Stability of square wave periodic solution for singularly perturbed delay differential equations (Q1840595)

Here, the author studies the following singularly perturbed delay differential equation \[ \varepsilon z'(t)=f(z(t),z(t-1)),\quad z(t)\in \mathbb{R}^m,\quad 0<\varepsilon\ll 1. \] Assuming that \(f\) has a pair of period doubling points \(p^1\) and \(p^2\), i.e. \(f(p^1,p^2)=0\) and \(f(p^2,p^1)=0\), he proves, under an additional technical assumption, that there exists a square-wave-like ( or slowly oscillating) periodic solution \(z_{\varepsilon}(t)\) of period \(2+2\varepsilon r\) alternating between \(p^1\) and \(p^2\). Scaling the time \(t\to-\varepsilon r t\) and introducing variables \(u(t)=z_{\varepsilon}(\varepsilon r t), v(t)=z_{\varepsilon}(-\varepsilon r t+1+\varepsilon r)\) he obtains the transition layer equations \(u'(t)=-rf(u(t),v(t-1)), v'(t)=-rf(v(t),u(t-1)).\) Assuming that the transition layer system has a heteroclinic solution connecting points \((p^2,p^1)\) and \((p^1,p^2)\), the author investigates a relationship between the stability of the square wave periodic solution and the spectrum of the variational operator associated to the heteroclinic solution.