  The controllable transition between the Turing and antispiral patterns is
studied by using time-delayed-feedback strategy in a FitzHugh-Nagumo model. We
treat the time delay as perturbation and analyze the effect of the time delay
on the Turing and Hopf instabilities near the Turing-Hopf codimension-two phase
space. Numerical simulations show the transition between the Turing patterns
(hexagon, stripe, and honeycomb), the dual-mode antispiral, and the antispiral
by applying appropriate feedback parameters. The dual-mode antispiral pattern
originates from the competition between the Turing and Hopf instabilities. Our
results have shown the flexibility of the time delay on controlling the pattern
formations near the Turing-Hopf codimension-two phase space.
