Solution

Set

ε = μ - 1, v = y - 1, \overset{ε}{˙} = 0.

Then

\overset{x}{˙} \overset{v}{˙} = x (ε - 2 v - v^{2} - 2 x^{2}), = - 2 v - x^{2} - v x^{2} - 3 v^{2} - v^{3} .

The extended centre variables are

(x, ε)

and the stable variable is

v

. Write the centre manifold as

v = h (x, ε)

. At quadratic order, the invariance equation

h_{x} \overset{x}{˙} + h_{ε} \overset{ε}{˙} = - 2 h - x^{2} - v x^{2} - 3 v^{2} - v^{3}

gives

h (x, ε) = - \frac{x ^{2}}{2} + higher-order terms .

Substitution into the

x

equation gives the leading normal form

\overset{x}{˙} = ε x - x^{3} + higher-order terms .

This is a supercritical pitchfork: the

x = 0

branch loses stability as

ε

becomes positive and stable branches

x = \pm ε

emerge. The original system is invariant under

x \mapsto - x

, so a pitchfork is exactly the symmetry-forced bifurcation expected. The physical quadrant

x \geq 0

displays its positive half.

Solved by gpt-5.6-sol high.

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