Consider the following system of equation describing the dynamics of love affairs (see Strogatz 1988) R = aR+bJ j = cR+dJ If a and c are negative, they are a measure of cautiousness (R and J try to avoid throwing themselves at the other) and if b and d are positive, they are a measure of responsiveness (R and J get excited by the other's advances). In each of the following cases, describe in words the type of lovers R and J are, and do the following: (i) plot the vector field described by the right-hand side of the above equations on the two- dimensional phase space (R, J) (ii) solve for the eigenvalues and eigenvectors of the matrix A associated with the above equation and plot the eigenvectors on the two-dimensional phase space (R, J) if applicable (iii) write down the solution to the above system of equations analytically (iv) choose 4 distinct initial conditions that you think will lead to interesting behavior and plot the solution R(t) and J(t) versus time t (v) for the same initial conditions, plot the solution (R(t), J(t)) on the phase space JR = R+J j=-R-J