Straight-line stabilization

For finite-dimensional maps, an unstable orbit in a neighborhood of an unst able fixed point can be stabilized by adjusting parameters so that the orbi t goes to the fixed point along the straight line connecting the orbit (at a given time) and the fixed point [Yang Ling, Liu Zengrong and Jian-min Mao , Phys. Rev. Lett. 83, 67 (2000)]. This is called straight-line stabilizati on. In this paper, we derive the expression for the region of stabilization , i.e., the region within which the straight-line stabilization method is v alid. For two-dimensional maps, the parameter adjustments needed by the sta bilization method are explicitly given for nine cases. Stabilization of uns table flows, with or without introducing a Poincare map, is also investigat ed.