We've learnt in class that if the determinant of the Jacobian matrix of an equation or ODE is zero, the system/equation has a singular point, and this means that it undergoes bifurcation in that point. I don't understand the reason, why this is true. Why does the zero Jacobian determinant show a bifurcation point?
2026-03-27 07:48:15.1774597695
The Jacobian and bifurcation
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The determinant of the appropriate Jacobian matrix for a parametrized system of equations/ODE's being zero is a necessary, but not sufficient condition for a bifurcation to occur in your system.
As an example, consider the equation $x + \mu = 0$ for any odd integer $n$. The Jacobian of this system is zero at $x,\ \mu = 0$, but there is no bifurcation present there.