Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 61, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.61 READING MULTIPLICITY IN UNFOLDINGS FROM ε-NEIGHBORHOODS OF ORBITS RENATO HUZAK, PAVAO MARDEŠIĆ, MAJA RESMAN, VESNA ŽUPANOVIĆ Abstract. We consider generic analytic 1-parameter unfoldings of saddle-node germs of an- alytic vector fields on the real line, their time-one maps and the Lebesgue measure of ε- neighborhoods of the orbits of these time-one maps. The box dimension of an orbit gives the asymptotics of the principal term of this Lebesgue measure and it is known that it is dis- continuous at bifurcation parameters. To recover continuous dependence of the asymptotics on the parameter, here we expand asymptotically the Lebesgue measure of ε-neighborhoods of orbits of time-one maps in a Chebyshev system, uniformly with respect to the bifurcation parameter. We use Écalle-Roussarie-type compensators. We show how the number of fixed points of the time-one map born in the universal analytic unfolding of the parabolic point cor- responds to the number of terms vanishing in this uniform expansion of the Lebesgue measure of ε-neighborhoods of orbits. 1. Introduction In this article we study a diffeomorphisms f(x), or rather families of diffeomorphisms fν(x) on the real line. We want to relate the multiplicity of a fixed point of a diffeomorphism f with the local dynamical properties of the density of its orbit. By the multiplicity of a fixed point of a diffeomorphism f , in a family fν , we mean the maximal number of fixed points born from the fixed point in the family of diffeomorphisms fν deforming f . The dynamical density properties of an orbit are encoded by the tube function ℓ(ε) = ℓ(Tε,ν), defined below using the Lebesgue measure ℓ of ε-neighborhoods of orbits converging to a fixed point for the family of diffeomorphisms fν . The idea of reading the multiplicity of a fixed point of a diffeomorphism from tube functions of its orbits was already explored in [4, 14, 22]. However, no uniform expansion of the tube function for the family fν was given. It is known that, for studying bifurcations of zeros, expansions have to be uniform with respect to the parameters. The problem of constructing a uniform expansion of the tube function, and relating its bifurcation properties to the bifurcations of fixed points is addressed in the present paper. Here, for simplicity, we use only the tail part Tε,ν of the ε-neighborhood of an orbit (see (1.3)), as it carries the same information as the full tube function. Given a diffeomorphism f defined in a real interval, we associate with it the displacement function g = id− f , where id is the identity function. Recall that for differentiable functions, the multiplicity of a fixed point in a sufficiently general unfolding is given by the number of terms vanishing in the Taylor expansion at the fixed point. In particular, this is the case for the displacement function x 7→ g(x). On the other hand, the tube function ε 7→ ℓ(Tε,ν) of an unfolding fν of f is not differentiable at ε = 0. If a family of real functions is not necessarily differentiable, but admits an asymptotic expansion in a Chebyshev scale (see Definition 1.3), uniform with respect to the parameter ν, then the multiplicity of a zero is given by the number of terms vanishing in this asymptotic expansion. 2020 Mathematics Subject Classification. 37G10, 34C23, 28A80, 37C45, 37M20. Key words and phrases. Unfoldings; epsilon-neighborhoods; compensators; Chebyshev scale. ©2025. This work is licensed under a CC BY 4.0 license. Submitted February 21, 2025. Published June 10, 2025. 1 2 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 We show in Theorem 3.10 that the tube function ℓ(Tε,ν), in the case studied in this paper, admits a uniform asymptotic expansion in a Chebyshev scale, multiplied by a common function I, see (3.20). However, by its definition as a Lebesgue measure of a set, ℓ(Tε,ν) cannot be zero for any ε > 0: the function I(ε, ν) explodes precisely in points where ℓ(Tε,ν) I(ε,ν) vanishes, thus canceling the zeros. By abuse, by multiplicity in such a family, we will mean the number of leading terms vanishing in the asymptotic scale. The motivation for this article lies in the conjecture that we formulate here: Conjecture 1.1. Let f be a local diffemorphism of the real line at a fixed point x = 0, which is attractive from one side. Let fν , ν ∈ Rk, be a generic deformation of the diffeomorphism f i.e. a deformation realizing the maximal multiplicity of the fixed point. Consider the restriction of fν , to ν ∈ V ⊂ Rk, such that for all ν ∈ V , fν has a fixed point born from x = 0. Consider a point x0 whose orbit by f0 converges to the fixed point x = 0. Let ℓ(Tν,ε) be the corresponding tube function of fν . Then, there exists a function I(ε, ν) and an asymptotic expansion of the length of the tail ℓ(Tν,ε) uniform in ν ∈ V , as ν → 0, see (3.21), in a Chebyshev scale multiplied termwise by I(ν, ε). The multiplicity of a fixed point of a diffeomorphism x 7→ f(x) in the family fν , ν ∈ V , coincides with the number of vanishing terms of ℓ(Tν,ε) at parameter value ν = 0 in this asymptotic scale. Note that the two function: gν(x) and ℓ(Tν,ε), related by the conjecture, live in completely different spaces: the phase space of values of x, for id − fν , and the parameter ε measuring the size of the ε-neighborhoods of orbits for the tube function ℓ(Tε,ν). Our principal result (Theorem 1.4) is the proof of the Conjecture 1.1 in the simplest non-trivial case of generic saddle-node bifurcations. That is, diffeomorphisms having a parabolic fixed point bifurcating into a diffeomorphism with two hyperbolic fixed points on the real line. Moreover, we assume that this parabolic diffeomorphism is given as time-one map of a vector field. 1.1. Outline of the results. Consider generic analytic 1-parameter unfoldings of real analytic saddle-node vector fields Xν , whose flow is given by dx dt = F (x, ν), x ∈ R, ν ∈ R, (x, ν) → (0, 0). Then, two hyperbolic singular points are born from the saddle-node singular point at ν = 0 in the so-called saddle-node bifurcation (see e.g. [3]). By fν , we denote the time-one map of Xν , which is an unfolding of a parabolic diffeomorphism. Let gν := id− fν be the corresponding displacement function. In the main Theorem 1.4 (Section 1.2), we give a 1− 1 correspondence between the asymptotic expansion of the time-one map fν (i.e. of the displacement function gν) in the phase space and the uniform asymptotic expansion of the tail tube function of its orbit in an appropriate compensator variable. Recall the reason for introducing compensators: in the ε-space the expansion of the tube function was not uniform with respect to the parameter ν. Theorem 1.4 shows that we can read the multiplicity of a fixed point in the given bifurcation (how many fixed points can bifurcate from it in the given family), from the number of terms vanishing in the uniform asymptotic expansion of the Lebesgue measure of the tail of an orbit of the time-one map. Proposition 3.4 and Theorem 3.10 give precisely these uniform asymptotic expansions of the tail tube functions ℓ(ε) in appropriate Chebyshev systems including compensators. They are used in the proof of the main Theorem 1.4 at the end of Section 3. Proposition 3.4 concerns the model vector field case, while Theorem 3.10 is for general vector fields of the form (1.1) under generic assumptions. Note that, due to computational reasons, the expansions are given for the continuous counterpart