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 Incoming links: Symmetry-forced pitchfork bifurcation

Extended centre-manifold reductions of the 2019 Cambridge reflection-symmetric system 2026-10-03

For

\overset{x}{˙} = x + y^{2} - a, \overset{y}{˙} = y (4 x - x^{2} - a),

the extended centre manifold for a parameter gives the local reduced equations

(a, x, y) (0, 0, 0) (3, 3, 0) (4, 2, y_{0}) local parameter and centre coordinate μ = a, y μ = a - 3, y μ = a - 4, v = y - y_{0} reduced equation \overset{y}{˙} = 3 μ y - 4 y^{3} + O (4), \overset{y}{˙} = - 3 μ y + 2 y^{3} + O (4), \overset{v}{˙} = - y_{0} (μ + 8 v^{2}) + O (v^{3}, μv),

where

y_{0} = \pm 2

. Thus the first two points undergo respectively supercritical and subcritical symmetry-forced pitchfork bifurcations, while each of the last two points undergoes a saddle-node bifurcation.

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 78 / 3 / d / Solution 2026-10-03

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