We present direct numerical simulations of the coupled Poisson-Nernst-Planck and Navier-Stokes equations for an electrolyte around a polarizable cylinder subject to an external electric field. For high fields, a novel chaotic flow phenomenon is discovered. Our calculations indicate significant improvement in the prediction of the mean flow relative to standard asymptotic models. These results open possibilities for chaos-enhanced mixing in microdevices and provide insight into barriers to efficient electrokinetic micropumps with broad applications in electrochemical and lab-on-a-chip systems.