙q = 0
Plot the phase portrait for this system. Let us now assume that we modify the system to allow us
to control the system as follows:
q¨+ sin(q) + ˙q = u
where u is the input. Now, let us provide the input
u = sin(q) − q + 1
Plot the Phase portrait of the system after providing this feedback control.