Global branches of periodic solutions for forced delay differential equations on compact manifolds

Abstract

We prove a global bifurcation result for T -periodic solutions of the T -periodic delay differential equation x (t) = λf (t, x(t), x(t − 1)) depending on a real parameter λ 0. The approach is based on the fixed point index theory for maps on ANRs.

On forced fast oscillations for delay differential equations on compact manifolds

“…(1.1) tackling the case 0 < T < 1, and we prove the same global bifurcation result as in [1] in the more restrictive assumption that M is boundaryless. The reason of this restriction is due to the fact that, because of the lack of compactness of the Poincaré operator, when 0 < T < 1 we need the fixed point index theory of Eells and Fournier [3] and Nussbaum [7] instead of the classical one.…”

“…If we prove that Σ is locally compact, then the result follows with exactly the same proof as in [1,Lemma 4.5] applying the Eells-Fournier-Nussbaum fixed point index instead of the classical index.…”

“…The proof is the same as in Theorem 4.6 of [1] and will be given for the reader's convenience. The following topological lemma is needed.…”

“…Under the assumptions that M is compact with nonzero Euler-Poincaré characteristic, that T 1, and that f satisfies a natural inward condition along ∂ M (when nonempty), in [1] we proved the existence of an unbounded (with respect to λ) connected branch of nontrivial T -periodic pairs whose closure intersects the set of the trivial T -periodic pairs in the so-called set of bifurcation points. That result extends an analogous one of the last two authors for the undelayed case (see [4,5]).…”

“…That result extends an analogous one of the last two authors for the undelayed case (see [4,5]). The approach followed in [1] consists in applying to a Poincaré-type T -translation operator the fixed point index theory for locally compact maps on ANRs. To this purpose, the assumption T 1 is crucial, since otherwise the compactness of the Poincaré operator fails.…”