Adjoining germs of exponentials to R_an, exp while preserving o-minimality — Salma Kuhlmann <salma.kuhlmann@uni-konstanz.de>

Let κ be a regular uncountable cardinal. We construct non-archimedean exponential logarithmic models (R, an, exp) of cardinality κ, for T_an, exp (the elementary theory of the reals with restricted analytic functions and exponentiation), which admit a family F of 2^κ exponentials of pairwise distinct growth rates. These model exhibits the following remarkable features: 1. For each exponential exp' in F, (R, an, exp') is a model of T_an, exp and is thus o- minimal. 2. All exponentials in F agree exactly on the convex valuation ring of R. In particular, the germs of these exponentials are incompatible, in the sense that the structure (R, an, exp, exp') is no longer o-minimal.

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An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems — Renato HUZAK <renato.huzak@uhasselt.be>

It is known that the number of limit cycles of piecewise linear (PWL) systems is bounded. We show, using Hopf and jump-breaking mechanisms, that the number of (canard) limit cycles in linear regularizations of PWL systems is unbounded.

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1. Plenary

Composition of transseries, monotonicity, and analyticity — Vincenzo Mantova <v.l.mantova@leeds.ac.uk>

Transseries generalise power series by including exponential and logarithmic terms, if not more, and can be interpreted as germs of a non-standard Hardy field by composition (for instance, on surreal numbers). I'll discuss a few results that must 'obviously' be true, yet their proofs are not obvious: that composition is monotonic in both arguments, only proved by Edgar for LE-series, that it satisfies a suitable Taylor theorem and that in fact composition is 'analytic with large radius of convergence' (joint with V. Bagayoko), something which appeared before in various special forms, but not in full generality. I'll discuss briefly what I cannot prove yet (convexity!). I'll show how monotonicity and Taylor can be used to prove some fairly general normalisation results for hyperbolic transseries (joint with D. Peran, J.-P. Rolin, T. Servi).

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Cyclicity of piecewise linear centers — Rafel J. Prohens <rafel.prohens@uib.cat>

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Determining the local cycle locus of a vector field with a Hopf singularity — María Martín Vega <martinvega@imj-prg.fr>

Let $\xi$ be an analytic vector field at $0\in \mathbb{R}^3$ with a Hopf singularity, i.e. with eigenvalues $\pm i, c$ with $c\in \mathbb R$. We describe the germ of the subanalytic set $\mathcal{C}(\xi)$ defined by the union of the cycles on small neighborhoods of the singularity. We prove $\mathcal{C}(\xi)$ is the union of a finite number of surfaces or the complement of a curve of singularities of $\xi$. We also prove that the set $\mathcal{C}(\xi)$ is formally determined by any given formal normal form of $\xi$. This talk is based on a joint work with Nuria Corral and Fernando Sanz Sánchez.

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Discrete hyperbolic dynamical systems and surreal numbers — Dino Peran <dino.peran@pmfst.hr>

Determining the normal form of a map $f=\lambda z+\cdots$, where $\lambda>0$ and $\lambda\neq 1$, is a classical problem in dynamical systems. The goal is to "simplify" $f$ by finding a parabolic change of coordinates $\varphi=z+\cdots$ such that $\varphi^{-1}\circ f\circ\varphi=f_0$, where $f_0$ is a chosen normal form. This problem has been successfully solved in several settings, including analytic diffeomorphisms, various classes of real maps, Dulac maps, and logarithmic transseries. In this work, we investigate the normal form problem in the broader framework of surreal numbers. We review the main techniques used in existing normal form constructions and discuss how these methods can be extended to the surreal number setting. The study of normal forms in this context is connected with the dynamics of analytic planar vector fields because such objects arise as asymptotic expansions of Poincar\'e maps associated with hyperbolic and semi-hyperbolic polycycles of such fields. This is connected to the classical Dulac problem concerning the non-accumulation of limit cycles near polycycles of analytic planar vector fields. A deeper understanding of the formal dynamics of these asymptotic expansions may provide valuable insight into the dynamics of the corresponding Poincar\'e maps and could contribute to further progress on understanding of the Dulac problem. This is joint work in progress with V. Mantova, J.-P. Rolin, and T. Servi.

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Infinitesimal and tangential center problems for planar hamiltonian vector fields — Maria Jesus Alvarez <chus.alvarez@uib.es>

The infinitesimal center problem concerns the persistence of a center under perturbations of a planar Hamiltonian differential system. Its first-order approximation is known as the tangential center problem. In this talk, we study the relationship between these two problems for systems whose origin is a non-degenerate center. We introduce an algorithm that yields necessary conditions for the tangential center problem and explain how its solutions can be employed to investigate the infinitesimal center problem. As an illustration of the method, we present a family of cubic systems for which the tangential center problem admits a complete solution.

