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We develop, analyze, and numerically simulate a model of a prototype, glucose- driven, rhythmic drug delivery device, aimed at hormone therapies, and based on chemomechanical interaction in a polyelectrolyte gel membrane. The pH-driven interactions trigger volume phase tran- sitions between the swollen and collapsed states of the gel. For a robust set of material parameters, we find a class of solutions of the governing system that oscillate between such states, and cause the membrane to rhythmically swell, allowing for transport of the drug, fuel, and reaction products across it, and collapse, hampering all transport across it. The frequency of the oscillations can be adjusted so that it matches the natural frequency of the hormone to be released. This work is linked to extensive laboratory experimental studies of the device built by Siegel's team. The thinness of the membrane and its clamped boundary, together with the homogeneously held conditions in the experimental apparatus, justify neglecting spatial dependence on the fields of the problem. Upon identifying the forces and energy relevant to the system, and taking into account its dissipative properties, we apply Rayleigh's variational principle to obtain the governing equations. The material assumptions guar- antee the monotonicity of the system and lead to the existence of a three-dimensional limit cycle. By scaling and asymptotic analysis, this limit cycle is found to be related to a two-dimensional one that encodes the volume phase transitions of the model. The identification of the relevant parameter set of the model is aided by a Hopf bifurcation study of steady state solutions.