Controls of dynamic flows with attractors

Abstract

Analytic and numerical results are obtained concerning the entrainment and migration of dynamic systems, which are governed by ordinary differential equations x\ifmmode \dot{}\else \.{}\fi{}=E(x) (x\ensuremath{\in}${\mathit{openR}}^{\mathit{n}}$=1,2,3), when they have attracting sets. Using the control x\ifmmode \dot{}\else \.{}\fi{}=E(x)+g\ifmmode \dot{}\else \.{}\fi{}-E(g) (t\ensuremath{\ge}0), the goal dynamics g(t), to which x(t) is entrained, ${\mathrm{lim}}_{\mathit{t}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$\ensuremath{\Vert}x(t)-g(t)\ensuremath{\Vert}, is confined to convergent regions of phase space g(t)\ensuremath{\in}${\mathit{C}}_{\mathit{k}}$={x\ensuremath{\Vert} \ensuremath{\parallel}\ensuremath{\lambda}(x)${\mathrm{\ensuremath{\delta}}}_{\mathit{i}\mathit{j}}$-\ensuremath{\partial}${\mathit{E}}_{\mathit{i}}$/\ensuremath{\partial}${\mathit{x}}_{\mathit{j}}$\ensuremath{\parallel}=0, Re\ensuremath{\lambda}0 ?\ensuremath{\lambda}; i,j=1,...,n}. These regions can be determined analytically, using the Routh-Hurwitz theorem, without explicitly determining the roots \ensuremath{\lambda}(x) of the characteristic determinant. The control is only initiated when the system is in the basin of entrainment x(0)\ensuremath{\in}BE{(g)}, which ensures entrainment. BE(${\mathit{g}}_{0}$) is proved to exist for any fixed-point goal ${\mathit{g}}_{0}$\ensuremath{\in}${\mathit{C}}_{\mathit{k}}$. It is conjectured that BE({g(t)}) exists for all g(t)\ensuremath{\in}${\mathit{C}}_{\mathit{k}}$ which are ``dynamically limited'': \ensuremath{\Vert}g\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\Vert}D(min[Re\ensuremath{\lambda}(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.