Control of the patterns by using time-delayed feedback near the codimension-three Turing-Hopf-Wave bifurcations

Control of the spatiotemporal patterns near the codimension-three Turing-Hopf-Wave bifurcations is studied by using time-delayed feedback in a three-variable Brusselator model. Linear stability analysis of the system shows that the competition among the Turing-, Hopf- and Wave-modes, the wavenumber, and the oscillation frequency of patterns can be controlled by changing the feedback parameters. The role of the feedback intensity P-u played on controlling the pattern competition is equivalent to that of P-w, but opposite to that of P-v. The role of the feedback intensity P-u played on controlling the wavenumber and oscillation frequency of patterns is equivalent to that of P-v, but opposite to that of P-w. When the intensities of feedback are applied equally, changing the delayed time could not alter the competition among these modes, however, it can control the oscillation frequency of patterns. The analytical results are verified by two-dimensional (2D) numerical simulations.