  This paper concerns the effects of additive non-Gaussian L\'evy noises on the
pitchfork bifurcation. We consider two types of noises, $\alpha$-stable process
and the truncated process. Under both $\alpha$-stable process and the truncated
process, the classical pitchfork bifurcation model exists a unique invariant
measure. The Lyapunov exponent associated with the invariant measure is always
negative for the system under the truncated case. While the stochastic
pitchfork bifurcation still occurs. In both cases, the attractivity uniformity,
the finite-time Lyapunov exponent, and the dichotomy spectrum behave varies
with the bifurcation parameter changing.
  Compared with Brownian motion, there is two key difficulties for the L\'evy
processes. The stationary density can not be solved explicitly, thus we have to
estimate it properly. This is overcome by the strong maximum principle. The
bilateral suprema for L\'evy processes need to be analyzed. This is acquired by
the strong Markov property. Based on them, we establish the main results
indicating stochastic pitchfork bifurcation.
