1.Find the x- and y-Nullclines of the system and sketch them. How many equilibria are there? Where are they located?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1.Find the x- and y-Nullclines of the system and sketch them.
How many equilibria are there? Where are they located?

2.The Nullclines separate the plane into regions where solutions
are generally moving in the same direction. For each region in your plot,
indicate the general direction that solutions are moving (e.g. “Northeast”
or “Down-and-right”).

3. A region is called “positively invariant” if solutions in that
region stay in that region as time increases. Which regions of your plot are
positively invariant? 

4. Use your answers to the previous parts (NOT the linearization
theorem) to classify each equilibria as a source, a sink, a saddle, or none
of the above.

5. describe the long-term behavior of every
solution in the plane. You should explain what can happen as t → ∞ and
qualitatively describe which solutions have each behavior.

a'(t) = x · (y + x² – 2)
y (t) = (x² – y)
Transcribed Image Text:a'(t) = x · (y + x² – 2) y (t) = (x² – y)
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