  A new type of instability - electrokinetic instability - and an unusual
transition to chaotic motion near a charge-selective surface was studied by
numerical integration of the Nernst-Planck-Poisson-Stokes system and a weakly
nonlinear analysis near the threshold of instability. Two kinds of initial
conditions were considered: (a) white noise initial conditions to mimic "room
disturbances" and subsequent natural evolution of the solution; (b) an
artificial monochromatic ion distribution with a fixed wave number to simulate
regular wave patterns. The results were studied from the viewpoint of
hydrodynamic stability and bifurcation theory. The threshold of
electroconvective movement was found by the linear spectral stability theory,
the results of which were confirmed by numerical simulation of the entire
system. The following regimes, which replace each other as the potential drop
between the selective surfaces increases, were obtained: one-dimensional steady
solution; two-dimensional steady electroconvective vortices (stationary point
in a proper phase space); unsteady vortices aperiodically changing their
parameters (homoclinic contour); periodic motion (limit cycle); and chaotic
motion. The transition to chaotic motion did not include Hopf bifurcation.
Numerical resolution of the thin concentration polarization layer showed
spike-like charge profiles along the surface, which could be, depending on the
regime, either steady or aperiodically coalescent.
  The numerical investigation confirmed the experimentally observed absence of
regular (near-sinusoidal) oscillations for the overlimiting regimes. There is a
qualitative agreement of the experimental and the theoretical values of the
threshold of instability, the dominant size of the observed coherent
structures, and the experimental and theoretical volt-current characteristics.
