Periodic perturbations of reducible scalar second order functional differential equations

. Using a topological approach we investigate the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to periodic forcing. More precisely, we consider a delay-type functional dependence involving a gamma probability distribution and, using a linear chain trick, we formulate a first order system of ODEs whose T -periodic solutions correspond to those of the functional equation.


Introduction and setting of the problem
Integro-differential and functional-delayed equations are often used to describe phenomena whose dynamics depends on the past states of the system of concern.We mention, in particular, scalar second order functional differential equation of the following form: where g : R 3 → R is a continuous map, φ : R 2 → R is continuous and K is some integral kernel.Such equations, often studied in the first order case, have been considered in many contexts.A comprehensive bibliography is beyond the scope of this paper, we only mention a few relevant papers and books, e.g., [14,22,32,34,35,40,41].
In this paper we focus on the case when the integral kernel in (1.1) is the gamma probability distribution γ b a , for a > 0 and b ∈ N \ {0} given by γ b a (s) = a b s b−1 e −as (b − 1)! for s ≥ 0, γ b a (s) = 0 for s < 0, ( with mean b/a and variance b/a 2 .Namely, we study the following equation: γ b a (t − s)φ(x(s), ẋ(s)) ds . (1.3) An equation as (1.3) is often called reducible because, unlike equation (1.1) whose dynamics is, in general, infinite dimensional (see, e.g., [31]), equation (1.3) can be essentially reduced to a system of ordinary differential equations by the so-called linear chain trick (see Section 2).Indeed, backtracking the linear chain trick procedure, one can see how equations of the form (1.3) may arise from the coupling of a nonlinear equation with a linear one (see, e.g., [35,Ch. 10]).We will briefly come back on this last topic in Section 8 on perspectives and further developments.Another of the reasons that motivates the interest in this particular type of kernels is that, at least for some linear functional equations containing a convolution-like term as in equation (1.1) with general kernel, it is possible to construct gamma-like kernels such that the solutions of the corresponding equations approximate those of the equation we started with (see [6,9,10]).
The particular dependence on the past of the solution that is considered in equation (1.3) naturally arises in many contexts (not necessarily for second order equations) see, e.g., [5,10,21,22,34,35,40,41].In the equation we examine, the delay is spread along the whole history but more concentrated at a given distance in the past.We can interpret this, from a probabilistic point of view, as a delay that follows a gamma-type distribution with a given mean (and variance).This approach seems to be reasonable in contexts when the delay cannot be measured with precision, but its mean value and variance are known, or it is genuinely spread in time as, e.g., in [18,19].Notice, in particular, that letting a and b tend to infinity in such a way that the quotient r := a/b remains constant (for instance, put a = nr and b = n and let n → ∞), one concentrates the memory effect close to the delay r.Indeed, at least in the sense of distributions, (1.3) approximates the second order delay differential equation ẍ(t) = g x(t), ẋ(t), φ x(t − r), ẋ(t − r) .
Observe that the function γ b a defined in (1.2) is continuous for b = 2, 3, . . .but not for b = 1.However, also in the latter case, assuming x in C 1 , the function a (t − s)φ(x(s), ẋ(s)) ds is continuous.Then the right hand side of (1.3) is continuous as well; consequently, any C 1 solution of (1.3) is actually of class C 2 .It is also worth noticing (and we will use this fact later) that, since in the integral one has t − s ≥ 0, then ae −a(t−s) φ(x(s), ẋ(s)) ds, so that the function t → t −∞ γ 1 a (t − s)φ(x(s), ẋ(s)) ds is actually in C 1 .Our main concern will be the structure of periodic solutions of (1.3) when subject to a periodic forcing.Given T > 0, we consider the following T-periodic perturbation of (1.3): in which we assume that the map f : R 3 → R is continuous and T-periodic in the first variable, and λ is a nonnegative real parameter.Our main purpose is to investigate the set of T-periodic solutions of (1.4).Here, given λ ≥ 0, by a T-periodic solution (on R) of (1.4) we mean a C 2 function x : R → R of period T that satisfies identically equality (1.4).In that case, we will call (λ, x) a T-forced pair of (1.4).The T-forced pairs of the type (0, x) with x constant will be called trivial.
