dx/dt=4x(1-2x)-xy dy/dt=2y-4xy (a) Find all critical (equilibrium) points. (b) Using the Jacobian matrix, classify (if possible) each critical (equilibrium) point as a stable node, a stable spiral point, an unstable node, an unstable spiral point, or a saddle point.

Consider the system described by i(t) = u(t) – sin (r(t)) y(t) = u(t) + cos (r(t)) a) Find all equilibrium points of the system. b) For each equilibrium point, determine whether or not the equilibrium point is (i) stable in the sense of Lyapunov; (ii) asymptotically stable; (iii) globally asymptotically stable. Explain your answers. c) Determine whether or not the system is bounded-input bounded-output stable. Explain your answer.

Consider a second-order nonlinear system described by i1 = x – 02, i2 = -1 – x1 + (1+x²)u, = X1 - y = x1 + 2x1x2. (a) Find all q equilibrium points xe,1, Xe,2, ..., xe,g of the system and the corresponding outputs Y0,1, Yo,2, . .. , Y0,g when the input u has a nominal value uo = 1. (b) For each i E {1,2, ..., q}, linearize the system around the operating point (xe,i, u0, Yo,i).

8. For the system below, find all the critical points, find the linearization at each crit- ical point and classify it. Also indicate if they are stable, asymptotically stable, or unstable. a' = (0.6 – 0.01y)x, y = (0.002r – 0.4)y

Consider a linear dynamical system whose dynamics are given by: Axt, t=1,2,... Xt+1 = True or false: The dimensions of ; must be the same as those of xj, where i, jare positive integers. False True