13.1 When can a bifurcation occur?
Theorem 13.1.1. Let \(f(x, \mu )\) be \(C^1\) on \(\mathbb R^n \times \mathbb R\) with \(f(x_0, \mu _0) = 0\). If the Jacobian \(D_xf(x_0, \mu _0)\) is invertible, then for \(\mu \) near \(\mu _0\) there is exactly one equilibrium \(x^*(\mu )\) near \(x_0\), it depends smoothly on \(\mu \), and if \(x_0\) is hyperbolic then \(x^*(\mu )\) is hyperbolic with the same stability type. So a local bifurcation of equilibria requires \(D_xf(x_0, \mu _0)\) to be singular, or, for a change in stability, an eigenvalue on the imaginary axis.
Proof. By the implicit function theorem, invertibility of \(D_xf\) gives a unique \(C^1\) solution \(x^*(\mu )\) of \(f(x, \mu ) = 0\) near \((x_0, \mu _0)\). The eigenvalues of \(D_xf(x^*(\mu ), \mu )\) depend continuously on \(\mu \), so if none lies on the imaginary axis at \(\mu _0\), none crosses it for \(\mu \) near \(\mu _0\), and the numbers of eigenvalues in each half-plane are unchanged; by Theorem 10.3.1 the local phase portrait is unchanged up to homeomorphism. □
There are therefore two generic ways for a hyperbolic equilibrium to bifurcate as one parameter varies: a real eigenvalue passes through \(0\), or a complex pair crosses the imaginary axis. The first gives the saddle-node, transcritical and pitchfork bifurcations; the second gives the Hopf bifurcation.
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