Putnam 1960 B3

The motion of the particles of a fluid in the plane is specified by the following components of velocity $$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\; \frac{dy}{dt}=-x.$$ Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$ , and justify your conclusion.