Roughly speaking, we explicitly construct a scalar function Φ whose change of sign implies the existence of a connected set, called a "branch", of nontrivial pairs (λ, x) that emanates out of the set of zeros of Φ and whose closure -in a suitable Banach space -is not compact.
We also give sufficient conditions yielding the multiplicity of T-periodic solutions of (1.4) for λ > 0 small.Such conditions are essentially based upon the notion of ejecting set (see, e.g., [26]).
The methods that we employ are topological in nature and are based on the Brouwer degree.Nevertheless, the use of our main results is accessible even without a familiarity with this notion since the application of the degree theory is restricted to the proofs and some preliminary results proved elsewhere.This is possible since we deal with scalar equations for which degree-theoretic assumptions can be replaced with more elementary ones.It would be not so if the vector case was considered, see Section 8 for a brief discussion.However, for completeness, we provide a short summary of degree theory in Section 4.
Our results are so-to-speak dual to those of [12], where periodic perturbations containing delay terms are applied to scalar, second order ODEs.The study, by means of topological methods, of the branching and multiplicity of periodic solutions of periodically perturbed equations, is now a well-investigated subject in the case of ODEs both in Euclidean spaces and on manifolds (see, e.g., [25,28]).The case of ODEs perturbed with delayed forcing terms is also studied in the literature, although not so broadly (see, e.g., [12,13]).However, the presence of delay terms in the unperturbed equation (1.3) is peculiar of the problem addressed here.In spite of the apparent similarities, a different approach is called upon in order to manage the unperturbed equation.
In the undelayed case, that is for periodically forced second-order scalar autonomous ODEs, both the problems of existence and multiplicity of periodic solutions are quite classical.However, this is still the subject of active research by the mathematical community.There are many approaches that have been successfully pursued to get multiplicity results: among the others let us mention here, e.g., the recent contributions [3,4,11,23], the survey papers [37,38] as well as the monograph [24] and the references therein.
The strategy adopted in this paper is inspired to [42] and can be summarized as follows.First we construct a system of b + 2, T-periodically perturbed first-order ordinary differential equations whose T-periodic solutions correspond to that of (1.4).We then use known results about these perturbed systems to ensure the existence of a branch as sought when the topological degree of the unperturbed field G has nonzero degree and finally, we show, by homotopy techniques, that the change of sign of Φ implies that G has nonzero degree.
It should be noted that the result just described does not guarantee the existence of forced oscillations even for very small values of λ.For this reason, we also provide a nonlocal condition, based on an inequality of J. A. Yorke [43], implying that the branch projects nontrivially onto [0, ∞).
In order to develop a better understanding of the nature of the branch of T-periodic solutions and its relation with multiplicity results we discuss, following [2] and [42], a method to visualize, in finite dimension, a homemorphic set that retains all the relevant properties.We illustrate the procedure with a numeric example.

The linear chain trick
As pointed out in the introduction, the purpose of this paper is to investigate the structure of the set of the T-periodic solutions of the parametric ODE where we make the following set of assumptions: i. g : R 3 → R is a continuous map; ii. φ : R 2 → R is a continuous map; iii. γ b a , for a > 0 and b ∈ N \ {0}, represents the gamma probability distribution (1.2); iv.f : R 3 → R is continuous and T-periodic in the first variable; v. λ ≥ 0 is a real parameter. (2.2) We start with a crucial remark; namely, the fact that the T-periodic solutions of (2.1) and those of the following system of b + 2 ordinary differential equations correspond in some sense: ξ = G(ξ) + λF(t, ξ), ( where ξ = (u, v 0 , v 1 , . . ., v b ) ∈ R b+2 , and the maps G : R b+2 → R b+2 and F : R × R b+2 → R b+2 are respectively defined as Let us show now how a "linear chain trick" (see, e.g., [41,42]) can be used to prove the correspondence between T-periodic solutions of the second order equation (2.1) and of the first order system (2.3).
