The sodium ion-hydrogen ion, hydrogen ion-sodium ion exchange titration curves of «-zirconium phosphate crystals, ( 04)2• 20, exhibit a hysteresis loop. This results from the presence of different phases in the forward and backward titrations. When the crystals are titrated with sodium hydroxide two solid phases are present up to 50% of exchange. They are the unexchanged «-zirconium phosphate and a half-exchanged phase whose formula is ZrCNaPChXHPCh) -5H20. This latter phase is the only one present at exactly 50% of exchange but at higher levels of exchange another two-phase region is obtained. The two phases are the half-exchanged crystals and a fully exchanged phase of composition Zr(NaP04)2.3H20. The interconversion of the half and fully exchanged phases is reversible. However, when the half-exchanged phase takes up protons it forms an unexchanged phase which is more highly hydrated than the original «-zirconium phosphate. Dehydration of the exchanged phases leads to the formation of several new phases. The phase changes during exchange and dehydration are explained on the basis of the swelling and contraction of the layers in the «-zirconium phosphate crystals.

We describe methods for automating the control and tracking of states within or near a chaotic attractor. The methods are applied in a simulation using a recently developed model of thermal pulse combustion as the dynamical system. The controlled state is automatically tracked while a parameter is slowly changed well beyond the usual flame-out point where the chaotic attractor ceases to exist because of boundary crisis. A learning strategy based on simple neural networks is applied to map-based proportional feedback control algorithms both with and without a recursive term. Adaptive recursive proportional feedback is found to track farther beyond the crisis (flame-out) boundary than does the adaptive non-recursive map-based control. We also found that a continuous-time feedback proportional to the derivative of a system variable will stabilize and track an unstable fixed point near the chaotic attractor. The positive results suggest that a pulse combustor, and other nonlinear systems, may be suitably controlled to reduce undesirable cyclic variability and extend their useful operating range.

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We develop and demonstrate an automated control strategy using an adaptive learning algorithm that can control and track periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The control system is designed to operate in real time, taking time series measurements of a single variable as input and providing as output the control parameter value required to stabilize the desired unstable periodic orbit ͑UPO͒. The control scheme directs the system to the fixed point itself rather than a stable manifold and works when the the unstable Lyapunov multipliers are relatively large (Ϸ6). The learning and control algorithm uses a time delay embedding with the full state vector collected within one period of the controlled orbit. Control is achieved by small perturbations of a single control parameter once each cycle using a control algorithm with one recursive term. A simulation is used to study the application of the control algorithm to the hyperchaotic Rössler system. The simulation demonstrates both control of a highly unstable UPO and tracking the UPO as system parameters slowly drift over a wide range. The difficulties encountered in tracking with recursive control are discussed.

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