ℓc(Tε,ν) of the length first and then, in Corollary 3.12, for the standard length ℓ(Tε,ν). For the standard length ℓ(Tε,ν), some additional oscillatory terms appear in the expansion due to the step function nature of the discrete critical time separating the tail and the nucleus, see Subsection 2.2. In Theorem 4.1 the expansions from Proposition 3.4 and Theorem 3.10 are regrouped so that (in general infinitely many) terms from the same group at ν ̸= 0 merge to the same asymptotic EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 3 term at the bifurcation value ν = 0. This illustrates the earlier mentioned phenomenon that the limit as ν → 0 does not commute with asymptotic expansion in ε of the tube function. Also, we obtain in Section 4.1 the asymptotic expansions in ε → 0 of ℓ(Tε,ν) in the case ν > 0 and ν = 0. The expansions have qualitatively different terms, which, in particular, results in a jump in the box dimension at the moment of bifurcation. Theorem 4.1 is also used in Remark 4.5 for reading the formal class of the unfolding from fractal data. Finally, note that the multiplicity (here 2) is one of the analytic invariants of saddle-node vector field unfoldings [7, 6]. For reading the other analytic invariants by tube function, see Remark 4.5. We recall also that in [5] parabolic germs were studied (not depending on parameters and not necessarily given by a time-one map of a saddle-node). It was shown how to read the analytic invariants of these germs from their orbit. 1.2. Main results. We consider an analytic germ of a system dx dt = F (x, ν), (1.1) with F real, analytic germ in x and in parameter ν, and with a non-hyperbolic singular point x = 0 at the bifurcation value ν = 0 (i.e. F (0, 0) = 0, Fx(0, 0) = 0), satisfying the generic assumptions: Fν(0, 0) ̸= 0, Fxx(0, 0) ̸= 0. (1.2) Under these assumptions, the saddle-node point at x = 0 bifurcates at ν = 0 into two hyperbolic points on the real axis: one attracting and one repelling, for ν ∈ (0, δ), or ν ∈ (−δ, 0) depending on the sign of Fν and Fxx in (1.2). For details, see Section 2.1. Take x0 in the attracting basin of the saddle-node point, sufficiently close to 0. Consider the time-one map fν of (1.1) and let gν = id− fν be the corresponding displacement function. Definition 1.2 (tube function ℓ(Tε,ν)). Let Ofν (x0) := {fnν (x0) : n ∈ N0} be the orbit with initial point x0. By Ofν (x0)ε, ε > 0, we denote its ε-neighborhood, i.e. the set of all points at distance less than ε from Ofν (x0). That is, it is an infinite union of intervals of length ε around points of the orbit. By [21], they form two disjoint parts. The non-overlapping intervals are called the tail Tε,ν , and the overlapping infinitely many intervals forming one interval are called the nucleus Nε,ν : Ofν (x0)ε = Tε,ν ∪Nε,ν . (1.3) We denote by ℓ the Lebesgue measure. We call the function ℓ(Tε,ν) the tube function of the family fν . In [14, 17], we studied the length ℓ(Ofν (x0)ε) = ℓ(Tε,ν) + ℓ(Nε,ν). However, the essential information is carried already by ℓ(Tε,ν) [9, 5]. Hence, we investigate here only this term. Moreover, the dependence on x0 is not essential. In Theorem 1.4 below, we use the notion of Chebyshev systems. Chebyshev systems are gen- eralizations of Taylor power monomial scales, on which the division-derivation algorithm can be performed, see [11]. More precisely: Definition 1.3 (Chebyshev system, [11]). Let I = (−r, r) or I = [0, r), r > 0. A finite sequence {u0 = 1, u1, u2, . . . , um}, m ∈ N, of continuous functions on I and m times differentiable on I \ {0} is called a Chebyshev system, if the functions Di(uk), i, k = 0, . . . ,m, are well defined on I inductively by the following division and differentiation algorithm: D0(uk) = uk, Di+1(uk) = (Di(uk)) ′ (Di(ui+1))′ , i = 0, . . . ,m− 1, (1.4) for every 0 ≤ k ≤ m, except possibly at x = 0, to which they are extended by continuity. 4 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 Note that by linearity, (1.4) extends to operators well-defined on the space of functions generated by {u0, . . . , um}. Now, we allow the functions ui to depend continuously on a parameter ν, but we require that the differential operators Di from (1.4) be well-defined on the interval I. Our study is local, in a neighborhood of the origin, so we allow the reduction of the interval I, if necessary. More generally, let u0 be a function different from 0 in I \ {0}, for all values of ν. Let the system {u0, . . . , um} be a Chebyshev system after division of all terms by u0. Let gν(x) := a0(ν)u0(ν, x)+ . . .+am(ν)um(ν, x), x ∈ I. Let g0(x) = ak(0)uk(0, x)+ . . .+am(0)um(0, x), where 0 ≤ k ≤ m and ak(0) ̸= 0. That is, let gν be an unfolding of g0 on I, generated by {u0, . . . , um}. If u0(ν, 0) ̸= 0, for all ν, then, by Rolle’s theorem, the maximal number of zeros that can be born from the origin in unfolding gν is bounded from above by k, i.e., by the number of missing Chebyshev terms in g0 before the leading one. Otherwise, adding the zero at the origin, the maximal number of zeros that can be born from 0 in unfolding gν is bounded from above by k+1. The latter will be the case with our expansions both of the displacement and of the tube function in Theorem 3.10 (Section 3), which is a more precise formulation of Theorem 1.4. We now state our main result. Theorem 1.4. Let (1.1) be a generic 1-parameter analytic unfolding of a parabolic fixed point, satisfying the generic assumptions (1.2). Assume unilateral values of parameter ν for which par- abolic point unfolds into two hyperbolic points: without loss of generality, ν ∈ [0, δ). Let fν be the time-one map and Ofν (x0) be its attractive orbit starting at x0. There exists a compensator variable η(ε, ν) and an asymptotic expansion of the length of the tail ℓ(Tν,ε) in a Chebyshev system, uniform in ν ∈ [0, δ), as η → 0. There is a 1 − 1 correspondence between the expansion of the length ℓ(Tν,ε) in the η variable and the Taylor expansion of the displacement function gν := id − fν in the phase x-variable, in the following sense. For every value of the parameter ν, the number of vanishing terms of the expansions of ℓ(Tν,ε) and gν , in the corresponding systems, is the same (here, for 1-parameter unfoldings, at most 1). Remark 1.5. The precise form of the asymptotic expansion of the tail ℓ(Tν,ε) in a Chebyshev system from Theorem 1.4 is given in Theorem 3.10, in equation (3.21). Recall that the number of terms in a Chebyshev expansion of a family of diffeomorphisms vanishing at the bifurcation value gives the multiplicity of the bifurcation. In our case, zero points of the displacement function gν correspond to the fixed points of the time-one map fν , that is, to the singularities of the vector field Xν . Therefore, as a direct consequence of Theorem 1.4, the multiplicity in the generic saddle-node bifurcation can be read from the expansion of tube function ℓ(Tε,ν) in the compensator variable η. Precisely, by expansions given in Theorem 3.10, the multiplicity is the number of vanishing Chebyshev terms of the tube function at the bifurcation value ν = 0 incremented by 1. Here it equals 2. Here we claim (and prove) the theorem only in the case of generic saddle-node bifurcation (i.e. multiplicity 2). However, we believe that it is true for any multiplicity, but the technical difficulties would increase dramatically. 