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Length of iterated integrals in Melnikov functions — Jessie Diana Pontigo Herrera <pontigo@ciencias.unam.mx>

Let $H\in \mathbb{R}[x,y]$, and assume that the Hamiltonian foliation $dH=0$ in $\mathbb{R}^2$ has a continuous family of cycles $\gamma(t)\subset \{H=t\}$. We consider a deformation $$ dH+\varepsilon\eta=0, $$ where $\eta$ is a polynomial 1-form and $\varepsilon$ is a small parameter. The question is then what happens to the family of cycles $\gamma(t)$ under this deformation. To study this problem, we complexify the foliations and consider the displacement map $$ \Delta(t,\varepsilon) =\varepsilon M_1(t)+\varepsilon^2 M_2(t)+\cdots, $$ where the functions $M_j(t)$ are analytic in a neighborhood of a regular value $t_0$ of $H$ and are called Melnikov functions (or Poincaré--Pontryagin functions). Depending on whether $\Delta\equiv0$ or $\Delta\not\equiv0$, the family either persists as periodic orbits or gives rise to limit cycles. In this context, the Melnikov functions provide essential information. It follows from Françoise's algorithm that if $\Delta\not\equiv0$, then the first nonzero Melnikov function $M_\mu$ can be expressed in terms of iterated integrals of length at most $\mu$. However, this bound depends explicitly on the deformation $\eta$. On the other hand, in 2018 we showed that there exists a constant $\kappa$, depending only on $H$ and on the orbit under monodromy of $\gamma(t_0)$, that bounds the length of the iterated integrals appearing in $M_\mu$. This constant was called the orbit depth. Later, however, we exhibited an example showing that the orbit depth can be infinite. This motivated us to develop new approaches for obtaining bounds on the length of the iterated integrals appearing in Melnikov functions. In this talk, I will explain the problem of bounding the length of Melnikov functions. The talk by P. Mardesic will continue this discussion and present recent joint work in this direction.

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1. Plenary

Multisummability relative to certain quasianalytic classes realted to Dulac's Problem — Patrick Speissegger <speisse@mcmaster.ca>

Using Tougeron’s characterization of multisummable series (in the positive real direction), the latter can be viewed as infinite series of convergent power series with radii of convergence shrinking to 0. In joint work with Jean-Philippe Rolin and Tamara Servi we showed that, if we replace “convergent power series” with “convergent generalized power series”, we obtain a larger class of multisummable series (again in the positive real direction). This class is shown to generate an o-minimal expansion $\mathbb{R}_{\mathcal{G}^*}$, whose expansion by the exponential function then defines the restrictions to some unbounded interval of both the Gamma and zeta functions. More recently, with Ilgwon Seo, we have been further generalizing this construction by replacing “convergent power series” with “almost regular generalized power series”. The resulting Hardy field is a first step towards filling the remaining gap in Ilyashenko’s proof of Dulac’s problem.

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Newton Diagram and Topological Invariance for $\mu$-constant Deformations of Generalized Curves — Jesus Alberto Palma Marquez <jpalma@im.unam.mx>

We prove that $\mu$-constant deformations of generalized curves; that is, non-dicritical plane holomorphic foliations with no saddle-nodes in their desingularization, are equisingular. Furthermore, under the classical convenience assumption on the Newton diagram, we show that there exists an analytic family of coordinates preserving the Newton diagram throughout the deformation. Thus, we extend both the L\^{e}--Ramanujam theorem and Oka's Newton stability to germs of plane holomorphic foliations.

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Noetherianity and Length of Melnikov Functions — mardesic pavao <mardesic.pavao@gmail.com>

We study foliations in $\mathbb{C}^2$ given by polynomial deformations of the form $dH+\epsilon \eta=0$, with $\gamma(t)\subset H^{-1}(t)$ a family of cycles. The Poincaré first return map is of the form $P(t)=t+\sum_j \epsilon^j M_j^\gamma(t).$ The functions $M_j^\gamma$ are called Melnikov functions and are given by iterated integrals of orbit length at most $j$. This length is a measure of the complexity of Melnikov functions. We show that, for each $k\in\mathbb{N}$, there exists a universal Noetherianity index $n_{ H,\gamma}(k)$, independent of the deformation $\eta$, such that, if $M_j^\gamma=0$, for $j=1,\ldots,n_{ H,\gamma}(k)$, then $M_j^\gamma$ is of orbit length $j-k$, for any Melnikov function $M_j^\gamma$. In order to prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt-Raudenbush differential algebra theorem. We calculate the universal Noetherianity index $n_{H,\gamma}(k)$ in various nontrivial examples. The presented work is a recent work which is a continuation of the work to be presented here by J. Pontigo-Herrera.

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Normal forms for planar homoclinic 1:1 saddle loops — Loïc TEYSSIER <teyssier@math.unistra.fr>

We solve the embedding problem for Poincaré maps appearing in foliations on abstract complex surfaces near 1-polycycles corresponding to homoclinic connections of a 1:1 saddle point. We particularly prove that every such foliation is biholomorphic to a foliated neighborhood of some unique model saddle-loop in $\mathbb{C}^{2}$, defined in a neighborhood of an explicit singular elliptic curve.

