We present a simple model describing the effect of a kinetic Alfvén wave on the motion of a test electron along a closed magnetic field line. We show that the electron motion can be strongly chaotic for a wide range of conditions. The chaos exhibits a broad‐band power spectrum in the frequency domain, although only three principal frequencies are involved in the model. Phase space density of electrons exhibits complex but physically interesting evolution in response to the KAW. Typical of a chaotic situation, the scale of the electron distribution in phase space descends rapidly over the scale hierarchy to give rise to a growing set of irregularities, which may in turn generate strong electrostatic turbulences. This chain of events suggests a fundamental mechanism that couples physical processes of different scales.