2. Notation and main objects 2.1. Normal forms for the saddle-node bifurcation. The saddle-node bifurcation is a generic 1-parameter bifurcation of 1-dimensional vector fields. Consider a system dx dt = F (x, ν), (2.1) with F real and analytic in x and in ν, having at ν = 0 a non-hyperbolic (Non-hyperbolic means that Fx(0, 0) = 0.) singular point x = 0 (i.e. undergoing a bifurcation of the singular point at ν = 0) and which satisfies generic assumptions Fxx(0, 0) ̸= 0 and Fν(0, 0) ̸= 0. By [7, 6], by a weak analytic change of variables, the system (2.1) can be brought to the form dx dt = Fmod(x, ν), Fmod(x, ν) := −x2 + ν 1 + ρ(ν)x , ν ∈ (−δ, δ). (2.2) EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 5 The two analytic invariants are the multiplicity k = 2 and the residual invariant ν 7→ ρ(ν), ν ≥ 0. Here, weak analytic change of variables is an analytic germ (germified at x = 0, ν = 0) of change of variables, fibered in ν [15, 6], i.e. of the form Φ(x, ν) = (φν(x), h(ν)), (2.3) where h is an analytic diffeomorphism such that h(0) = 0 and φν ∈ R{x}, ν ∈ (−δ, δ), is an analytic diffeomorphism conjugating one to the other. For the corresponding time-one maps it holds that fmod h(ν) = φν ◦ fν ◦ φ−1 ν . Note that the adjective weak refers to the possible bijective reparametrization h(ν) of the param- eter ν. For ν < 0 there are no real singular points of the model. For ν > 0, there are two singular hyperbolic points: attracting √ ν and repelling − √ ν. For ν = 0 the point zero is a saddle-node singular point, attracting from the right and repelling from the left. Choose an initial point x0 ̸= 0. For small values of ν ∈ [0, δ), δ > 0, x0 lies outside [− √ ν, √ ν], but stays in the attracting resp. repelling basin of √ ν resp. − √ ν, [15]. We suppose x0 > 0, so that (for ν sufficiently small and positive) it lies in the attractive basin for the bifurcation. Otherwise, if we choose x0 < 0, we consider the opposite field, ẋ = x2 − ν, so that x0 lies again in the basin of attraction. 2.2. Continuous-time length of the tail of orbits. In this subsection we show how to calculate the length ℓ(Tε,ν) of the tail of the ε-neighborhood of the orbit Ofν (x0). Recall [21], that the dis- crete critical time nνε ∈ N, separates the order of iterates of fν , for which ε-neighborhoods of points in the orbit are not overlapping, from the order for which overlapping of these ε-neighborhoods starts. It is determined by the inequalities: f nν ε ν (x0)− f nν ε+1 ν (x0) ≤ 2ε, f nν ε−1 ν (x0)− f nν ε ν (x0) > 2ε. The ε-neighborhood of an orbit Ofν (x0) consists of the nucleus Nε,ν and the tail Tε,ν . The nucleus is the overlapping part of ε-neighborhood of the orbit and the tail consists of nνε nonintersecting intervals each of length 2ε. Hence ℓ(Tε,ν) = nνε2ε. Since the critical time ε 7→ nνε is a step function, for a fixed ν, the function ε 7→ ℓ(Tε,ν) does not have a full asymptotic expansion as ε → 0. Therefore, we replace nνε by the so-called continuous critical time τνε ∈ R satisfying f τν ε ν (x0)− f τν ε +1 ν (x0) = gν(f τν ε ν (x0)) = 2ε. Here, {f tν : t ∈ R} is the flow of the field Xν = F (x, ν) d dx given by (2.1). Note that fν := f1ν is the (germ of) time-one map of Xν . The continuous critical time τνε can be understood as the time needed to move along the field from the initial point x0 to the point x whose displacement function value gν(x) is exactly equal to 2ε, for every ε > 0. Note that, as ε tends to 0, we choose the positive time τνε (tending to +∞), such that the flow f τν ε ν (x0) approaches the attracting singular point √ ν from the side of the initial point x0. That is, although gν is multivalued around the singular point, we take the inverse of the unilateral, strictly increasing restriction of gν containing x0. We now define the continuous-time length of Tε,ν (see [13]) by ℓc(Tε,ν) := τνε · 2ε. (2.4) Equivalently, ℓc(Tε,ν) = (Ψν(g −1 ν (2ε))−Ψν(x0)) · 2ε, (2.5) where the time coordinate germ Ψν , defined up to an additive constant, is the trivialization coordinate for the flow of field Xν = F (x, ν) d dx from (2.1), satisfying Ψν(f t ν(x0))−Ψν(x0) = t, t ∈ R, (2.6) 6 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 or, equivalently, Ψ′ ν(x) = 1 F (x, ν) . For details of the definition and the relation between the two definitions, see [13]. We consider only the case when ν ≥ 0. In the case that ν < 0, the fixed points lie on the imaginary axis, and are not hyperbolic but indifferent, that is, their linear part is a rotation. On the real line the vector field passes from −∞ to +∞ in a finite time, and there are no singular points on the real line. This case will be a subject of future research, see Section 5. All our theoretical results in the following sections will first be given for the continuous length ℓc(Tε,ν). However, from the orbit it is natural to read the standard length ℓ(Tε,ν), as the sum of the lengths of 2ε-intervals centered at points of the orbit of the time-one map fν before they start overlapping. By [12, 18], for a fixed ν, this function does not allow the full asymptotic expansion in ε → 0, due to oscillatory terms. Nevertheless, by Corollary 3.12, the Chebyshev system given in Proposition 3.4 and Theorem 3.10 can easily be adapted for the standard length ℓ(Tε,ν). 3. Uniform asymptotic expansions of the length functions ℓ(Tε,ν) and ℓc(Tε,ν) The Subsections 3.2 (Proposition 3.4) and 3.3 (Theorem 3.10 and Corollary 3.12) respectively give the Chebyshev systems for ℓ(Tε,ν) and ℓ c(Tε,ν) for the model family and for generic saddle- node families respectively. The model case is used in the proof of the general case by performing an analytic change of variables. 3.1. Compensators. In the sequel we first define three compensators that we use in the uniform expansions in Proposition 3.4. We use the name compensator for elementary expressions in variable x and parameter ν, i.e. expressions that cannot be further asymptotically expanded uniformly in ν. Definition 3.1 ([20]). Let ν and x be small. The function ω(x, ν) := x−ν − 1 ν is called the Écalle-Roussarie compensator. Note that, pointwise, ω(x, ν) → − log x, as ν → 0. The convergence becomes uniform in x, if we multiply by xδ, δ > 0. Definition 3.2. For x > 0 and ν ∈ (−δ, δ), we call the function α(x, ν) := 1 ν log ( 1 + ν x ) the inverse compensator. The name comes from Definition 3.1 and the fact that α(x, ν) = − log ◦ω−1 ( 1 x , ν ) , where ω−1 is the inverse of ω with respect to the variable x. Pointwise, α(x, ν) → 1 x , as ν → 0. For every δ > 0, we obtain x1+δα(x, ν) → xδ, as ν → 0, uniformly in x > 0. The asymptotic behavior, as x→ 0, is qualitatively different in the case ν = 0 and ν ̸= 0: α(x, ν) = { 1 x , ν = 0, 1 ν (− log x) + log ν ν + Rν [[x]], ν ̸= 0. (3.1) Here and in the sequel R[[x]] denotes a formal expansion in powers of x and R{x} an analytic germ in x. The notation Rν [[x]] resp. Rν{x} stands for a formal resp. analytic series in x with coefficients analytic germs in ν, that is, for R{ν}[[x]] resp. R{ν}{x}. EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 7 Definition 3.3. For ν small by absolute value and x > 0, we define η̃(x, ν) := √ x+ ν − √ ν, and call η̃ the square root-type compensator. The asymptotic expansion of η̃, as x→ 0, changes qualitatively as ν changes from zero: η̃(x, ν) = {√ x, ν = 0, x√ ν + √ ν x2 ν2 R{x ν }, ν > 0, x→ 0. (3.2) Note that η̃ is small, for small x. Moreover, it can easily be checked that η̃(x, ν) → √ x uniformly in x, as ν → 0+. Let a(ν) := 1− e−ν ν , |ν| < δ. Note that a ∈ R{ν} and a(0) = 1. Therefore a is bounded. 