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O-minimality of some almost regular multisummable germs — Ilgwon Seo <seoi@mcmaster.ca>

The main goal of this project is to establish the o-minimality of an algebra containing multisummable functions and almost regular germs. An o-minimal structure is a framework for studying sets and functions with tame geometric behavior: in particular, every one-dimensional definable set is a finite union of points and intervals. This finiteness property leads to various uniform boundedness results and is a central source of tameness. Roughly speaking, the proof of o-minimality proceeds in two steps. The first is to construct a quasianalytic algebra of generalized variables. The second is to identify a suitable class of power series with coefficients in this algebra that remains stable under the operations needed in the construction. In this talk, I will describe the current progress of the project and explain the main ideas behind these two steps.

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On the number of normally hyperbolic limit Tori in 3D polynomial vector fields — Lucas Arakaki <lucas.queiroz@unesp.br>

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the maximum number $N(m)$ of limit tori that can occur in spatial polynomial vector fields of degree $m$. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$, if finite, increases strictly with $m$. More precisely, we prove that $N_h(m+1)\geq N_h(m)+1$. Our proof relies on the torus bifurcation phenomenon observed in spatial vector fields near Hopf-Zero equilibria. While conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of a torus bifurcation assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach circumvents the need for intricate computations involving higher-order normal forms.

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Rigidity of saddle loops — Maja Resman <mresman@math.hr>

We define an abstract complex saddle loop in $\mathbb C^2$ as a pair $(F,R)$ of a hyperbolic normalized saddle foliation $F$ with a corner Dulac map D and a regular map $R\in\mathrm{Diff}(\mathbb C,0)$. Up to an appropriate equivalence relation that corresponds to different determinations of complex Dulac and to transversal changes, the first return map is given by $F=RD$ on the universal cover of the standard quadratic domain. We show that such Poincar\` e maps are rigid, in the sense that their non-ramified formal conjugacy implies the analytic conjugacy.

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Strongly linear algebra and topological methods for algebraic problems — Vincent Bagayoko <bagayoko@imj-prg.fr>

Strongly linear algebra is an enrichment of linear algebra that allows infinite sums of a formal flavor. I will give some applications of this approach, which can be seen as extensions of topological methods, to the algebra of generalised formal series (such as transseries or multivariate formal series in commuting or non-commuting variables), and operators on these structures. The first application is a formal version of the Lie correspondence that applies to objects that are "formally nilpotent" without being nilpotent or topologically nilpotent. The second (related) application is a general result for treating the problem of normalisation of formal vector fields using asymptotic differential algebra. _This will be based on joint work with Lothar Sebastian Krapp, Salma Kuhlmann, Daniel Panazzolo, Michele Serra, and Vincenzo Mantova._

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The First Example of a Completely Integrable System with an A _2 Singularity — Gabriela Gutierrez <gabrielajgg4@gmail.com>

Completely integrable systems are Hamiltonian systems with “enough” first integrals and were originally introduced to model the phase spaces of mechanical systems with symmetries. Although the local structure of these systems is well understood, their global structure is far from being understood, particularly in dimensions greater than or equal to six. In this talk, I will give a geometric introduction to completely integrable systems and present a six-dimensional system for which we have established the existence of an A_2 singularity, a type of singularity that had not been observed before in this setting. I will give a complete description of the topology of its singular fibers and explain how Hamiltonian monodromy can be studied for this system. I will conclude with some perspectives arising from this work, which is joint with K. Efstathiou, P. Mardešić, and D. Sugny.

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Trajectories of vector fields asymptotic to formal invariant curves — Fernando Sanz Sánchez <fsanz@uva.es>

In this talk we present the following result obtained jointly with O. Le Gal, that generalizes to any dimension a result by F. Dumortier and P. Bonckaert in 1986 concerning the realizability of formal invariant curves of three-dimensional vector fields: Let $\xi$ be a $C^{\infty}$ vector field at $(\mathbb R^{n},0)$ and suppose that it has a formal invariant curve $\Gamma$. Then, as soon as the Taylor expansion of $\xi$ is not identically zero along $\Gamma$ (a necessary condition), there is a trajectory $\gamma\subset \mathbb R^{n}$ of $\xi$ which has $\Gamma$ as an asymptotic expansion at the origin. In fact, we realize the family of all trajectories which are asymptotic to $\Gamma$: we construct an invariant $C^0$ manifold in some open horn around $\Gamma$, entirely composed of asymptotic trajectories, and containing the germ of any such trajectory. Furthermore, if $\xi$ is analytic, we prove that there exists a trajectory $\gamma$ asymptotic to $\Gamma$ which is, moreover, non-oscillating with respect to subanalytic sets.

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