Chapter 7.2, Problem 16P

In each of Problems $1$ through $20:$

(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.

(e) Draw a sketch of, or describe in words, the basin of attraction of each asymptotically stable critical point.

$\begin{array}{l}dx/dt=y+x\left(1-x^2-y^2\right),\\ dy/dt=-x+y+\left(1-x^2-y^2\right)\end{array}$