3.2. Model family. Consider the model family (2.2), for ν ∈ [0, δ). Let fmod ν be its time-one map and gmod ν := id− fmod ν its displacement function. Let Ψmod ν be the time-coordinate for fmod ν , as defined in (2.6) for fν . Let ℓc(Tε,ν), ν ∈ [0, δ), be the continuous lengths of the tails of the ε-neighborhoods of or- bits of time-one maps fmod ν for the unfolding (2.2), with initial condition x0 > 0, as defined in Subsection 2.2. Let θc(x) = x+ c denote the translation by c ∈ R. Proposition 3.4 (Chebyshev system for the model family). Let η(2ε, ν) := θ− √ ν ◦ (gmod ν )−1(2ε), ε > 0. (3.3) In the compensator variable η ≥ 0, the continuous length η 7→ ℓc(Tε,ν), admits an asymptotic expansion, uniform in the parameter ν ∈ [0, δ), as η → 0, in the system{ I(ν, η)η, I(ν, η)η2, I(ν, η)η3, . . . } , which becomes Chebyshev after division by the first term I(ν, η)η. Here, I(ν, η) := α(η, 2 √ ν) + ρ(ν) 2 log ( η2 + 2 √ ν · η ) −Ψmod ν (x0), (3.4) where the determination of Ψmod ν from (3.10) is used. Furthermore, for M > 0 there exist δ, d > 0 such that I(ν, η) > M for η ∈ [0, d) and ν ∈ [0, δ). More precisely, for η → 0+ the expansion is ℓc(Tε,ν) = I(ν, η)gmod ν (η + √ ν) (3.5) = ( 1− e − 2 √ ν 1−ρ(ν) √ ν ) I(ν, η)η (3.6) + e − 2 √ ν 1−ρ(ν) √ ν a ( 2 √ ν 1− ρ(ν) √ ν ) 1 + ρ(ν) √ ν (1− ρ(ν) √ ν)2 I(ν, η)η2 + oν(I(ν, η)η 2). (3.7) Here, ρ(ν) is the residual invariant from the normal form (2.2), and oν(I(ν, η)η 2) means that limη→0 oν(I(ν,η)η 2) I(ν,η)η2 = 0 uniformly in ν ∈ [0, δ). Remark 3.5. Note that the number of vanishing terms in the uniform asymptotic scale for the tube function in Proposition 3.4 and forthcoming Theorem 3.10 at the bifurcation value ν = 0, which we call multiplicity of the expansion of the tube function in the given asymptotic scale, does not imply the number of zero points of the tube function that bifurcate from ε = 0. Indeed, the tube function ℓ(Tε,ν) measures the length of the ε-neighborhood of the orbit and is therefore strictly positive for all ε > 0 and zero only at ε = 0. The reason for that is the term I(ν, η) which is common to all terms of the scale for the expansion, which is strictly positive but explodes to ∞ exactly at zero points of the quotient ℓ(Tε,ν) I(ν,η) . Despite these singularities of the strictly positive common factor I(ν, η), in Proposition 3.4 and Theorem 3.10, by abuse, we will nevertheless call the asymptotic scale for ℓ(Tε,ν) Chebyshev. The number of vanishing terms of the expansion of the tube function at ν = 0 gives only an 8 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 upper bound on the number of zero points of ℓ(Tε,ν), none of which are really zero points of the tube function, due to singularities of the common factor. However, this upper bound equals the multiplicity of the generic unfolding. A similar analysis can be done for k-unfoldings, where we obtain multiplicity of the expansion of the tube function equal to k, although only the point η = 0 i.e. ε = 0 is the true zero point of the tube function for all values of the parameter. It is again due to the term Iν,η that vanishes at all singular points of the field (=zero points of the displacement function) in the unfolding. Remark 3.6. Note that a similar expansion is obtained if we choose the initial point x0 < 0, and the inverse orbit converging to the other (repelling) fixed point − √ ν. Then gmod ν = id− (fmod ν )−1. The variable η is a small variable (in the sense that it tends to 0 as (ε, ν) → 0), and behaves essentially as a square root compensator η̃ from Definition 3.3. The precise asymptotic equivalence statement between the two compensator variables η and η̃ is given by (4.3) in Lemma 4.2 below. The proof of Proposition 3.4 is given at the end of the subsection. For the proof, we need Lemmas 3.7 and 3.9. Lemma 3.7. The time coordinate for family (2.2) is (up to an additive constant) equal to Ψmod ν (x) = α(x− √ ν, 2 √ ν) + ρ(ν) 2 log(x2 − ν) = ( α(x, 2 √ ν) + ρ(ν) 2 · log(2 √ ν · x+ x2) ) ◦ θ−√ ν(x). (3.8) Moreover, Ψmod ν (x) ∼x→ √ ν { 1 x , ν = 0,(ρ(ν) 2 − 1 2 √ ν ) log(x− √ ν), ν ̸= 0. (3.9) Here and in the sequel the relation ∼ between two functions means that their quotient tends to 1 at the limit, which is considered. More generally, it is also used to denote an asymptotic expansion. Proof. The time coordinate Ψmod ν is computed as antiderivative in variable x (determined up to an additive constant) of 1 Fmod(ν,x) . We obtain Ψmod ν (x) = 1 2 √ ν log x+ √ ν x− √ ν + ρ(ν) 2 log(x2 − ν) = 1 2 √ ν log ( 1 + 2 √ ν x− √ ν ) + ρ(ν) 2 log(x2 − ν), (3.10) and substitute α from Definition 3.2. In case ν ̸= 0, we have log(x2 − ν) = ( log(x− √ ν) + log(2 √ ν) + log ( 1 + x− √ ν 2 √ ν )) = log(x− √ ν) +Oν(1), (3.11) as x → √ ν. Here and in the sequel, we denote by Oν(h(x)) a function, parametrized by ν, such that the absolute value of its quotient with h(x) is bounded from above, for x → 0. We define analogously oν(h(x)). In general, if unspecified, we do not request uniformity in ν. Therefore, by (3.10) and (3.11), we deduce (3.9). Indeed, for ν ̸= 0, (3.10) transforms to Ψmod ν = − 1 2 √ ν log(x− √ ν) + ρ(ν) 2 log(x2 − ν) +Oν(1), x→ √ ν. (3.12) Now (3.11) and (3.12) give (3.9) for ν ̸= 0. □ Remark 3.8. Note that the above formula (3.10) is valid, for all values of ν ∈ (−δ, δ). In case ν < 0, √ ν = √ |ν|i is pure imaginary. The formula can be rewritten as Ψmod ν (x) =  1 x + ρ(0) · log x, ν = 0, 1 2 √ ν log ( 1 + 2 √ ν x− √ ν ) + ρ(ν) 2 log(x2 − ν), ν > 0, − 1√ |ν| arctan x√ |ν| + ρ(ν) 2 log(x2 − ν), ν < 0. EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 9 Lemma 3.9 (Time-one map and displacement germ). For the family (2.2), the following Taylor expansions at the fixed point x = √ ν hold, for ν ∈ [0, δ): (A similar expansion can be obtained near the other, symmetric fixed point x = − √ ν.) (1) For the time-one map fmod ν ∈ Diff(R, 0): fmod ν (x) ∼ √ ν + e − 2 √ ν 1−ρ(ν) √ ν · (x− √ ν) − e − 2 √ ν 1−ρ(ν) √ ν · a ( 2 √ ν 1− ρ(ν) √ ν ) 1 + ρ(ν) √ ν (1− ρ(ν) √ ν)2 (x− √ ν)2 + (x− √ ν)3Rν{(x− √ ν)}, x→ √ ν, (3.13) (2) For the displacement function gmod ν := id− fmod ν ∈ Diff(R, 0): gmod ν (x) ∼ ( 1− e − 2 √ ν 1−ρ(ν) √ ν ) · (x− √ ν) + e − 2 √ ν 1−ρ(ν) √ ν a ( 2 √ ν 1− ρ(ν) √ ν ) 1 + ρ(ν) √ ν (1− ρ(ν) √ ν)2 (x− √ ν)2 + (x− √ ν)3Rν{(x− √ ν)}, x→ √ ν. (3.14) The Taylor coefficients of fmod ν and gmod ν belong to R{ √ ν} (are analytic at 0 in √ ν). Proof. It can be checked by the operator exponential formula fmod ν = exp ( Fmod(x, ν) d dx ) id that the coefficients ak(ν) of monomials xk, k ≥ 0, in the Taylor expansion of fmod ν (x) =∑∞ k=0 ak(ν)x k converge towards coefficients ak(0) of f mod 0 (x) = ∑∞ k=0 ak(0)x k, as ν → 0. There- fore, to obtain (3.13), it suffices to obtain the Taylor expansion of fmod ν at √ ν in the case when ν > 0. The time-one map fmod ν is obtained from the time coordinate, by (2.6). It follows that fmod ν = (Ψmod ν )−1(Ψmod ν + 1). First we compute (the first few terms) of the inverse (Ψmod ν )−1. When ν ̸= 0 and x→ √ ν, by (3.10) Ψmod ν admits the following expansion, as x→ √ ν, Ψmod ν (x) = (ρ(ν) 2 − 1 2 √ ν ) log(x− √ ν) + (ρ(ν) 2 + 1 2 √ ν ) log(2 √ ν) + (ρ(ν) 2 + 1 2 √ ν ) log(1 + x− √ ν 2 √ ν ) ∼ (ρ(ν) 2 − 1 2 √ ν ) log(x− √ ν) + (ρ(ν) 2 + 1 2 √ ν ) log(2 √ ν) + Rν [[(x− √ ν)]]. Denote the above coefficients by: K±(ν) := ρ(ν) 2 ± 1 2 √ ν , K(ν) := K+(ν) log(2 √ ν). Then, for the expansion of the inverse (where y K−(ν) → −∞), we obtain (Ψmod ν )−1(y) ∼ √ ν + e y−K(ν) K−(ν) − K+(ν) K−(ν) · 2 √ ν e 2 y−K(ν) K−(ν) + e 3 y−K(ν) K−(ν) Rν [[ e y−K(ν) K−(ν) ]] . Therefore, fmod ν (x) = (Ψmod ν )−1(1 + Ψmod ν (x)) ∼ √ ν + e − 2 √ ν 1−ρ(ν) √ ν · (x− √ ν) − e − 2 √ ν 1−ρ(ν) √ ν a ( 2 √ ν 1− ρ(ν) √ ν ) 1 + ρ(ν) √ ν (1− ρ(ν) √ ν)2 · (x− √ ν)2 + (x− √ ν)3Rν{(x− √ ν)}, x→ √ ν. Note that ρ(ν) is a