We study local and global Hamiltonian dynamical behaviors of some Lagrangian submanifolds near a Lagrangian sphere S in a symplectic manifold X. When dim S = 2, we show that there is a one-parameter family of Lagrangian tori near S, which are nondisplaceable in X. When dim S = 3, we obtain a new estimate of the displacement energy of S, by estimating the displacement energy of a one-parameter family of Lagrangian tori near S.

In the 2-dimensional case, the proof relies on a computation of the bulkdeformed Floer cohomology of the one-parameter family of Lagrangian tori near S. In the 3-dimensional case, due to the absence of a local bulk cycle, we establish a version of deformed Floer cohomology by using bulk chains with boundary as S. We also make some computations and observations of the classical Floer cohomology by using the symplectic sum formula and Welschinger invariants.