Saddle-Node Bifurcation
We have introduced above the Saddle-Node bifurcation without naming it. This bifurcation is associated to the differential equation:
, where
and
are the control parameters. For
we are speaking about subcritical bifurcation. Lets
, the equilibrium points are easy to determine and they are immediately obtained :
, this means that the equilibrium point exist only for
. We can summarize the result in the following table:
Equilibrium point |
|
|
| doesn't exist | stable |
| doesn't exist | unstable |
The visualization can be done in the bifurcation diagram:
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from pylab import *
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def xeq1(mu):
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return sqrt(mu)
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def xeq2(mu):
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return -sqrt(mu)
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domain = linspace(0, 10)
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plot(domain, xeq1(domain), 'b-', linewidth = 3)
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plot(domain, xeq2(domain), 'r--', linewidth = 3)
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plot([0], [0], 'go')
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plot([0,0],[-5,5], "--")
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plot([-10,10],[0,0], "--")
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axis([-10, 10, -5, 5])
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xlabel('$r$', fontsize = 18)
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ylabel('$x_{eq}$', fontsize = 18)
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show()16
For
we are speaking about super-critical bifurcation.