Let us introduce some notation.By C n T (R s ), n = 0, 1, 2, we will denote the Banach space of the T-periodic C n maps x : R → R s with the the standard norm Here, x (i) denotes the i-th derivative of x, in particular x (0) coincides with x.Theorem 2.1.Assume (2.2) and suppose x 0 is a T-periodic solution of (2.1), and let y 0 (t) := ẋ0 (t), for t ∈ R.Then, (x 0 , y 0 , y 1 . . ., y b ) is a T-periodic solution of (2.3).
Proof.First notice that, since x 0 ∈ C 2 T (R) and φ : R 2 → R is continuous, it is not difficult to prove that the functions y i , i = 0, . . ., b, are in C 1 T (R).Moreover, we claim that, for any t ∈ R, Indeed, the first equality follows by definition.Since x 0 is a solution of (2.1), for λ ≥ 0 we have = g x 0 (t), y 0 (t), y b (t) + λ f t, x 0 (t)y 0 (t) , ∀t ∈ R. Now observe that, when i = 1, we have for any t ∈ R, Thus, taking the derivative under the integral sign, e −a(t−s) φ(x 0 (s), y 0 (s)) ds so that ẏ1 (t) = a φ(x 0 (t), y 0 (t)) − y 1 (t) .
Also, for i = 2, . . ., b, we get for any t ∈ R, proving our claim.Finally, to see that (x 0 , y 0 , y 1 . . ., y b ) is T-periodic, observe that so are x 0 and y 0 , and with the change of variable σ = s − T we get for i = 1, . . ., b and any t ∈ R, and this completes the proof.
The proof of Theorem 2.2 is based on the following technical lemma on linear systems of ODEs, see [42,Lemma 3.3], cf. also [41,Prop. 7.3].The proof is omitted.Lemma 2.3 ([42], Lemma 3.3).Given any continuous and bounded function z 0 : R → R and any a > 0, there exists a unique C 1 solution z = (z 1 , . . ., z b ), which is bounded in the C 1 norm.This solution is given by Remark 2.4.Observe in particular that, for i = 1 in the previous lemma, one has so that z 1 is actually a C 1 function, and thus so are all the z i 's for i = 2, . . ., b.

Branches of T-pairs
This section investigates the structure of the set of T-periodic solutions of (2.1).We begin by recalling some notation and basic facts.
Consider the following first-order parameterized ODE on R k : where λ ≥ 0, the maps g : We say that a pair (λ, x is a T-periodic solution of (3.1) corresponding to λ.If λ = 0 and x is constant then the T-pair is said trivial.It is not hard to see that the trivial T-periodic pairs of (3.1) correspond to the zeros of g.From now on, given any p ∈ R k , we will denote by p the constant map We have the following fact concerning the T-pairs of (3.1), see [27,Theorem 3.3].The statement of Theorem 3.1 involves the Brouwer degree of the map g, see Section 4 for a definition and related notions.Let us now go back to the second-order equation (2.1).Roughly speaking we will state a global bifurcation result, analogous to Theorem 3.1, whose assumptions do not involve the topological degree, but rely only on sign-changing properties of the real-valued function Φ : R → R, given by To make a precise statement we need some further notation.The pairs (λ, x) ∈ [0, +∞) × C 1 T (R), with x : R → R a T-periodic solution of (2.1) corresponding to λ, will be called Tforced pairs (of (2.1)), and we denote by X the set of these pairs.Among the T-forced pairs we shall consider as trivial those of the type (0, x) with x constant.Given an open subset Ω of [0, ∞) × C 1 T (R), we will denote by Ω := u ∈ R : (0, u) ∈ Ω , where by u we mean the constant map t → u, t ∈ R.
