CONTROLS OF DYNAMIC FLOWS WITH ATTRACTORS

Analytic and numerical results are obtained concerning the entrainment and migration of dynamic systems, which are governed by ordinary differential equations x = E(x) (x is-a-member-of R(n) = 1,2,3), when they have attracting sets. Using the control x = E(x) + g-E(g) (t greater-than-or-equal-to 0), the goal dynamics g(t), to which x(t) is entrained, lim t --> infinity \x(t)-g(t)\, is confined to convergent regions of phase space g(t) is-a-member-of C(k) = {x\ parallel-to lambda(x)delta-ij - partial derivative E(i)/partial derivative x(j) parallel-to = 0, Re-lambda < 0 FOR-ALL-lambda; i,j = 1, ..., n}. These regions can be determined analytically, using the Routh-Hurwitz theorem, without explicitly determining the roots lambda(x) of the characteristic determinant. The control is only initiated when the system is in the basin of entrainment x(0) is-a-member-of BE {(g)}, which ensures entrainment. BE(g0) is proved to exist for any fixed-point goal g0 is-a-member-of C(k). It is conjectured that BE({g(t)}) exists for all g(t) is-a-member-of C(k) which are "dynamically limited": \g\ < D(min[Re-lambda(x)], maxg), where the function D is system specific. This dynamic limitation is illustrated for the Duffing oscillator. Basins of entrainment are explicitly determined in one-dimensional flows and for the van der Pol limit cycle (n = 2) in the Lienard phase space. This example is used to show that convergent regions are not topologically invariant. The convergent regions are obtained for both the Lorenz and Rossler systems (n = 3). The global character of the basin of entrainment for a class of goals is analytically proved for the Lorenz system. The transfer of systems between different attractors in multiple attractor systems (MAS) is demonstrated both in one-dimensional flows and in the Lorenz system, where the transfers between stable fixed points and from a strange attractor to a stable fixed point are illustrated.