The number of Kaplan-Yorke periodic solutions for delay differential equations

This paper is devoted to the number of Kaplan-Yorke periodic solutions (i.e. periodic solutions with period 4) for the delay differential equation x(center dot)(t) = - f (x (t - 1 )), where f (x) = ax + bx (3) + cx (5) and xf (x) > 0 (x not equal 0). By making the polar coordinate change, we develop a new technique to give some sufficient conditions for determining that the system has no, one or two Kaplan- Yorke periodic solutions, respectively. Meanwhile we obtain that the lower bound of the maximum number of Kaplan-Yorke periodic solutions of the above system is four. Moreover, we conjecture that the maximum number of Kaplan-Yorke periodic solutions of the above system is also four with numerical simulations. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.