As pointed out in Section 2, there is a correspondence between T-periodic solutions of the second order equation (2.1) and of the first order system in (2.3).A similar correspondence holds between their "T-periodic pairs" in a sense that we are going to specify.We point out that, since equation (2.3) can be seen as a special case of (3.1), in accordance with the notation introduced above, a pair (λ, 3) corresponding to λ.We also recall that the set of T-forced pairs of (2.1) is regarded as a subset of [0, +∞) × C 1 T (R).As a first remark we relate the corresponding "trivial T-periodic pairs".This observation will be deduced from the following link between the zeros of the map G, defined in (2.4), and those of Φ, introduced in (3.2).
Let us now establish a general correspondence between the sets of T-forced pairs and of T-pairs, that preserves the notion of triviality.Let J : [0, +∞) ) be the map defined as follows: where ξ := (x 0 , y 0 , y 1 . . ., y b ) is given by y 0 (t) := ẋ0 (t), As above, denote by X the set of the T-forced pairs of (2.1), and let ) be the set of the T-pairs of (2.3).
Lemma 3.4.Let J | X : X → Y be the restriction to X of J .Then J | X is a homeomorphism of X onto Y, which establishes a bijective correspondence between the trivial T-forced pairs in X and the trivial T-pairs in Y.
Proof.Since φ is continuous, the continuity of J is obtained by construction.Whence we get the continuity of the restriction J | X .Injectivity and surjectivity of J | X : X → Y follow from Theorems 2.1 and 2.2.The inverse map ( J | X ) −1 can be seen merely as the projection onto the first two components, so it is continuous.Finally the last part of the assertion follows by Remark 3.3.
The following fact will be crucial for the proof of our main result.Its proof heavily relies on the properties of the topological degree, it is therefore postponed to Section 4 where this concept and its features are discussed.Given a bounded open interval (α, Theorem 3.5.Assume (2.2), let G be as in (2.4) and let Φ be the real-valued function defined in (3.2).Suppose that Φ(α) • Φ(β) < 0. Then G is admissible for the degree in W * and we have deg(G, W * ) ̸ = 0.
We are now in a position to state and prove our main result concerning the set of T-forced pairs of (2.1).Theorem 3.6.Consider equation (2.1) and assume that (2.2) hold.Denote by X ⊆ [0, ∞) × C 1 T (R) the set of its T-forced pairs.Let Φ be the real-valued function defined in (3.2), and suppose that (α, be open and such that Ω = (α, β).Then, in X ∩ Ω there is a connected subset Γ of nontrivial T-forced pairs of (2.1) whose closure relative to Ω is not compact and intersects the set We claim that Γ is not contained in any compact subset of Ω.By contradiction, assume that there exists a compact set K ⊆ Ω containing Γ. Since X is closed, K ∩ X is compact.Thus, its image J (K ∩ X) is a compact subset of O containing Υ, which is a contradiction.The assertion follows.
We remark that, as in the well-known case of the resonant harmonic oscillator this unbounded branch is possibly contained in the slice {0} × C 1 T (R).We will discuss some conditions preventing this "pathological" situation in Section 5.

Computation of the degree 4.1 Brouwer degree in Euclidean spaces
We will make use of the Brouwer degree in R k in a slightly extended version (see e.g.[1,20,39]).Let U be an open subset of R k , f a continuous R k -valued map whose domain contains the closure U of U, and q ∈ R k .We say that the triple ( f , U, q) is admissible (for the Brouwer degree) The Brouwer degree is a function that to any admissible triple ( f , U, q) assigns an integer, denoted by deg( f , U, q) and called the Brouwer degree of f in U with target q.Roughly speaking, deg( f , U, q) is an algebraic count of the solutions in U of the equation f (p) = q.In fact, one of the properties of this integer-valued function is given by the following computation formula.Recall that, if f : U → R k is a C 1 map, an element p ∈ U is said to be a regular point (of f ) if the differential of f at p, d f p , is surjective.Non-regular points are called critical (points).The critical values of f are those points of R k which lie in the image f (C) of the set C of critical points.Any q ∈ R k which is not in f (C) is a regular value.Therefore, q is a regular value for f in U if and only if det(d f p ) ̸ = 0, ∀p ∈ f −1 (q) ∩ U. Observe, in particular, that any element of R k which is not in the image of f is a regular value.
Computation formula.If ( f , U, q) is admissible, f is smooth, and q is a regular value for f in This formula is actually the basic definition of the Brouwer degree, and the integer associated to any admissible triple (g, U, r) is defined by where f and q satisfy the assumptions of the Computation Formula and are, respectively, "sufficiently close" to g and r.It is known that this is a well-posed definition.
The more classical and well-known definition of Brouwer degree is usually given in the subclass of triples ( f , U, q) such that f : U → R k is continuous, U is bounded and q / ∈ f (∂U).However, all the standard properties of the degree, such as homotopy invariance, excision, additivity, existence, are still valid in this more general context.For a detailed list of such properties we refer, e.g., to [30,33,39].
Since in this paper the target point q will always be the origin, for the sake of simplicity, we will simply write deg( f , U) instead of deg( f , U, 0).In this context, we will say that an element p ∈ f −1 (0) is a nondegenerate zero (of f ) if det(d f p ) ̸ = 0; this means, equivalently, that p is a regular point.Accordingly, we will also say that ( f , U) is an admissible pair (or that the map f is admissible in U) if so is the triple ( f , U, 0).Observe that deg( f , U) can be regarded also as the degree (or characteristic, or rotation) of the map f seen as a tangent vector field on R k .
In what follows we will make use of the following elementary fact whose proof we include for the sake of completeness.In particular deg ψ, (a, b) ̸ = 0. s) is an admissible homotopy with the identity map translated by b+a 2 .Hence If, otherwise, one has ψ(b) < 0, it is possible to define an admissible homotopy H by setting

The degree of the map G
The last part of this section is devoted to the proof of Theorem 3.5.Roughly speaking we will relate the degree of the map G, defined in (2.4) by with that of the function Φ : R → R in (3.2), given by Φ(u) = g(u, 0, φ(u, 0)).
To simplify the computation of the degree, it is convenient to introduce the next map on R b+2 : As in Remark 3.2, we have that the zeros of G and those of Let G and G be as above, and let V ⊆ R b+2 be open.Suppose that one of the maps G or G is admissible for the Brouwer degree in V; then, so is the other and they are admissibly homotopic in V.
Proof.Proof.First notice that, by Remark 4.2 the zeros of G and those of Φ correspond.Assume now that ξ := ( ū, 0, w . . ., w) is a nondegenerate zero of G.We represent the differential dG ξ of G at ξ as the following Jacobian matrix: where "∂ i ", i = 1, 2, 3, represents the partial derivation with respect to the i-th variable.By direct computation based on Laplace expansion we obtain Thus ū is a nondegenerate zero of the function Φ.The proof of the converse implication is analogous.
Let us now compute the degree of the map G in the smooth case.Proof.By Remark 4.2 we have that all zeros of G are of the form ξ := ( ū, 0, w . . ., w) with Φ( ū) = 0.Moreover, they are all nondegenerate.In particular, if Φ is admissible in U then Consider now the case in which Φ is merely continuous, or the zeros of Φ in (α, β) may not be nondegenerate.Clearly the above Lemmas 4.4 and 4.5 are not directly applicable, nevertheless it is possible to prove the assertion by applying some well-known approximation theorems and the properties of the Brouwer degree.More precisely, using the so-called smooth Urysohn Theorem and Sard's Theorem, we will construct a real-valued function Φ r : R → R, depending on a suitable small parameter r, satisfying the assumptions of Lemma 4.5 in (α, β).
For this purpose, observe first that, as a consequence of Remark 4.2, the map G is admissible in W * so that deg(G, W * ) is well-defined.Let W ⊆ W * be an open relatively compact neighborhood of the compact set G −1 (0) ∩ W * .Without loss of generality, one can choose W with the following properties: Note that V is bounded and contains W; in particular, G −1 (0) ∩ W * ⊆ W ⊆ V.
By Lemma 4.3 and the excision property of the degree, By a smooth approximation argument similar to the one used in the definition of the degree (cfr.[30,Ch. 5, §1], see in particular [30, Theorem.2.6]), we can choose smooth approximations g and φ of g and φ, respectively, such that, setting the map G has the following properties: G −1 (0) ∩ W * ⊆ W, all the zeros of G are nondegenerate, we have deg(G, In particular, for such an r, all the zeros of this map are nondegenerate.Let σ : R b+2 → [0, 1] be a smooth function that vanishes identically on R b+2 \ V and is identically equal to 1 in W.
The existence of such a function follows from the so-called smooth Urysohn Theorem; see, e.g., [29, Ch. 1, §8, Exercise 15].Define Observe that (4.2)-(ii) implies that Φ r coincides with the map in (4.7) in [α, β]; consequently, all of its zeros in this interval are nondegenerate.From (4.6) it follows Φ r (α) Φ r (β) < 0, which implies that Φ r is admissible in (α, β).Let now where σ is the scalar function introduced above.Applying Lemma 4.1, Lemma 4.4 and Lemma 4.5 to Φ r and G r we obtain that G r is admissible in Furthermore, by construction, the map G r coincides with Actually, as a consequence of (4.4)-(ii), (4.5) and the choice of r, G r −1 (0) ∩ W * ⊆ W. Thus, from (4.8) and the excision property of the degree Moreover, note that max and consider the homotopy Observe that, by construction, hence, the compact set Finally, from (4.3) and (4.10) we get and the assertion follows from inequality (4.9).

Ejecting points and small perturbations
As we have seen by the simple example at the end of Section 3, it may happen that the branch Γ of T-forced pairs of equation (2.1), as in the assertion of Theorem 3.6, is completely contained in the slice {0} × C 1 T (R).In this section we show simple conditions ensuring that this is not the case.Such conditions are based on the key notion of ejecting set (see, e.g., [26]).
Let us first introduce the following notation: let Below, we adapt to our context the definition of ejecting set of [26]: T (R) be the set of T-forced pairs of (2.1), and let X 0 be the slice of X at λ = 0. We say that A ⊂ X 0 is an ejecting set (for X) if it is relatively open in X 0 and there exists a connected subset of X which meets A and is not contained in X 0 .In particular, when A = {p 0 } is a singleton we say that p 0 is an ejecting point.
We now discuss a sufficient condition for an isolated point of Φ to be ejecting for the set X of T-forced pairs of (2.1).This condition is based on a result by J. Yorke ( [43], see also [7]) concerning the period of solutions of an autonomous ODE with Lipschitz continuous right-hand side.
We point out that an analysis of (2.3), for λ = 0, linearized at its zeros leads to a different, not entirely comparable, approach.A discussion of the latter technique, which is based on the notion of T-resonance and requires the knowledge of the spectrum of the linearized equation is outside the scope of the present paper (see, e.g., [17,Ch. 7] and [16, Ch. 2], a similar idea can be traced back to Poincaré see, e.g., [38]; see also [2] for an application of this idea to multiplicity results).
The aforementioned result of Yorke is the following: Theorem 5.2 ([43]).Let ξ be a nonconstant τ-periodic solution of where Then the period τ of ξ satisfies τ ≥ 2π/L.
Inequalities on the period like that of Theorem 5.2 have been object of study in different contexts, see e.g., the introduction of [8] for an interesting discussion.
Since we are going to use Theorem 5.2, besides our standard assumptions (2.2), we need to assume the following fact on the maps g and φ throughout this and the following sections: g and φ are such that G, defined in (2.4), is (locally) Lipschitz. (5.1) Observe that taking g and φ locally Lipschitz allows us to apply Theorem 5.2 in a suitable set W ⊆ R b+2 .In particular, since we are going to apply such a result only to unperturbed part of (2.3), no further hypothesis on f is necessary.From Theorem 5.2, we immediately deduce the following facts concerning the branch of T-forced pairs of (2.1) given by Theorem 3.6: Corollary 5.3.Consider equation (2.1), assume that (2.2) hold and let (α, β), Ω and Φ be as in Theorem 3.6.Suppose further that g and φ are such that G is Lipschitz with constant L G .If the period T of the forcing term f satisfies T < 2π/L G , then the set Γ of Theorem 3.6 cannot intersect the slice {0} × C 1 T (R).
