  Topological characteristics in quantum systems typically determine the ground
state, while the corresponding quantum phase transition (QPT) can be identified
through quenching dynamics. Based on the exact results of extended Kitaev
chains, we demonstrate that the system dynamics can be comprehended through the
precession of an ensemble of free-pseudo spins under a magnetic field. The
topology of the driven Hamiltonian is determined by the average winding number
of the non-equilibrium state. Furthermore, we establish that the singularity of
the dynamical quantum phase transition (DQPT) arises from two perpendicular
pseudo-spin vectors associated with the preand post-quenched Hamiltonians.
Moreover, we investigate the distinct behaviors of the dynamic pairing order
parameter in both topological and non-topological regions. These findings offer
valuable insights into the non-equilibrium behavior of topological
superconductors, contributing to the understanding of the resilience of
topological properties in driven quantum systems.
