Active polymers exhibit very rich dynamic behaviors due to their deformable long-chain architecture. In this work, we perform Langevin dynamics simulations to study the behavior of a single self-propelled polymer chain in a plane (two dimensions) whose activity can be tuned by external field. We consider a spatially on-off periodic field along the x direction, i.e. the plane is patterned into stripes of alternating active region and passive region. The width d of the stripe (half period length) plays a key role in determining the kinetic behavior of a flexible polymer chain. When d\gg 2R_\rmg0 ( R_\rmg0 is the radius of gyration of the passive flexible chain in the random coil state), the polymer chain can stay for a long time in either the active region or the passive region and moves mainly by slow Brownian diffusion; when 2R_\rmL < d < 2R_\rmg0 ( R_\rmL is the radius of the spiral formed by the self-propelled polymer chain), the polymer chain could stay entirely in one region but cross-regional motion happens frequently; when d < 2R_\rmL , the polymer chain does not stay entirely in one region but keeps moving cross-regionally accompanied by the stretching of the parts in active regions. With the kinetic behavior of the polymer chain changing as d varies, the long-time diffusive coefficient changes by as great as two orders of magnitude and other statistical quantities such as spatial density distribution, mean total propelling force, characteristic size and orientation all show non-monotonic variations. In addition, we find four typical processes of the cross-regional motion of a flexible chain. For a semiflexible polymer chain, the cross-regional motion is accompanied by buckling behavior and the width d affects greatly the degree of buckling and the continuity of the motion. Our work suggests a new idea for tuning and controlling the dynamic behavior of active polymers and provides a reference for the design and the potential applications of chain-like active materials.