ABSTRACT
For finite-dimensional maps, an unstable orbit in a neighborhood of an unstable fixed point can be stabilized by adjusting parameters so that the orbit 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. 84, 67 (2000)]. This is called straight-line stabilization. In this paper, we derive the expression for the region of stabilization, i.e., the region within which the straight-line stabilization method is valid. For two-dimensional maps, the parameter adjustments needed by the stabilization method are explicitly given for nine cases. Stabilization of unstable flows, with or without introducing a Poincare map, is also investigated.
ABSTRACT
For a finite-dimensional dynamical system, whose governing equations may or may not be analytically available, we show how to stabilize an unstable orbit in a neighborhood of a "fully"unstable fixed point (i.e., a fixed point at which all eigenvalues of the Jacobian matrix have modulus greater than unity). Only one of the unstable directions is to be stabilized via time-dependent adjustments of control parameters. The parameter adjustments can be optimized.
