Limiting profiles for periodic solutions of scalar delay differential equations (Q2764654)

Let \(f(\cdot,\lambda): \mathbb{R}\to\mathbb{R}\) be given so that \(f(0,\lambda)= 0\) and \(f(x,\lambda)= (1+ \lambda)x+ ax^2+ bx^3+o(x^3)\) as \(x\to 0\). The author characterizes those small values of \(\varepsilon> 0\) and \(\lambda\in \mathbb{R}\) for which there are periodic solutions of periods approximately \(1/k\), with \(k\in\mathbb{N}\), to the delay equations NEWLINE\[NEWLINE\varepsilon\dot x(t)= -x(t)+ f(x(t- 1), \lambda).NEWLINE\]NEWLINE When \(a=0\), these periodic solutions approach square waves if \(b< 0\) or pulses if \(b> 0\) as \(\varepsilon\to 0\). These results are similar to those obtained by Chow et al. and Hale and Huang, where the case of \(f(x,\lambda)= -(1+\lambda) x+ ax^2+ bx^3+ o(x^3)\) as \(x\to 0\) is considered. However, when \(a\neq 0\), all these periodic solutions approach pulses as \(\varepsilon\to 0\); an interesting phenomenon that cannot happen in the case considered by Chow et al. and Hale and Huang.