Qualitative Analysis: Example 6

Example: Consider the system

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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

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From the first equation, we get y=0. The second equation gives x=0 or x=1. Hence the equilibrium points are (0,0) and (1,0). The x-nullcline are given by the equation y = 0 which is the x-axis and the y-nullcline are given by the equation tex2html_wrap_inline56 , which reduces to the two vertical lines x=0 (the y-axis) and x=1. In the picture below, we draw the vector field as well as the nullclines.

Clearly, the solutions spiral around the equilibrium point (1,0) and get away from the other equilibrium point (0,0)(see the picture below)

A closer look at the solutions around the equilibrium point (1,0) gives

Clearly the solutions are close to cycles. The graphs of x versus t as well as the graphs of y versus t illustrate better this remark:

and

[Differential Equations] [First Order D.E.]
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Author: Mohamed Amine Khamsi

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