An attractivity region for characteristic multipliers of special symmetric solutions of \(\dot x(t)=\alpha f(x(t-1))\) near critical amplitudes
Consider the differential delay equation \[ x'(t) = \alpha f(x(t-1), \quad t \in \mathbb{R}, \] where the \(C^2\)-function \(f\) is odd about \(x = 0\), even about some value \(x = a/2\), convex on the interval \([0,a]\), and satisfies \(f'(0) = -1\). It was proved in [\textit{J. L. Kaplan} and \textit{J. A. Yorke}, J. Math. Anal. Appl. 48, 317-324 (1974; Zbl 0293.34102)] that this problem has a branch of four-periodic solutions with extra symmetry properties, parametrized by their amplitude \(z\), for any \(z < a\) and some \(\alpha =\alpha(z)\), which can be obtained from a related system of ordinary differential equations. As \(z\) approaches the critical amplitude \(a\), these solutions look like step functions with values alternating between \(-a, 0, a\), and \(\alpha(z) \sim c_0 - 2 \log(a-z)\). The author examines the spectrum of the linearization of the Poincaré map of the induced flow about these special periodic solutions, as \(z \to a\). In a sense that is made precise in the paper, it is shown that in this limit the spectrum of the linearization of the half-period Poincaré map is contained in the cardioid region \(\{ z = r e^{i \varphi} | r < \sin^2 (\varphi/2) \}\) plus possibly the isolated point \(z=1\). The proof uses Kaplan and Yorke's related system of ordinary differential equations and a detailed analysis of the corresponding characteristic function derived from its linearization. The paper contains an interesting conjecture about secondary bifurcation off this branch of periodic solutions occurring at infinitely many amplitudes \(z_k\) that accumulate at \(z = a\).
scientific article; zbMATH DE number 150624