bifurcations in 2dtranscritical and pitchfork bifurcations in 2d i figure 8.1.5 is similar to 8.1.1....
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� 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.