13.2 Zero eigenvalue: saddle-node, transcritical and pitchfork

When one real eigenvalue vanishes, the dynamics near the bifurcation reduce (by the centre manifold theorem, which we do not prove) to a scalar equation \(x' = g(x, \mu )\) along the corresponding eigenvector. Three normal forms cover the common cases (Figure 17).

1.: Saddle-node: \(x' = \mu - x^2\). For \(\mu < 0\) there are no equilibria; at \(\mu = 0\) one; for \(\mu > 0\) two, \(\pm \sqrt \mu \), the positive one stable. It occurs whenever \(g_x = 0\), \(g_\mu \neq 0\) and \(g_{xx} \neq 0\) at the bifurcation point; it is the generic way for equilibria to be created or destroyed, as in the fishery of Example 1.6.2.
2.: Transcritical: \(x' = \mu x - x^2\). The equilibria \(0\) and \(\mu \) cross at \(\mu = 0\) and exchange stability. It is typical when an equilibrium exists for all \(\mu \), such as the zero population of Problem 1.8.7.
3.: Pitchfork: \(x' = \mu x - x^3\) (supercritical) or \(x' = \mu x + x^3\) (subcritical). It is typical in systems with the symmetry \(x \mapsto -x\). In the supercritical case, the equilibrium \(0\) loses stability at \(\mu = 0\) and two stable equilibria \(\pm \sqrt \mu \) branch off: a symmetric beam under increasing load buckles to one side or the other. In the subcritical case the branches \(\pm \sqrt {-\mu }\) are unstable and exist before the bifurcation; past it there is no nearby stable state, and the system jumps to a distant one.

μxsμxtμxp∗a∗r∗idatcdnhlescf-ronitroik:dcealx::′ x x =′′μ = =x μμ−−x x−3x2x2 — Figure 17: The three bifurcations with a zero eigenvalue, in their normal forms. Each panel shows the equilibria \(x^*\) against the parameter \(\mu \): solid lines stable, dashed unstable. A saddle-node creates a pair of equilibria from nothing; a transcritical bifurcation exchanges the stability of two equilibria that cross; a supercritical pitchfork splits one stable equilibrium into two.

[next] [prev] [prev-tail] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.