Bifurcations: Answer to Example 1

Example: Consider the autonomous equation

displaymath19

with parameter a.

1.
Draw the bifurcation diagram for this differential equation.
2.
Find the bifurcation values, and describe how the behavior of the solutions changes close to each bifurcation value.


Answer:

1.
First, we need to find the equilibrium points (critical points). They are found by setting tex2html_wrap_inline37 . We get

displaymath39.

This a quadratic equation which solves into

displaymath41,

we have

This clearly implies that the bifurcation occurs when tex2html_wrap_inline45 , or equivalently tex2html_wrap_inline51, which gives tex2html_wrap_inline53 . The bifurcation diagram is given below. The equilibrium points are pictured in white, red colored areas are areas with "up" arrows, and blue colored areas are areas with "down" arrows.


2.
The bifurcation values are a = 4 and a = -4. Let us discuss what is happening around a=4 (similar conclusions hold for the other value):

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Author: Helmut Knaust
Last Update 6-23-98

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