Multiple periodic orbits of high-dimensional differential delay systems

In this paper, we consider differential delay systems of the form x′(t)=−∑s=12k−1(−1)s+1∇F(x(t−s)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'(t)=-\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr), $$\end{document} in which the coefficients of the nonlinear terms with different hysteresis have different signs. Such systems have not been studied before. The multiplicity of the periodic orbits is related to the eigenvalues of the limit matrix. The results provide a theoretical basis for the study of differential delay systems.