Consider the nonlinear dynamical system that models the van der Pol oscillator: (t) + (x(t) 1)i(t)...

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Consider the nonlinear dynamical system that models the van der Pol oscillator: (t) + (x(t) 1)i(t) + x(t) = 0, > 0 (1) (a) Solve for the equilibrium points. (b) Determine their stability according to the values of . (c) Plot the van der Pol oscillator trajectories r(t) with t between 0 and 50 for = {0, 1, 2, 3). The initial conditions are x(0) = 0.5, (0) = 0. Are the trajectories stable? (d) Plot the phase portrait of r(t) and (t) with t between 0 and 50 for = {0, 1, 2, 3}. The initial conditions are x(0) = 0.5, (0) = 0. Describe the flow in the phase portrait for the four values of and compare to each other. (e) Learn the concept of limit cycle and describe it. Explain the connection with parts (c) and (d). Note: In MATLAB, solve the equation using ode23s. Consider the nonlinear dynamical system that models the van der Pol oscillator: (t) + (x(t) 1)i(t) + x(t) = 0, > 0 (1) (a) Solve for the equilibrium points. (b) Determine their stability according to the values of . (c) Plot the van der Pol oscillator trajectories r(t) with t between 0 and 50 for = {0, 1, 2, 3). The initial conditions are x(0) = 0.5, (0) = 0. Are the trajectories stable? (d) Plot the phase portrait of r(t) and (t) with t between 0 and 50 for = {0, 1, 2, 3}. The initial conditions are x(0) = 0.5, (0) = 0. Describe the flow in the phase portrait for the four values of and compare to each other. (e) Learn the concept of limit cycle and describe it. Explain the connection with parts (c) and (d). Note: In MATLAB, solve the equation using ode23s.