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A1) is obtained when the dynamics is reduced to the most unstable mode, which has the regular zigzag structure (x n = n/?,y n = (?1) n y) Then the exact normal form for the supercritical pitchfork bifurcation is obtained when the second-order differential equation Eq. (A2) is further reduced to its center manifold, dropping the slave mode which is attenuated with time constant ? ?1, pp.62135-62148 ,