bounded function on ν ∈ [0, δ), so ρ(ν) √ ν → 0, as ν → 0+. □ Proof of Proposition 3.4. We use the formula for the continuous tail (2.5): ℓc(Tε,ν) = ( Ψmod ν ((gmod ν )−1(2ε))−Ψmod ν (x0) ) · 2ε, (3.15) 10 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 and change the variable from ε to η given by (3.3), i.e. verifying 2ε = gmod ν (η + √ ν). Note that ε ∈ [0, d) corresponds strictly increasingly to η ∈ [0, b), for some b > 0. We denote I(η, ν) := Ψmod ν ( (gmod ν )−1(2ε) ) −Ψmod ν (x0) = Ψmod ν (η + √ ν)−Ψmod ν (x0). (3.16) Now using expression for Ψmod ν given by (3.8) in Lemma 3.7, we obtain I(η, ν) as in (3.4). Then, putting 2ε = gmod ν (η+ √ ν) and I(η, ν) in (3.15), expansion (3.5) follows from (3.14) in Lemma 3.9. Using (3.9) and boundedness of ρ(ν) for ν ∈ [0, δ), we see that Ψmod ν (x) → +∞, as x → √ ν+, uniformly in ν ∈ [0, d). As a consequence, as η → 0+ (i.e. as η+ √ ν → √ ν+), from (3.16) we have that, for a big M > 0, there exist δ, d > 0 such that I(η, ν) > M , for all η ∈ [0, d) and ν ∈ [0, δ). The scale is now obviously Chebyshev since, apart from the common nonzero factor I(ν, η), it is the classical monomial scale. The result now follows. Finally, limη→0 oν(I(ν,η)η 2) I(ν,η)η2 = 0 uniformly in ν ∈ [0, δ), since the family of displacement functions gmod ν depends analytically on ν ∈ [0, δ). □ 3.3. Generic saddle-node families. Let Φ(ν, x) := (φν(x), h(ν)) be the analytic change of variables conjugating (1.1) to its analytic model (2.2), where h(0) = 0 and h is an analytic diffeomorphism at 0, and φν(x) = a0(ν) + a1(ν)x + oν(x 2) an analytic diffeomorphism at x = 0, with a1(ν) ̸= 0, ν ∈ [0, δ). Here, by [15], the change of variables in analytic for ν ∈ (0, δ) and continuous at ν = 0 (see the notion of weak conjugacy in [15]) and therefore uniform in ν ∈ [0, δ), i.e. oν(x 2) means that limx→0 oν(x 2) x2 = 0 uniformly in ν ∈ [0, δ). Then, by (2.3), the initial time-one map fν satisfies fν = φ−1 ν ◦ fmod h(ν) ◦ φν . Let xν1,2 denote the fixed points of fν . It holds that x ν 1,2 → 0, ν → 0. Without loss of generality, we assume that the fixed point xν1 is positive and attractive. For the germ at 0 of the time coordinate, it holds Ψν = Ψmod h(ν) ◦ φν . (3.17) As in the introduction, without loss of generality, we assume that Fν(0, 0) > 0 so that h(ν) > 0, for ν > 0. Note that φν(x ν 1) = √ h(ν), or − √ h(ν) and we suppose that φν(x ν 1) = √ h(ν). Let ℓc(Tε,ν), ν ∈ [0, δ), be the continuous length of the tail of the ε-neighborhood of the orbit Ofν (x0), for x0 > 0 sufficiently small. Then, there exists δ > 0 such that x0 is attracted by xν1 , for all sufficiently small ν ∈ [0, δ). Analogously, we could have considered initial point x0 < 0 repelled from xν2 (i.e. attracted to it by f−1 ν ), ν ∈ [0, δ), and the orbit of the inverse Of−1 ν (x0). Let Φ = (h, φν), ν ∈ [0, δ), be the normalizing germ of change of variables reducing a given saddle-node field (1.1) to its model (2.2), and let C(ν) := φ′ ν(x ν 1) ̸= 0 be the coefficient of the linear term of φν at the point xν1 . Let kν := θ− √ h(ν) ◦ φν ◦ θxν 1 . (3.18) Then kν(x) = C(ν)x+ oν(x), x ∈ (−d, d), x→ 0, quadν ∈ [0, δ), is an analytic germ (i.e. germ of an analytic family, analytic in x ∈ (−d, d) and in ν ∈ [0, δ)). Theorem 3.10 (Chebyshev system for generic cases). Let η(2ε, ν) := θ−xν 1 ◦ g−1 ν (2ε), ε ∼ 0, (3.19) where gν := id − fν . In the variable η ≥ 0, the continuous length η 7→ ℓc(Tε,ν) admits a uniform asymptotic expansion in the system {I(h(ν), kν(η))η, I(h(ν), kν(η))η2, I(h(ν), kν(η))η3, . . .}, (3.20) as η → 0, where I(ν, η) is as given in (3.4), which becomes Chebyshev after division by the common term I(h(ν), kν(η))η. For M > 0, there exist δ, d > 0 such that the common term I(h(ν), kν(η)) > M , for η ∈ [0, d) and ν ∈ [0, δ). EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 11 More precisely, the expansion is ℓc(Tε,ν) = I(h(ν), kν(η)) · gν(η + xν1) = ( 1− e − 2 √ h(ν) 1−ρ(h(ν)) √ h(ν) ) I(h(ν), kν(η))η + c2(ν) · I(h(ν), kν(η))η2 + oν(I(h(ν), kν(η))η 2), η → 0. (3.21) Here, c2(0) ̸= 0, and notation oν(.) means that the limit is uniform in ν ∈ [0, δ). The following lemma is used in the proof of Theorem 3.10. Lemma 3.11. There is no constant term in the expansions of gν and gmod h(ν), at x ν 1 and √ h(ν), respectively. The coefficient of the linear term in the expansion of gν at xν1 is the same as the coefficient of the linear term in the expansion of gmod h(ν) at √ h(ν). Moreover, the coefficient of the quadratic term in the expansion of gν at xν1 at the bifurcation value ν = 0 is nonzero. Proof. Since φ′ 0(0) ̸= 0, due to the continuity of (x, ν) 7→ φ′ ν(x), it follows that φ ′ ν(x ν 1) ̸= 0, for ν sufficiently small. Therefore, the following expansion holds: φν(x) = √ h(ν) + C(ν)(x− xν1) + oν(x− xν1), C(ν) ̸= 0. (3.22) It follows by (2.3) and (3.22) that f ′ν(x ν 1) = (φ−1 ν )′(fmod h(ν) ◦ φν)(x ν 1)(f mod h(ν) ) ′(φν(x ν 1)) · φ′ ν(x ν 1) = 1 φ′ ν(x ν 1) (fmod h(ν) ) ′( √ h(ν)) · φ′ ν(x ν 1) = (fmod h(ν) ) ′( √ h(ν)), f ′′0 (0) = φ′′ 0(0) C(0) ( (fmod 0 )′(0)− (fmod 0 )′(0)2 ) + C(0)(fmod 0 )′′(0) = C(0)(fmod 0 )′′(0) = 2C(0) ̸= 0. (3.23) The last line follows since (fmod 0 )′(0) = 1 (tangent to the identity). The fact that the expansion, i.e. the notation oν is uniform in ν ∈ [0, δ) follows from the fact that the expansion of fmod ν (x) is uniform in ν ∈ [0, δ) since it is an analytic family, and that the change of variables φν is analytic in ν ∈ (0, δ) and continuous at ν = 0, and therefore also expands uniformly in ν ∈ [0, δ). As a consequence, the family gν(η) expands uniformly in ν ∈ [0, δ), as η → 0. □ Proof of Theorem 3.10. We have ℓc(Tε,ν) = (Ψν(g −1 ν (2ε))−Ψν(x0))2ε. (3.24) Put η := θ−xν 1 ◦ g−1 ν (2ε), as in (3.19). Therefore, g−1 ν (2ε) = θxν 1 ◦ η. By (3.17), we obtain Ψν(g −1 ν (2ε)) = Ψmod h(ν) ◦ φν(θxν 1 ◦ η). (3.25) Let kν = C(ν)y + oν(y) be as defined in (3.18). We then have, by (3.4) and (3.8): Ψν(g −1 ν (2ε)) = ( Ψmod h(ν) ◦ θ√h(ν) ) (kν(η)) = I(h(ν), kν(η)) + Ψmod h(ν)(x0). (3.26) On the other hand, 2ε = gν(η+ xν1). Using Lemma 3.11 to get the first terms of the expansion of gν and inserting it together with (3.26) in (3.24), we obtain the expansion (3.21). Finally, since h(0) = 0, kν(η) = O(η), h and kν are diffeomorphisms on some positive open neighborhoods of 0, kν depends continuously on ν and I(ν, η) > M > 0, for η ∈ [0, d), ν ∈ [0, δ), the same bound holds, for (ν, η) 7→ I(h(ν), kν(η)), possibly in smaller neighborhoods. □ Note that it is more convenient in applications to consider the standard length ℓ(Tε,ν) instead of the continuous length ℓc(Tε,ν). Let G : [0,+∞) → [0,+∞) be the periodic function of period 1, on [0, 1) given by G(s) = 1− s, s ∈ (0, 1) and G(0) = 0. We have the following corollary. 