Qualitative Analysis: Example 5

Example: Consider the system

Find the equilibrium points and the nullclines. Draw the vector field. Sketch some solutions and specially the solutions around the equilibrium points.

Answer. The equilibrium points are given by the algebraic system

It is easy to see that we must have x=0 (from the second equation) and therefore y=0. Hence the system has one equilibrium point (0,0). The x-nullcline are given by the equation

which is the graph of the function . The y-nullcline are given by the equation

which reduces to the line x=0 (the y-axis). In the picture below, we draw the vector field as well as the nullclines.

Clearly, the solutions spiral around the equilibrium point (see the picture below)

Notice that the solutions spiral and get closer to a cycle. We can see this better by looking at the graphs of x versus t as well as the graphs of y versus t.

and

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