� Bifurcations in 2D

I Bifurcations of fixed point in 1D have analogs in 2D (and higher).

I But action is confined to 1D subspace where bifurcations occur,while flow in other D is attraction or repulsion from 1D subspace.

I Saddle-node bifurcation is basic mechanism for creation and de-struction of fixed points.

I Prototypical example for saddle-node bifurcation in 2D:

x = µ− x2 y = −y

� Saddle-Node Bifurcations in 2D

I 2D system with parameter µ:

x = f (x, y, µ) y = g(x, y, µ).

I Intersections of nullclines are fixed points.

I Saddle-node bifurcations:

• Nullclines pull away as µ varies, becomes tangent at µ = µc.

• Fixed points approach each other and collide at µ = µc.

• Nullclines pull apart, no intersections. Fixed points disappear.

� Example of Saddle-Node Bifurcation in 2D

I Analyse saddle-node bifurcation for genetic control system

x = −ax + y y = (x2/1 + x2)− by

I To compute ac, find where fixed points coalesce (ac = 1/2b).

I Unstable manifold trapped in narrow channelbetween nullclines.

I Stable manifold separate 2 basins of attraction.It acts as biochemical switch: gene turns on or off, dependingon intial values of x, y.

� Transcritical and Pitchfork Bifurcations in 2D

I Figure 8.1.5 is similar to 8.1.1. Saddle node bifurcation is 1Devent. Fixed points slide toward each other.

I Prototypical examples of transcritical and Pitchfork bifurcations:

x = µx− x2, y = −y.

x = µx− x3, y = −y.

I Consider pitchfork bifurcation: for µ < 0, stable node at origin.

I For µ = 0, origin is still stable with very slow decay along x-axis.

I For µ > 0, origin loses stability, gives birth to 2 new stable fixedpoints at (x∗, y∗) = (±√µ, 0)

� Example of Pitchfork Bifurcation in 2D

I Analyse pitchfork bifurcation for

x = µx + y + sin x y = x− y.

I Symmetric w.r.t origin (invariant if x → −x, y → −y).

I Origin is fixed point for all µ. Stable if µ < −2, saddle if µ > −2(τ = µ,4 = −(µ + 2)). Pitchfork may occur at µc = −2.

I Find 2 new fixed points at x∗ ≈√

6(µ + 2) for µ > −2.Supercritical pitchfork occurs at µc = −2

I Zero-eigenvalue bifurcations: Saddle-node, transcritical, pitchfork.