**Example:** Consider the system

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Draw the nullclines and find all the equilibrium points. Use this information to determine the fate of the solutions corresponding to the following initial conditions

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**Answer:** Note that this system is non linear. The x-nullclines are given by

We recognize the two lines *x*=0 (y-axis) and the line -*x*-*y*+1=0.

The y-nullclines are given by

We recognize the line *y*=0 (x-axis) and the circle (centered at (0,0) with radius 2). In the graph below we give the nullclines:

The equilibrium points may be obtained as the intersection between the x-nullclines and y-nullclines. There are 6 points (see the above graph). The solutions corresponding to the given points are drawn in white in the graph below:

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