Stabilizing and characterizing unstable states in high-dimensional systems from time series

Abstract

An algorithm for stabilizing, characterizing, and tracking unstable steady states and periodic orbits in multidimensional dynamical systems is presented. The algorithm requires only one variable to be monitored and only one parameter to be perturbed for the stabilization of states with many unstable degrees of freedom and possibly an infinite number of stable degrees of freedom. The system is identified in terms of a linear recursive model with coefficients determined from successive readings of the variable subject to small random perturbations of the parameter. These coefficients determine the appropriate perturbations for control and also provide a direct route to the eigenvalues of the autonomous system. Spatially extended systems with an infinite number of degrees of freedom can be reduced to n effective dimensions that involve all the unstable manifolds and the weakly attracting stable manifolds. The remaining highly attracting manifolds are treated as one lumped eigenvector with an eigenvalue close to zero. The algorithm also allows the effective dimension of the state to be determined.