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We study global continuation properties of the set of T-periodic solutions of parameterized second order delay differential equations with constant time lag on smooth manifolds. We apply our results to get multiplicity of T-periodic solutions. Our topological approach is mainly based on the notion of degree of a tangent vector field.

When the plane is pie-sliced in n ≤ 4 parts (with nonempty interior and common vertex at the origin) our main result provides sufficient conditions for any map L, that is continuous and piecewise linear relatively to this slicing, to be invertible. This result cannot be plainly extended to a greater number of slices. Also, some examples show that the assumptions cannot be relaxed too much. Our result is proved by a combination of linear algebra and topological arguments.

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré T-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.

In this paper we associate a semigroup to a locally maximal subset of complete controllability, i.e., a local control set. This fundamental semigroup is based on equivalence classes under homotopies in the set of trajectories. It reflects the structure of the set of closed (trajectory) loops in the local control set. We discuss the relations between different local control sets and prove a Van Kampen-type theorem for their unions and intersections.

It is well known that some of the properties enjoyed by the fixed point index can be chosen as axioms, the choice depending on the class of maps and spaces considered. In the context of finite-dimensional real differentiable manifolds, we will provide a simple proof that the fixed point index is uniquely determined by the properties of normalization, additivity, and homotopy invariance.

The local controllability behavior near an equilibrium is discussed. If the Jacobian of the linearized system is hyperbolic, uniqueness of local control sets is established.

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.¶In particular, the genericity of the following property is investigated: if the differentiable manifold Mis compact, then the equation π=h(x,)+f(t,x,) on Mhas |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f.

This paper illustrates how the notion of an ejecting setrecently introduced by the authors can be used to get multiplicity results for forced oscillations. The motion problem of a mass point constrained to a one-dimensional differentiable manifold and acted on by a periodic force is treated in detail.

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