The aim of this exercise is to practice some of the advantages and difficulties of the Lyaounov stability approach. Consider the second order system ẋ_1=x_1(x_1+x_2-3) ẋ_2=x_1(x_1-x_2-1) Find equilibrium points and analyze them. Plot a state-plane portrait. Now try to estimate the domain of attraction and the rate of convergence around an equilibrium point which is at least locally exponentially stable. Find (at least) three estimated regions, X_3 ⊂X_2 ⊂X_1 (the origin is in X_3) where there is (i) the rate convergence in X, is asymptotic but non-exponential. (ii) In X_2 for some β≥ 1 and α > 0, β||x(0)||e^(-tα)≥ ||x(t)|| ∀≥ 0. (iii) In X_3 for some β, and α' (where β>β'> 1 or & α'> α > 0) β'||x(0)||e^(-tα')≥ ||x(t)|| ∀≥ 0. Plot the (at least) three estimated regions you found. Relate to the state-plane portrait. Solve it step by step