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{∈}openRⁿ=1,2,3), when they have attracting sets. Using the control x ̇ \.{}{}=E(x)+g ̇ \.{}{}-E(g) (t{≥}0), the goal dynamics g(t), to which x(t) is entrained, lim_t→∞{}x(t)-g(t){}, is confined to convergent regions of phase space g(t){∈}Cₖ={x{} {∥}{λ}(x)δᵢⱼ-{∂}Eᵢ/{∂}xⱼ{∥}=0, Re{λ}0 ?{λ}; i,j=1,...,n}. These regions can be determined analytically, using the Routh-Hurwitz theorem, without explicitly determining the roots {λ}(x) of the characteristic determinant. The control is only initiated when the system is in the basin of entrainment x(0){∈}BE{(g)}, which ensures entrainment. BE(g₀) is proved to exist for any fixed-point goal g₀{∈}Cₖ. It is conjectured that BE({g(t)}) exists for all g(t){∈}Cₖ which are ``dynamically limited'': {}g ̇ \.{}{}{}D(min[Re{λ}(x)],max g), 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 Li\'enard 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 R\"ossler 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.