12 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 Corollary 3.12 (Expansion of the standard length ℓ(Tε,ν)). Under assumptions of Theorem 3.10, the length η 7→ ℓ(Tε,ν) admits a uniform asymptotic expansion in the Chebyshev system (3.20), but with I(h(ν), kν(η)) replaced by (id +G) (I(h(ν), kν(η))) , which is also bounded away from zero, for η ∈ [0, d), d > 0, uniformly in ν ≥ 0. The asymptotic expansion is given by (3.21), up to the same modification. Proof. This follows directly using the relation between continuous τνε and discrete critical time nνε , which is its integer part nνε − τνε = G(τνε ), ν ∈ [0, δ). For details, see Subsection 2.2 and [12]. Therefore, ℓ(Tε,ν)− ℓc(Tε,ν) = G(τνε ) · 2ε, where, as in the proof of Theorem 3.10, τνε = I(h(ν), kν(η)). By Theorem 3.10 and its proof, for M > 0 there exist δ, d > 0 such that I(h(ν), kν(η)) > M , for all η ∈ [0, d) and ν ∈ [0, δ). Moreover, G is bounded inside [0, 1]. Therefore, for M > 0 there exist δ > 0 and d > 0 such that (id+G) ( I(h(ν), kν(η)) ) > M for all η ∈ [0, d) and ν ∈ [0, δ). □ Proof of Theorem 1.4. After the change of variables η := θ−xν 1 ◦ g−1 ν (2ε), we have ℓc(Tε,ν)(η) = (Ψν(η + xν1)−Ψν(x0))gν(η + xν1). Since the first factor in brackets is strictly bigger than some positive constant for all sufficiently small non-negative ν ≥ 0 and η ≥ 0 (see the proof of Theorem 3.10), the multiplicity (from the right) of the zero point 0 of ℓc(Tε,0) in the variable η in the unfolding is the same as the multiplicity (from the right) of the zero point 0 of the displacement function g0(η) in the unfolding, which corresponds to the multiplicity (from the right) of the singular point 0 of the saddle-node vector field in the unfolding (1.1). By Corollary 3.12, ℓ(Tε,ν) = ( id + G ) (Ψν(η + xν1) − Ψν(x0)) · gν(η + xν1). As above, the first factor is bounded from below by some positive constant for all sufficiently small non-negative η, ν, and the same conclusion about multiplicities follows for ℓ(Tε,ν) instead of ℓc(Tε,ν). □ 4. Precise forms of the expansions of the length function ℓc(Tε,ν) for all parameter values The expansion (3.21) in Theorem 3.10 is valid for the whole bifurcation, because of the presence of compensator variables. By Lemma 4.2 and (4.3) below, the function η given in (3.19) behaves essentially in the same way as the simpler square root compensator η̃ defined in Definition 3.3. The function η(2ε, ν) is therefore a compensator that behaves, from the qualitative point of view, asymptotically differently, as ε→ 0, depending on the case ν = 0, or ν > 0, see (3.2). In Lemma 4.2, we expand η from (3.19) in a simpler square root compensator variable η̃ and expand ℓc(Tε,ν) in a Chebyshev system in this simpler compensator variable η̃ instead of η. In Theorem 4.1 we then re-group the terms of this new expansion so that, for ν > 0, all terms in the same block merge to the same term of the asymptotic expansion in ε at the bifurcation value ν = 0. Hence, we show that confluence of singularities leads to confluence of asymptotic terms in the expansion of ε 7→ ℓc(Tε,ν), as ν → 0. In particular, from Theorem 4.1, in Subsection 4.1, we deduce the expansions of the continuous length ℓc(Tε,ν) in ε, as ε → 0, for each of the qualitatively different cases, ν > 0 and ν = 0. Theorem 4.1 is then used in Remark 4.5 for reading the formal class of the unfolding from fractal data. In Theorem 4.1, we use another compensator variable, κ(x, ν) := 1 x+ ν . (4.1) It is related to the compensator α from Definition 3.2 by the formula d dx α(x, ν) = − 1 x κ(x, ν). EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 13 Evidently, κ(x, ν) → 1 x , as ν → 0, moreover, for every δ > 0, xδκ(x, ν) → x−1+δ, uniformly, as ν → 0. Let ν 7→ h(ν), ν 7→ C(ν), ν 7→ c2(ν), ν ∈ [0, δ), be as in Theorem 3.10, and let r(ν) := 1− e − 2 √ ν 1−ρ(ν) √ ν 2C(0) , ν ∈ [0, δ). Theorem 4.1. The continuous length ℓc(Tε,ν) in the variable η̃ := η̃ ( 2ε C(0) , r 2(h(ν)) ) , as η̃ → 0, can be written in the form ℓc(Tε,ν) ∼ ( 1− e − 2 √ h(ν) 1−ρ(h(ν)) √ h(ν) ){ α(C(ν)η̃, 2 √ h(ν)) · η̃ + ρ(h(ν)) 2 ∞∑ k=0 ak(ν) [ log η̃ · η̃k+1 + log ( η̃ + 2 √ h(ν) C(ν) ) · η̃k+1 ] + ∞∑ k=1 [ ak(ν) · α ( C(ν)η̃, 2 √ h(ν) ) η̃k+1 +Nν k ( η̃, κ ( C(ν)η̃, 2 √ h(ν) ))]} + c2(ν) · { [α(C(ν)η̃, 2 √ h(ν)) · η̃2] + ρ(h(ν)) 2 ∞∑ k=0 bk(ν) [ log η̃η̃k+2 + log ( η̃ + 2 √ h(ν) C(ν) ) η̃k+2 ] + ∞∑ k=1 [ bk(ν) · α ( C(ν)η̃, 2 √ h(ν) ) η̃k+2 +Mν k+1 ( η̃, κ ( C(ν)η̃, 2 √ h(ν) ))]} . (4.2) Here, c2(0) ̸= 0, a0(ν) = 1 and β0(ν) = 1, for all ν ∈ [0, δ), and Nν k , M ν k are homogenous polynomials of degree k whose coefficients depend on ν. The expansion is written in such a form that the terms that give the same power-logarithmic asymptotic term in η̃ for ν = 0, with their respective coefficients in ν, are grouped together as a block inside square brackets. Note that, for a fixed ν > 0, each block is possibly infinite in the sense that it can be further expanded asymptotically in a convergent power-logarithmic series in η̃, as η̃ → 0. For simplicity, in Theorem 4.1 each block is written in a closed form, as a true function of η̃. Moreover, by (3.2), η̃ = √ 2 C(0)ε 1/2 for ν = 0, and η̃ expands as an integer power series in ε, for ν > 0, so complete expansions in the original variable ε→ 0, for ν = 0 and ν > 0 are given in Subsection 4.1. In the proof of Theorem 4.1 we need the following lemmas: Lemma 4.2 (Compensator variable η expressed by η̃). Let η be as defined in (3.19) for the field (1.1) and let η̃ be as in Definition 3.3. Then η(2ε, ν) = χν ( η̃ ( 2ε C(0) , r2(h(ν)) )) , ν ∈ [0, δ), where r(ν) := 1−e − 2 √ ν 1−ρ(ν) √ ν 2C(0) , and χν is an analytic germ of a real diffeomorphism tangent to the identity. Consequently, η possesses a Taylor expansion in the variable η̃ ( 2ε C(0) , r 2(h(ν)) ) , and lim (ε,ν)→(0,0) η(2ε, ν) η̃( 2ε C(0) , r 2(h(ν))) = 1. (4.3) 14 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 Proof. From (3.14) and Lemma 3.11, we write gν as gν(x) = (( 1− e − 2 √ h(ν) 1−ρ(h(ν)) √ h(ν) ) (x− xν1) + C(0)(x− xν1) 2 ) ◦ ( (x− xν1) + ∞∑ i=2 ci(ν)(x− xν1) i ) = Pν ◦ ψν ◦ θ−xν 1 (x), (4.4) where Pν(x) = ( 1− e − 2 √ h(ν) 1−ρ(h(ν)) √ h(ν) ) x+ C(0)x2, and ψν := id + ∑∞ i=2 ci(ν)x i is a germ of a real diffeomeorphism, tangent to the identity at 0. Note that 1− e − 2 √ h(ν) 1−ρ(h(ν)) √ h(ν) is the linear coefficient of the expansion of gν around its zero point xν1 , and C(0) the quadratic coefficient of the expansion of g0 around its zero point 0. The coefficients ci(ν) are explicitly determined by the above equality and the coefficients of the expansion of gν . Inverting explicitly, we obtain P−1 ν (2ε) = √ r2(h(ν)) + 2ε C(0) − r(h(ν)) = η̃ ( 2ε C(0) , r2(h(ν)) ) , where η̃ is as defined before in Definition 3.3, and r(ν) := 1−e − 2 √ ν 1−ρ(ν) √ ν 2C(0) . By (4.4), η(2ε, ν) = θ−xν 1 ◦ g−1 ν (2ε) = χν ( η̃ ( 2ε C(0) , r2(h(ν)) )) , (4.5) where χν := ψ−1 ν ∈ Diff id(R, 0) is a diffeomorphism tangent to the identity. Note that ψν is analytic, since the above equality (4.4) can equivalently be written as P−1 ν ◦ gν ◦ θxν 1 = ψν . Now, P−1 ν ◦ gν ◦ θxν 1 is an analytic germ at 0, tangent to the identity, for every ν ∈ [0, δ). Indeed, for ν = 0, P−1 0 = √ x, and it follows by the binomial expansion, since g0 is an analytic germ of multiplicity 2. For ν > 0, gν is an analytic germ tangent to the identity at xν1 , and Pν is an analytic diffeomorphism tangent to the identity at 0, so the composition P−1 ν ◦ gν ◦ θxν 1 is an analytic diffeomorphism at 0 tangent to the identity. □ Lemma 4.3 (Properties of the compensator κ). For all integer k ≥ 1, the following properties of the compensator κ defined in (4.1) hold (i) d dx ( κ(x, ν)k ) = −kκ(x, ν)k+1; (ii) dk dxk α(x, ν) = Pk+1 ( 1 x , κ(x, ν) ) , where Pk is a homogenous polynomial in two variables of degree k, with coefficients inde- pendent of ν; (iii) d dx log(x+ ν) = κ(x, ν), dk+1 dxk+1 log(x+ ν) = (−1)kk! · κ(x, ν)k+1. Moreover, for every homogenous polynomial Pk of degree k ≥ 1, it holds that Pk ( 1 x , κ(x, ν) ) → pk 1 xk pointwise, as ν → 0, where pk ∈ R. The proof of the above lemma consists of simple computations, so we omit it. EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 15 Lemma 4.4. Let I(ν, η) be as in (3.4) and let kν , C(ν), h(ν) be as defined in Theorem 3.10. Let η̃ be as in Theorem 4.1. The following expansion in η̃ holds I(h(ν), kν(η)) = α(C(ν)η̃, 2 √ h(ν)) + ρ(h(ν)) 2 ( log η̃ + log ( C(ν)η̃ + 2 √ h(ν) )) + ∞∑ k=0 Nν k ( η̃, κ ( C(ν)η̃, 2 √ h(ν) )) . Here, Nν k are homogenous polynomials of degree k, whose coefficients depend on ν, k ≥ 0. Proof. By (3.4), we have I(h(ν), kν(η)) = α(kν(η), 2 √ h(ν)) + ρ(h(ν)) 2 log ( k2ν(η) + 2 √ h(ν) · kν(η) ) −Ψmod h(ν)(x0). Recall, from Theorem 3.10, that kν(η) = C(ν)η+oν(η) is a real analytic germ of a diffeomorphism, analytic also in ν, with asymptotic expansion as η̃ → 0+ in R{ν}[[η̃]]. On the other hand, by Lemma 4.2, η = χν(η̃), where χν is a diffeomorphism tangent to the identity. Therefore, putting Kν := kν ◦ χν , we obtain kν(η) = Kν(η̃), Kν = C(ν) · id + h.o.t., where Kν is an analytic germ of a diffeomorphism, for every ν ∈ [0, δ), i.e. with asymptotic expansion as η̃ → 0+ in R{ν}[[η̃]]. We expand, using Lemma 4.3 and denoting by ∂1 the partial derivative with respect to the first variable α(kν(η), 2 √ h(ν)) = α(Kν(η̃), 2 √ h(ν)) = α(C(ν)η̃, 2 √ h(ν)) + ∂1α(C(ν)η̃, 2 √ h(ν))(Kν(η̃)− C(ν)η̃) + 1 2 ∂21α(C(ν)η̃, 2 √ h(ν))(Kν(η̃)− C(ν)η̃)2 + oν((Kν(η̃)− C(ν)η̃)2) = α(C(ν)η̃, 2 √ h(ν)) + ∞∑ k=1 Pk+1 ( 1 C(ν)η̃ , κ ( C(ν)η̃, 2 √ h(ν) )) (Kν(η̃)− C(ν)η̃)k = α(C(ν)η̃, 2 √ h(ν)) + ∞∑ k=0 Hν k ( η̃, κ ( C(ν)η̃, 2 √ h(ν) )) . Here, the coefficients of Pk do not depend on ν. The last line is obtained after re-grouping the terms triangularly, where Hν k are homogenous polynomials of degree k whose coefficients depend on ν, k ≥ 0. Furthermore, by Taylor expansion of the logarithmic term and Lemma 4.3, we obtain log ( k2ν(η) + 2 √ h(ν) · kν(η) ) = log ( K2 ν (η̃) + 2 √ h−1(ν) ·Kν(η̃) ) = log (Kν(η̃)) + log(Kν(η̃) + 2 √ h(ν)) = log(η̃) + rν(η̃) + log ( C(ν)η̃ + 2 √ h(ν) ) + κ ( C(ν)η̃, 2 √ h(ν) ) (Kν(η̃)− C(ν)η̃) + ∞∑ k=2 (−1)k−1(k − 1)! · κk ( C(ν)η̃, 2 √ h(ν) ) (Kν(η̃)− C(ν)η̃) k = log η̃ + rν(η̃) + log ( C(ν)η̃ + 2 √ h(ν) ) + ∞∑ k=3 Mν k ( η̃, κ ( C(ν)η̃, 2 √ h(ν) )) . 16 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 Here, rν is an analytic germ of diffeomorphism, with asymptotic expansion as η̃ → 0+ in R[[η̃]], for every ν ∈ [0, δ). The coefficients of the expansion are analytic germs at ν = 0, as also is Cν . Also, Mν k are homogenous two-variable polynomials of degree k with coefficients depending on ν. □ Proof of Theorem 4.1. We use the expansion (3.21) from Theorem 3.10, the fact that η = η̃ + Oν(η̃ 2), which follows from Lemma 4.2, and Lemma 4.4. The expansion follows after regrouping in a same block the terms (with their respective coefficients in ν) that merge to the same asymptotic term in η̃ for ν = 0. Note that c2(0) ̸= 0, so c2(ν) ̸= 0, for ν ∈ [0, δ), by continuity. Therefore, all terms in expansion (3.21) after c2(ν) · I(h(ν), kν(η))η2 can be factored through c2(ν). □ 4.1. Expansions in cases ν = 0 and ν > 0. We now use the expansion (4.2) from Theorem 4.1 to get expansions in ε, as ε→ 0. In the case ν = 0, η̃ = √ 2ε C(0) , and (4.2) immediately becomes ℓc(Tε,0) ∼ c2(0) C(0) η̃ + ρ(0) ∞∑ k=0 bk(0)η̃ k+2 log η̃ + ∞∑ k=1 ckη̃ k+2 = c2(0) √ 2 C(0)3/2 ε 1 2 + ρ(0) ∞∑ k=0 ckε k+2 2 log ε+ ∞∑ k=1 dkε k+2 2 , ε→ 0, ck, dk ∈ R. Note that the logarithmic terms appear only thanks to nontrivial residual invariant ρ(0) of the parabolic time-one map for ν = 0. In the case ν > 0, with the notation of Theorem 4.1, by (3.2) we have η̃ = 2ε C(0) · r(h(ν)) + o(ε) ∈ Rν{ε}. Furthermore, α(C(ν)η̃, 2 √ h(ν)) ∼ − 1 2 √ h(ν) log η̃ + 1 2 √ h(ν) log 2 √ h(ν) C(ν) + η̃Rν{η̃} ∼ − 1 2 √ h(ν) log ε+ 1 2 √ h(ν) log C(0)r(h(ν)) √ h(ν) C(ν) + εRν{ε}, log ( η̃ + 2 √ h(ν) C(ν) ) ∼ log 2 √ h(ν) C(ν) + C(ν) 2 √ h(ν) η̃ + η̃2Rν{η̃} ∼ log 2 √ h(ν) C(ν) + 2C(ν) C(0)r(h(ν)) √ h(ν) ε+ ε2Rν{ε}, κ(C(ν)η̃, 2 √ h(ν)) ∼ 1 2 √ h(ν) − C(ν) 4h(ν) η̃ + η̃2Rν{η̃} ∼ 1 2 √ h(ν) − C(ν) 2C(0)h(ν)r(h(ν)) ε+ ε2Rν{ε}, ε→ 0. (4.6) Inserting (4.6) in (4.2), we see that there are no non-integer powers of ε in the expansion, but there are additional logarithmic terms that are not related to non-zero residual invariant, coming from compensators. The monomials in the expansion are εk, k ∈ N0, and ε k log ε, k ≥ 1. More precisely, using the above calculations and proof of Corollary 3.12 for the relation between ℓ(Tε,ν) and ℓ c(Tε,ν), we have ℓ(Tε,ν) = ( 1√ h(ν) − ρ(h(ν)) C(0) ) ε(− log ε) + o(ε log ε), ε→ 0. (4.7) Note that the first term ε(− log ε) in the case ν > 0 exists even if the residual invariant ρ(ν) is zero. It is related to the hyperbolic nature of the orbit, as compared with the parabolic nature in the case ν = 0, where logarithms appear only thanks to the nonzero residual term. EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 17 Remark 4.5. This paper was concerned with reading the analytic invariant k (the multiplicity of 0 in the unfolding) of a generic analytic unfolding of vector field (1.1), (1.2) with a saddle-node singular point, from the Chebyshev system expanding the length of the ε-neighborhoods of orbits in the unfolding. It is equal to the number of terms of this Chebyshev system that disappear at the bifurcation value ν = 0. Note that also the other invariant ν 7→ ρ(ν), ν ≥ 0, can be read from ε-neighborhoods of hyperbolic orbits in the unfolding. Indeed, under the assumption that the unfolding is already prenormalized by a constant homothecy φν(x) = ax, ν ≥ 0, we may assume that φ′ 0(0) = 1, that is, that C(0) = 1. In that case, from the first term, expanded asymptotically as function of ν > 0, of the hyperbolic expansion as ε → 0 of ε 7→ ℓ(Tε,ν), we read h(ν) and then the other invariant ρ(ν) of the unfolding, as can be seen by the formula (4.7), ℓ(Tε,ν) ∼ ( 1√ h(ν) − ρ(h(ν)) ) ε(− log ε), ε→ 0, ν > 0. Note that h(ν) → 0, as ν → 0, and that ν 7→ h(ν) and ν 7→ ρ(ν) are analytic germs at ν = 0 (since the unfolding is analytic in the variable and in the parameter, so the analytic normal form (2.2) is analytic in ν). Let h(ν) = ∑ i≥1 aiν i, ai ∈ R, and ρ(ν) = ρ(0) + ∑ i≥1 biν i, bi ∈ R, be their Taylor expansions. Note that ρ(0) = limν→0 ρ(h(ν)) is the invariant of the saddle-node point for bifurcation value ν = 0. Expanding the first coefficient of (4.7) in ν, as ν → 0, 1√ h(ν) − ρ(h(ν)) we recover ai triangularly from coefficients of rational powers k − 1 2 , k ∈ N0, of ν, and, simulta- neously, bi from coefficients of integer powers of ν. In other words, the coefficient of ε(− log ε), as function of ν > 0, can be decomposed in a unique way as a sum of an analytic function and a negative square root of an analytic function. The function under the negative square root is then h(ν) and ρ(ν) is the other analytic function postcomposed by h−1(ν). Note that this reconstruction of h(ν) and ρ(ν) relies heavily on the fact that normal forms are analytic in the parameter. On the other hand, just the multiplicity k in the unfolding can be reconstructed in the same way for smooth unfoldings, if the order of smoothness is bigger than the multiplicity [6]. 