SOLVED EE6415 Tutorial 1

  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. 3