Multiplicity results on period solutions to higher dimensional differential equations with multiple delays
The authors consider the multi-dimensional differential delay equation \[ x^{\,\prime}(t)=-f(x(t-1))-f(x(t-2))-\dots-f(x(t-2n+1)),\quad x\in\mathbb R^N,\tag{\(*\)} \] under the following basic assumptions {\parindent=8mm \begin{itemize}\item[(i)] \(f\in C(\mathbb R^N,\mathbb R^N)\) and \(f\) is odd, \(f(-x)=-f(x)\; \forall x\in\mathbb R^N\); \item[(ii)] there exists a continuously differentiable function \(F\) satisfying \(F(0)=0\) and \(\nabla F\equiv f\); \item[(iii)] \[ f(x)=A_0x+o(|x|)\;\text{as}\; |x|\to 0, \quad f(x)=A_{\infty}x+o(|x|)\text{ as } |x|\to\infty, \] where \(A_0\) and \(A_{\infty}\) are symmetric \(N\times N\) matrices. \end{itemize}} Sufficient conditions for the existence of periodic solutions of period \(4n\), including multiple periodic solutions, satisfying the additional ``symmetry'' condition \(x(t-2n)\equiv-x(t)\) are given. The paper is built upon and continues the prior work of the authors done for the particular case of \(n=1\) (see [J. Differ. Equations 218, No. 1, 15--35 (2005; Zbl 1095.34043)]). The pioneering work by \textit{J. L. Kaplan} and \textit{J. A. Yorke} of 1974, including a conjecture therein, is another motivation for this work (see [J. Math. Anal. Appl. 48, 317--324 (1974; Zbl 0293.34102)]).
higher dimensional symmetric differential delay equations with multiple delays
Kaplan-Yorke differential delay equation and conjecture
\(S ^{1}\) geometrical pseudo index theory
scientific article; zbMATH DE number 5996179
scientific article