2. Consider the first-order system x=rx-sinx, with r as a real parameter. (i) (ii) (iii) (iv) For the case of r=0, find all equilibrium points and their stability by linearization, and

then sketch the vector field. When r> 1, show that there is only one fixed/equilibrium point. What kind of fixed point it is, i.e. find its stability? As r decreases from 1 to 0, determine the equilibrium points, study their stability, and characterize the bifurcations that occur. Hint: a graphical analysis will be useful. For 0 0, that is T is the period]. Consider two different cases for a(t) given by (i) a (t) = 3b + ct, (0 1, show that there is only one fixed/equilibrium point. What kind of fixed point it is, i.e. find its stability? As r decreases from 1 to 0, determine the equilibrium points, study their stability, and characterize the bifurcations that occur. Hint: a graphical analysis will be useful. For 0 < r <<1, find an approximate formula for values of r at which bifurcations occur. See: Prob. 3.4.11, Nonlinear Dynamics and Chaos - Steven Strogatz. 3. (a) Consider the first-order time-periodic systems x = a (t) x, [where a (t)= a (t + T), T > 0, that is T is the period]. Consider two different cases for a(t) given by (i) a (t) = 3b + ct, (0