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Differential Equations and Bifurcation

For the one-parameter family dy/dt = e^-y^2 + a, find the bifurcation values of a and describe the bifurcation that takes place at each such value.

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The equilibrium lines are such at which dy/dt = 0

The values of y at which this equilibrium is possible are

y_{±} = ± [- ln (-α) ]^{1/2} (1)

As y must be real, the equilibrium is only possible when ln (-&#945;) is real and non-positive, that is only when 0 < -&#945; &#8804; 1, that is

-1 &#8804; &#945; < 0 (2)

We therefore conclude that &#945; = 0 and &#945; = -1 are the ...

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