ON THE CONTROL OF COMPLEX DYNAMIC-SYSTEMS

A method is described for the limited control of the dynamics of systems which generally have several dynamic attractors. associated either with maps or first order ordinary differential equations (ODE) in R(n). The control is based on the existence of 'convergent' regions, C(k) (k = 1,2,...), in the phase space of such systems, where there is 'local convergence' of all nearby orbits. The character of the control involves the 'entrainment' and subsequent possible 'migration' of the experimental system from one attractor to another. Entrainment means that lim(t --> infinity)\x(t) - g(t) \ = 0, where x(t) is-an-element-of R" is the systems controlled dynamics, and the goal-dynamics, g(t) is-an-element-of G(k), has any topological form but is limited dynamically and to regions of phase space, G(k), contained in some C(k), G(k) subset-of C(k). The control process is initiated only when the system enters a 'basin of entrainment', BE(k) superset-of G(k), associated with the goal-region G(k). Aside from this 'macroscopic' initial-state information about the system, no further feedback of dynamic information concerning the response of the system is required. The experimental reliability of the control requires that the regions, BE(k), be convex regions in the phase space, which can apparently be assured if G(k) subset-of C(k). Simple illustrations of these concepts are given, using a general linear and a piecewise-linear ODE in R(n). In addition to these entrainment-goals, 'migration-goal' dynamics is introduced, which intersects two convergent regions G intersection C(i) not-equal empty-set, G intersection C(j) not-equal empty-set (i not-equal j), and permits transferring the dynamics of a system from one attractor to another, or from one convergent region to another. In the present study these concepts are illustrated with various one-dimensional maps involving one or more attractors and convergent regions. Several theorems concerning entrainment are derived for very general, continuous one-dimensional maps. Sufficient conditions are also established which ensure 'near-entrainment' for a system, when the dynamic model of the system is not exactly known. The applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies.