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The propagation of chemical waves through narrow channels has been investigated. Thin layers of excitable Belousov-Zhabotinsky mixtures are connected by precision-bore capillary tubes of different internal diameters.A wave initiated on one side of an otherwise impenetrable barrier enters and travels through the capillary tube, forming a hemisphere of excited solution at the exit. When the tube diameter is greater than a critical value, the excitation serves to initiate a circular wave in the second compartment; otherwise, the hemisphere collapses and no wave is initiated. Electrochemically generated periodic wave trains give rise to resonance patterns characterized by firing numbers l/n, where n = 1, 2, etc. is the number of waves entering the tube for every wave exiting. These firing numbers correspond to one branch of a Farey tree; higher periodic resonances in modeling calculations indicate that more fully developed Farey sequences may also occur. A one-dimensional mapping procedure is proposed to describe the appearance and ordering of the resonance patterns.

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We report numerical and experimental results indicating successful stabilization of unstable steady states and periodic orbits in an electrochemical system. Applying a continuous delayed-feedback technique not only periodic and chaotic oscillations are suppressed via stabilization of steady-state solutions but also the chaotic dynamics can be converted to periodic behavior. In all cases the feedback perturbation vanishes as a target state is attained.

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