Mode Selection Rule for Three-Delay Systems

Abstract

We study the mode selection rule for a three-delay system to determine which oscillation mode is first excited by the Hopf bifurcation with increasing control parameter. We use linear stability analysis to detect an oscillating mode excited by the first bifurcation. There are two conditions, relevant and irrelevant conditions, determined by the ratios of three delay times, t1, t2, and tf, where tf is fixed and t1 and t2 are set as 0 < t1 < tf and 0 < t2 < tf. In a neighborhood of the relevant condition defined such that both t1/tf = n1/m1 and t2/tf = n2/m2 are ratios of odd to odd, oscillations nearly equal to the \(\tilde{m}\)th-harmonic mode are excited, where \(\tilde{m}\) is the least common multiple of m1 and m2. In the parameter space \((t_{1},t_{2})\), there are irrelevant lines each of which is determined by a rational dependence of t1, t2, and tf, and does not allow any relevant condition. Extremely high order modes are observed along both sides of the irrelevant line. In particular, the line t2 ...