SOLVED EE6415 Tutorial 2

 Identify the equilibrium point(s) for each of the following, and linearize the systems about the equilibrium point(s). 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 1 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. 2 (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 + 2x2)x 2 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 3