Differential Equations and Bifurcation
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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 (-α) is real and non-positive, that is only when 0 < -α ≤ 1, that is
-1 ≤ α < 0 (2)
We therefore conclude that α = 0 and α = -1 are the ...
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