5. Concluding remark: the case ν < 0 Throughout this paper, we restricted the study of the unfolding (1.1) to parameter values ν ∈ [0, δ). We explain here the reasons for this restriction. For ν > 0, there are two real singular points. The orbit of x0 > 0 accumulates at one of these singular points. If ν < 0, there are no real singular points and the real orbit passes near zero and goes through to −∞ in a finite real time. However, it can be seen that, the smaller the |ν|, the more and more iterations are defined on the real line (of order of growth |ν|−1/2). That is, as ν → 0, the density of iterates around 0 increases. Therefore, for a small ε > 0 and |ν| sufficiently small with respect to ε, it is possible to define the critical time, and, consequently, the tail, as the union of ε-neighborhoods of finitely many first iterations from x0, as long as the distance between the consecutive two points remains larger than or equal to 2ε. However, the tail will thus be defined only in a region of (ε>0, ν)-half-plane. It is a full neighborhood of the origin in the first quadrant, but not in the fourth quadrant. Additional difficulty is that, unlike the case ν > 0, for ν < 0 there are no fixed points of the time-one map on the real line to which the orbits on the real line converge. Therefore, we do not have a good choice of a point for the asymptotic expansions as in Proposition 3.4 and Theorem 3.10. For ν ≥ 0, we have calculated the critical time using the time coordinate. To have a uniform expansion, we need some continuity of the time coordinate with respect to ν (continuity of the domain of definition and of the function), that was obtained by use of the compensators. For ν > 0, we use the time coordinate defined around one (right-most) hyperbolic singular point, which extends until the left singular point, where it ramifies. Its domain and the time 18 R. HUZAK, P. MARDEŠIĆ, M. RESMAN, V. ŽUPANOVIĆ EJDE-2025/61 coordinate itself converge to the ’global’ time coordinate 1 z +ρ(0) log z as ν → 0 (see also Glutsyuk [2].) For ν > 0 the fundamental domains are annuli around singular points on the real line. For ν < 0, the real orbit of x0 > 0 lies in the passage between the two (complex) indifferent singular points with rotational linear part. Here, the natural space of orbits is a crescent-like fundamental domain with the two tips at the two complex critical points. This approach was studied by Lavaurs in [10] and resumed in [15]. In [15], the authors precise the difference between the two charts which they call Lavaurs and Glutsyuk charts. Opening of the passage between the two singular points for ν < 0 gives a mapping between the two domains: as each crescent-like fundamental domain, for ν < 0, corresponds holomorphically (by passing to the quotient) to a Riemann sphere with two marked points, a global mapping between these crescents corresponds to a global mapping of CP1 preserving two points (0 and ∞). Hence, to a linear mapping determined by one number, which is called the Lavaurs period. Note that the Lavaurs period does not have a limit as ν tends to zero. The study of the critical time for ν < 0 may be related to the concept of Lavaurs period. This research was supported by the Croatian Science Foundation grant IP-2022-10-9820. P. Mardešić, M. Resman, and V. Županović were supported by DSYREKI HORIZON - MSCA-2023- SE, and by bilateral Hubert-Curien Cogito grants 2021-22 and 2023-2024. Pavao Mardešić and Maja Resman want to express their gratitude to the Fields Institute for supporting their stay in the scope of the Thematic Program on Tame Geometry, Transseries and Applications to Analysis and Geometry 2022. References [1] K. Falconer; Fractal Geometry, Mathematical Foundations and Applications, John Wiley& Sons Ltd., Chich- ester, 1990. [2] A. A. Glutsyuk; Confluuence of singular points and the nonlinear Stokes phenomenon, Tr. Mosk. Mat. Obs. 62 (2001), 54–104. English translation in: Trans. Moscow Math. Soc. 2001, 49–95. [3] J. Guckenheimer, P. Holmes; Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer New York, 1983. [4] L. Horvat Dmitrović; Box dimension and bifurcations of one-dimensional discrete dynamical systems, Discrete and continuous dynamical systems, 32 (2012), 4; 1287-1307. [5] M. Klimeš, P. Mardešić, G. Radunović, M. Resman; Reading analytic invariants of parabolic diffeomorphisms from their orbits, Annali della scuola normale superiore di pisa-classe di scienze, 2025 [6] M. Klimeš, C. Rousseau; On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem, Qual. Theory Dyn. Syst. 19, 80 (2020), DOI: 10.1007/s12346-020-00416-y. [7] V. P. Kostov; Versal deformations of differential forms of degree α on a line, Funktsional. Anal. i Prilozhen., 18:4 (1984), 81–82; Funct. Anal. Appl., 18:4 (1984), 335–337. [8] Y. A. Kuznetsov; Elements of Applied Bifurcation Theory, second edition, Applied Mathematical Sciences, vol. 112, Springer-Verlag, New York, 1998. [9] M. Lapidus, M. van Frankenhuijsen; Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. Second edition. Springer Monographs in Mathematics. Springer, New York, 2013. xxvi+567 pp. [10] P. Lavaurs; Systèmes dynamiques holomorphes: explosion de points périodiques paraboliques, Thesis, Univer- sité de Paris-Sud, 1989. [11] P. Mardešić; Chebyshev systems and the versal unfolding of the cusp of order n, Hermann, Éditeurs des Sciences et des Arts, Paris, 1998. [12] P. Mardešić, G. Radunović, M. Resman; Fractal zeta functions of orbits of parabolic diffeomorphisms. Anal. Math. Phys. 12 (2022), no. 5, Paper No. 114, 70 pp. [13] P. Mardešić, M. Resman, J.-P. Rolin, V. Županović; Tubular neighborhoods of orbits of power-logarithmic germs. J. Dynam. Differential Equations 33 (2021), no. 1, 395–443. [14] P. Mardešić, M. Resman, V. Županović; Multiplicity of fixed points and growth of ε-neighborhoods of orbits. J. Differential Equations 253 (2012), no. 8, 2493–2514. [15] P. Mardešić, R. Roussarie, C. Rousseau; Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms. Mosc. Math. J., 4 (2004), no. 2, 455–502, 535. [16] M. El Morsalani, A. Mourtada; Degenerate and non-trivial hyperbolic 2-polycycles: Appearance of two inde- pendant Ecalle-Roussarie compensators and Khovanskii’s theory, Nonlinearity 7 (1994), 1593–1604. [17] M. Resman; ε-neighborhoods of orbits and formal classification of parabolic diffeomorphisms. Discrete Contin. Dyn. Syst. 33 (2013), no. 8, 3767–3790. EJDE-2025/61 READING MULTIPLICITY IN UNFOLDINGS 19 [18] M. Resman; ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations. Nonlinearity 27 (2014), no. 12, 3005–3029. [19] R. Roussarie; On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Mat 17, 67–101 (1986). DOI: 10.1007/BF02584827. [20] R. Roussarie; Bifurcations of planar vector fields and Hilbert’s sixteenth problem, Birkhäuser Verlag, Basel, 1998. [21] C. Tricot; Curves and fractal dimension. Translated from the 1993 French original. Springer-Verlag, New York, 1995. xiv+323 pp. [22] D. Žubrinić, V. Županović; Poincaré map in fractal analysis of spiral trajectories of planar vector fields, Bull. Belg. Math. Soc. Simon Stevin, 15 (5) (2008), 947–960. Renato Huzak Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, Diepenbeek 3590, Belgium Email address: renato.huzak@uhasselt.be Pavao Mardešić Université Bourgogne Europe, CNRS, IMB UMR 5584, 21000 Dijon, France. University of Zagreb, Faculty of Science, Department of Mathematics, Bijenička cesta 30 / Horvatovac 102a, 10000 Zagreb, Croatia Email address: pavao.mardesic@u-bourgogne.fr Maja Resman University of Zagreb, Faculty of Science, Department of Mathematics, Bijenička cesta 30 / Horvatovac 102a, 10000 Zagreb, Croatia Email address: maja.resman@math.hr Vesna Županović University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croa- tia Email address: vesna.zupanovic@fer.unizg.hr 1. Introduction 1.1. Outline of the results 1.2. Main results 2. Notation and main objects 2.1. Normal forms for the saddle-node bifurcation 2.2. Continuous-time length of the tail of orbits 3. Uniform asymptotic expansions of the length functions (T,) and c(T,) 3.1. Compensators 3.2. Model family 3.3. Generic saddle-node families 4. Precise forms of the expansions of the length function c(T,) for all parameter values 4.1. Expansions in cases =0 and >0 5. Concluding remark: the case <0 References