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Consider a system:

$dx/dt = x(1-x)-\frac{kxy}{kx+1}$

$dy/dt = ry(1-\frac{y}{x})$

For values of r as 0.15, 0.11, and 0.05, which of the corresponding phase portraits displays limit cycle behavior? Is the cycle an attractor or a repeller?

I found the three portraits as:

r = 0.15

enter image description here

r = 0.11

enter image description here

r = 0.05

enter image description here

But I don't know how to tell which one exhibits "limit cycle behavior." Can anyone explain what this means and how I can know, by looking at each phase portrait, which is the correct answer (and whether it's a repeller or attractor)?

1 Answers 1

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For a limit cycle, In the limit as $t \to \infty$, the response tends to some cycle (e.g., some curve parameterizable by some periodic form).

In your first plot, the curve keeps spiralling inward. In your second, it sticks to a "race track" pattern. The second is a limit cycle oscillation.

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    Great, succinct explanation. Thanks for the help.2012-12-06