## Supercritical pitchfork bifurcation

Example 3: Consider

For p >0, there are two nontrivial equilibria, x = iy/p. The transition stability is illustrated in Figure 6 (left). In eqn [4], there is a symmetry: replacing x !—x yields the same differential equation - that is, the equation is invariant with respect to the transformation x !—x. The symmetry of the differential equation is also found in the stationary solutions. Apart from x !—x the two half-branches emanating from (x0, p0) = (0, 0) are identical and thus can be identified. Symmetry plays a crucial role for several bifurcation phenomena.

 - - J V

Figure 6 Pitchfork bifurcation. Left: supercritical bifurcation of x = px-x3; Right: subcritical bifurcation of x = px+x3.

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