  The dynamical instability of weakly interacting two-component Bose--Einstein
condensates with coaxial quantized vortices is analytically investigated in a
two-dimensional isotopic harmonic potential. We examine whether complex
eigenvalues appear on the Bogoliubov--de Gennes equation, implying dynamical
instability. Rather than solving the Bogoliubov--de Gennes equation
numerically, we rely on a perturbative expansion with respect to the coupling
constant which enables a simple, analytic approach. For each pair of winding
numbers and for each magnetic quantum number, the ranges of inter-component
coupling constant where the system is dynamically unstable are exhaustively
obtained. Co-rotating and counter-rotating systems show distinctive behaviors.
The latter is much more complicated than the former with respect to dynamical
instability, particularly because radial excitations contribute to complex
eigenvalues in counter-rotating systems.
