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A Hamiltonian with periodic orbits having several delays. (English) Zbl 1163.34386

Summary: The authors prove the existence of periodic solutions of the so-called Kaplan-Yorke type for delayed differential equations of the form

${x}^{\text{'}}\left(t\right)=-f\left(x\left(t-1\right)\right)-f\left(x\left(t-2\right)\right)-\cdots -f\left(x\left(t-n\right)\right),$

where $f:ℝ\to ℝ$ is an odd, orientation preserving homeomorphism, differentiable at the origin and at infinity, with ${f}^{\text{'}}\left(0\right),{f}^{\text{'}}\left(\infty \right)$ satisfying some appropriate bounds. The work in this paper generalizes famous results by J. L. Kaplan and J. A. Yorke [J. Math. Anal. Appl. 48, 317–324 (1974; Zbl 0293.34018)] and R. D. Nussbaum [Proc. Roy. Soc. Edinburgh Sect. A 81, No. 1–2, 131–151 (1978; Zbl 0402.34061)], where only the cases $n=1$, $n=2$ and the case $n=3$, respectively, were studied. In order to construct these Kaplan-Yorke periodic solutions, and following a well-known approach in the literature, the key idea is to prove the existence of a family of periodic solutions for a coupled $n+1$-dimensional Hamiltonian system. Using geometric and algebraic arguments only, the authors first show the existence of $\left[\left(n+1\right)/2\right]$ closed orbits for linear Hamiltonians (corresponding to a linear $f$), and then obtain the general result for nonlinear Hamiltonian vector fields by considering homotopic deformations of those closed orbits, yielded by a homotopy from $f$ to the identity function. Moreover, the “delay” of the periodic solutions is shown to be $m/\left(2\left(n+1\right)\right)$ times the period, for $m$ odd, $1\le m\le n$. Hence, under appropriate conditions on ${f}^{\text{'}}\left(0\right)$, ${f}^{\text{'}}\left(\infty \right)$, the existence of a solution of period $2\left(n+1\right)/m$ ($m$ odd, $1\le m\le n$) and delay 1 is derived.

For the special case of three delays, the authors compare their results to other criteria established in the literature. The stability of the periodic solutions is not addressed in the paper.

##### MSC:
 34K13 Periodic solutions of functional differential equations
##### Keywords:
delay equation; Hamiltonian