Proof.Assume by contradiction that there exists a nontrivial T-forced pair (0, x 0 ) ∈ Γ.Then x 0 corresponds (in the sense of Theorem 2.1) to a T-periodic solution of ξ = G(ξ) with T < 2π/L G , violating Theorem 5.2.
A similar argument yields the following result: Corollary 5.4.Consider equation (2.1), suppose that (2.2) hold and let (α, β), Ω and Φ be as in Theorem 3.6.Let p ∈ (α, β) be an isolated zero of Φ and assume that g and φ are such that G is locally Lipschitz in a neighborhood W of the corresponding zero P = p, 0, φ(p, 0), . . ., φ(p, 0) of G. Suppose that a set Υ of T-pairs for (2.3) contains 0, P .Let Λ = J −1 (Υ) be the corresponding set of T-forced pairs of (2.1).As above, call L G the Lipschitz constant of G in W.Then, the trivial T-forced pair (0, p) is isolated in the slice Λ 0 .In particular, if T < 2π/L G , the set Λ cannot be contained in the slice {0} × C 1 T (R).
Observe that, unlike Γ in Corollary 5.3, the set Υ in Corollary 5.4 does not necessarily consist of nontrivial T-pairs.Consequently, Λ does not necessarily consist of nontrivial Tforced pairs as well.Notice also that since p is an isolated zero of Φ then 0, P is isolated in the set of trivial T-pairs of (2.3) and, similarly, (0, p) is isolated in the set of trivial T-forced pairs of (2.1).
Proof of Corollary 5.4.By construction of the map J , the T-forced pairs of (2.1) that lie in the slice {0} × C 1 T (R) corresponds bijectively to the T-pairs of (2.3) contained in {0} × C T (R b+2 ).Theorem 5.2 implies that there are not nonconstant T-periodic solutions of (2.3) contained in W. Thus Υ cannot be contained in the slice {0} × C T (R b+2 ).Hence, the same is true for Λ = J −1 (Υ).
Remark 5.5.Suppose that g and φ are such that G is C 1 in a neighborhood of a nondegenerate zero, then it is locally Lipschitz in this neighborhood.Then by Corollary 5.4 we have that if the frequency of the forcing term is sufficiently high, then a nondegenerate zero of the unperturbed vector field is necessarily an ejecting point of nontrivial T-forced pairs of (2.1).

Ejecting sets and multiplicity
This section is devoted to the illustration of sufficient conditions on f , g, and φ yielding the multiplicity of T-periodic solutions of (2.1) for λ > 0 small.As above, let G be as in (2.4), and let X ⊆ [0, ∞) × C 1 T (R) be the set of nontrivial T-forced pairs of (2.1).
To prove the last assertion, observe that by part (2) there exists λ * > 0 such that each {λ} × U i , contains at least one T-forced pair, say (λ, x λ i ) for any λ ∈ [0, λ * ].Hence, for j, k = 1, . . ., n and j ̸ = k, the images of x λ j and x λ k are confined to the disjoint sets I i and I k .
Restricting to a neighborhood of the set of zeros of G corresponding to p 1 , . . ., p n , we can give a somewhat less technical and perhaps more elegant formulation of our multiplicity result.Corollary 6.2.Consider equation (2.1) and suppose that (2.2) and (5.1) hold.Assume that Φ changes sign at the isolated zeros p 1 , . . ., p n ∈ (α, β).Then, for sufficiently high frequency of the perturbing term f and sufficiently small λ > 0, equation (2.1) has at least n solutions of period T whose images are pairwise disjoint.

