Period doubling in singularly perturbed delay equations
The authors consider periodic solutions of the delay differential equation \((\epsilon> 0)\) \(\epsilon x'(t)= -x(t)+ f(x(t- 1),\lambda)\) assuming that \(\lambda= 0\) corresponds to a generic period doubling for the map \(x\to f(x, 0)\). Specifically, it is assumed that \[ f(x, \lambda)= -(1+ \lambda) x+ ax^ 2+ bx^ 3+ o(x^ 3),\quad x\to 0, \] where the constant \(a\), \(b\) satisfy \(\beta= a^ 2+ b\neq 0\). Under this assumption on \(f\), and if \(\lambda\beta> 0\) there are periodic points \(d_{1\lambda}\), \(d_{2\lambda}\) of period 2 of the map \(f(.,\lambda)\) and such that \(d_{1\lambda}, d_{2\lambda}\to 0\) as \(\lambda\to 0\). If \(\beta> 0\) \((<0)\) the bifurcation is said to be supercritical (subcritical). The supercritical case with \(a= 0\) has earlier been investigated by \textit{S.-N. Chow} and the authors [Proc. Soc. Edinb., Sect. A (to appear)]. In the present paper the supercritical case with \(a\neq 0\) and the subcritical case are analyzed. The result proved is the same: Theorem: Let \(f(x,\lambda)\) be as above. Then there is a neighborhood \(U\) of \((0,0)\) in the \((\lambda, \epsilon)\) plane and a sectorial region \(S\) in \(U\) such that, if \((\lambda, \epsilon)\in U\), then there is a periodic solution \(x_{\lambda, \epsilon}\) of the above equation with period \(2\tau(\lambda, \epsilon)= 2+ 2\epsilon+ O(|\epsilon| (|\lambda|+ |\epsilon|))\) as \((\lambda, \epsilon)\to (0,0)\) if and only if \((\lambda, \epsilon)\in S\). This solution is unique.
scientific article; zbMATH DE number 696878