We establish the Lyapunov stability of the equilibrium (trivial) solution for the discrete
diamond–alpha difference operator, using the imaginary diamond–alpha ellipse. This
unifies and extends equilibrium analysis for first-order forward (Delta) and backward
(nabla) difference equations with a constant complex coefficient. We prove that for
coefficients with negative elliptical real part the equation is asymptotically stable,
with zero elliptical real part the equation is stable, and with positive elliptical real part
the equation is unstable, except in the critical case of alpha equals one half. We also
provide an asymptotic stability result for the origin in the case of the corresponding
inhomogeneous equation.