Visual representation of branches
We briefly discuss a method allowing to represent graphically the infinite dimensional set of T-forced pairs of (2.1).In other words, as in [2], we create a homeomorphic finite dimensional image of the set Γ yielded by Theorem 3.6 and show a graph of some relevant functions of the point of Γ as, for instance, the sup-norm or the diameter of the orbit of the solution x in any T-forced pair (λ, x).
In this section we assume g and φ as well as the perturbing term f to be at least Lipschitz continuous, so that continuous dependence on initial data of (2.3) holds.
Note that Φ changes sign at the zeros 0 and 1, the corresponding zeros of G being P 0 := (0, 0, 0, 0) and P 1 := (1, 0, −1, −1).Observe that G is locally Lipschitz.In particular, sufficiently small neighborhoods of P 0 and P 1 can be chosen where the Lipschitz constant L of G is smaller than 2π.The validity of this statement can be checked by computing the operator norms of the Jacobian matrices of G at P 0 and P 1 that actually turn out to be strictly smaller than 4. Thus, by Theorem 6.1 there are connected sets Γ 0 and Γ 1 of nontrivial T-forced pairs of (7.1) emanating, respectively, from (0, 0) and (0, 1).In particular, by Theorem 6.1 it follows that there exists λ * > 0 such that for λ ∈ [0, λ * ) there are two periodic solutions of (7.1) with disjoint images.Corollary 6.2 shows that the same is true (with possibly different values of λ * ) for all perturbations with sufficiently high frequency.Figure 7.1 shows sup-norm and diameter of the solutions of the T-forced pairs of (7.1). Figure 7.2, instead, shows the projections of Σ on the plane (λ, q) and on the 3-dimensional space (q, p 0 , λ).Indeed, Figures 7.1 and 7.2 suggest that, for Γ 0 and Γ 1 maximal, Γ 1 = Γ 0 .The figures suggest that the value of λ * in Theorem 6.1 is about 0.25.

Perspectives and future developments
The results of this paper can be naturally extended along two lines.The first and more direct one is to consider systems of equations, say in R n , allowing φ to take vector values, say in R k .In this case one considers (1.3) with g : R n × R n × R k → R n and φ : R n × R n → R k .This generalization that, for example, finds application to coupled nonlinear oscillators with memory, has been considered in [36].Here we only mention that the analogous of the function Φ that has to be constructed in this more general environment is a map of an open set of R n in R n , hence the sign change hypothesis has to be replaced with an assumption about the degree.
A second extension that can be considered is the case when more than one distributed delay is allowed.This situation is considered in some models (see, e.g., [35,41]), indeed it arises naturally in some situations where a nonlinear equation is coupled with a linear "subsystem".To illustrate this point we consider the following example: and assume that deg(g, O) is well defined and nonzero.Then there exists a connected set Γ ⊆ O of nontrivial T-pairs whose closure in O is not compact and meets the set of trivial T-pairs contained in O, namely the set: (0, p) ∈ O : g(p) = 0 .
Thus, Theorem 3.5 implies that deg(G, O) is well-defined and nonzero.So, there exists a connected set Υ of nontrivial T-pairs for (2.3) as in Theorem 3.1.Let Γ := J −1 (Υ); by Lemma 3.4, this is a connected set, it is made up of nontrivial T-forced pairs of (2.1), and its closure relative to Ω must intersect the set (3.3) of the trivial T-forced pairs contained in Ω.

Lemma 4 . 5 .
Assume that g and φ are of class C 1 .Assume further that Φ is admissible on an open set U ⊆ R and all its zeros are nondegenerate.Then, G is admissible in U * = U × R b+1 and deg(G, U * ) = (−1) b−1 deg(Φ, U).

Figure 7 . 2 :
Figure 7.2: Projections of Σ: Initial values and speed of x for (λ, x) ∈ Γ ). Hence H is an admissible homotopy.Now we prove that if Φ is smooth and the zeros of Φ are nondegenerate, so are those of G. Assume that g and φ are of class C 1 .Then, all zeros of Φ are nondegenerate if and only if all zeros of G are nondegenerate.