The talks take place on Mondays at 15:00 (BCN time, CET). You can add the calendar of the seminar to your Google Calendar by clicking here.

**Coming talks **

**March 27, 2023, 15:00****Alfonso Ruiz Herrera,**University of Oviedo, Spain**Title: Topology of attractors and periodic points.**.**Abstract. The dynamics of a dissipative and area contracting planar homeomorphism is described in terms of the attractor. This is a subset of the plane defined as the maximal compact invariant set. We prove that the coexistence of two fixed points and an $N$-cycle produces some topological complexity: the attractor cannot be arcwise connected. The proofs are based on the theory of prime ends. Joint work with Rafael Ortega**

Recording of the talk

Past talks

**March 20, 2023, 15:00****Changjian Liu,**Sun Yat-Sen University (China)**Title: The number of limit cycles of Josephson equation****Abstract. In this talk, we will study the Josephson equation $\beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi=\alpha$, which can be transformed to the following Liénard systems on the cylinder: $$\frac{dx}{d \tau}=y, \frac{dy}{d \tau}=-(\sin x-a)-(b+c \cos x)y.$$ We are concerned with the non-contractible limit cycles, which are the isolated $2\pi$-periodic solutions $y=y(x)$.**We show that these limit cycles cannot intersect the $x$-axis. Then by the transformation $y \rightarrow \frac{1}{y}$, the system becomes an Abel equation $$\frac{dy}{dx}=（b+\cos x）y^2+(\sin x-a) y^3.$$

The problem is changed to study the non-zero limit cycles of the above Abel equation. By the theory of rotated vector fields and studying the multiplicity of limit cycles, we will show that at most two non-zero limit cycles can appear and that the configurations $(2, 0)$ and $(1, 1)$ can be realized. Here we denote the configuration of the limit cycles by $(i, j)$, if $i$ limit cycles belong to the half plane $y>0$ and $j$ limit cycles belong to the half plane $y<0$.

This is a joint work with Xiangqin Yu and Hebai Chen and our work can be viewed as a step to solve the following open problem raised by A. Gasull:

Open problem: $\frac{dx}{dt}=(a_0+a_1\sin t+a_2\cos t)x^3+(b_0+b_1\sin t+b_2\cos t)x^2$ have at most three limit cycles (The trivial limit cycle $y=0$ is included).

Recording of the talk**March 6, 2023, 15:00****Renato Huzak,**Hasselt University, Belgium**Title: A new approach for detection of nonzero Lyapunov coefficients**The goal of my talk is to introduce a new fractal formula for detection of the first nonzero Lyapunov quantity in degenerate slow-fast Hopf bifurcations. It is coordinate-free and there is no need to compute derivatives of higher order.**Abstract.**Recording of the talk Slides of the talk**February 20, 2023, 15:00****M. V. Demina****,**HSE University, Russian Federation**Title:**Invariant algebraic manifolds of ordinary differential equations**Abstract.**The talk is devoted to the problem of finding polynomials thаt produce invariant algebraic manifold of a differential equation or a differential system. In detail we shall consider invariant algebraic manifolds of codimension n-1, where n is the order of the equation or the dimension of the related system. Our aim is to present a method of finding invariant algebraic manifolds in an explicit form. Further, we shall discuss a generalization of the method to other codimensions.

We shall pay some attention to applications of the theory of invariant algebraic manifolds including the problem of finding exact solutions and the integrability problem.

Recording of the talk Slides of the talk

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**February 13, 2023, 15:00****Jianfeng Huang****,**Jinan University, Guangzhou, People's Republic of China**Title:**On the study of limit cycles in piecewise smooth generalized Abel equations via a new Chebyshev criterion**Abstract.**In this topic, we first present our study of the limit cycle bifurcation of a kind of generalized Abel equation, where the coefficients are piecewise trigonometrical polynomials of degree $m$ with two zones separated by a vertical straight line. We focus on the maximum number of positive and negative limit cycles (i.e., positive and negative isolated periodic solutions) that the equation can have, and the problem that how this maximum number, denoted by $H(m)$, is affected by the location of the separation line. The main tools are the higher order analysis using the theories of Melnikov functions and a new Chebyshev criterion that we developed recently. In the second part we also show some other examples of planar polynomial differential systems which can be study applying this Chebyshev criterion.

Recording of the talk Slides of the talk

**January 30, 2023, 15:00****Xiang Zhang****,**Shanghai Jiao Tong University, China**Title:**Gevrey integrability of differential systems.**Abstract.**In this talk we first survey some known results on local integrability of analytic differential systems near a singularity, and then we introduce our recent results on local first integrals and their Gevery regularity of differential systems near an equilibrium, with emphasis in the case that one eigenvalue is zero and others are nonresonant.

Recording of the talk Slides of the talk

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**January 16, 2022, 15:00****Joan Saldaña****,**Universitat de Girona, Spain**Title:**Epidemic models with spread of awareness.**Abstract.**Since the early 2000s, there has been an increasing interest in the study of epidemic models that incorporate preventive behavioural responses in their formulation. A basic assumption in many approaches is that self-initiated individual responses are triggered by the awareness of the risk of infection. In this talk, we consider the question whether this type of responses is sufficient to prevent future flare-ups from low endemic levels if awareness can be transmitted but decays over time. Using an epidemic model, we will see that periodic solutions can arise either from a Hopf bifurcation of an endemic equilibrium, or from a saddle-node bifurcation of limit cycles when some rates in the model depend on the disease prevalence. Stochastic simulations of epidemic spreading and awareness dissemination on networks confirm these two predicted scenarios.

Slides of the talk

**December 19, 2022, 15:00****Mercè Ollé****,**Universitat Politècnica de Catalunya, Spain**Title:**Ejection/collision orbits in the RTBP: why, what and how.**Abstract.**We present some results on the so called n-ejection-collision orbits in the restricted three-body problem, both from analytical and numerical points of view. The interaction between ejection (collision) orbits and other invariant objects is also discussed.

Recording of the talk Slides of the talk**December 12, 2022, 15:00****Laura Gardini****,**University of Urbino Carlo Bo, Urbino, Italy**Title:**Border collision bifurcations in a PWL stock market model.**Abstract.**We consider a behavioral stock market model in which a market maker adjusts stock prices with respect to the orders of chartists, fundamentalists and sentiment traders. We prove that the mere presence of sentiment traders, i.e. traders who optimistically buy stocks in rising markets and pessimistically sell stocks in falling markets, compromises the stability of the fundamental equilibrium. The system is described by a two-dimensional map, piecewise linear and discontinuous. The bifurcations leading to oscillatory dynamics are analytically detected, and are related to the collision of periodic points with the discontinuity lines of the model (Border collision bifurcations). The periodicity regions associated with attracting cycles are issuing from the set related to a center bifurcation of the fixed point. Moreover, several regimes of multistability are evidenced.

Recording of the talk Slides of the talk

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**November 28, 2022, 15:00****Martha Alvarez Ramírez****,**Universidad Autónoma Metropolitana-Iztapalapa, México**Title:**Equilibrium points and their linear stability in the planar equilateral restricted four-body problem:a review and new results

**Abstract.**We consider the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration.

In this talk we will show some known results and some new ones about the existence and linear stability of the equilibrium points of this problem, which have been obtained earlier, either as relative equilibria or a central configuration of the planar restricted (3+1)-body problem.

This is a joint work with José Alejandro Zepeda Ramírez.

Recording of the talk Slides of the talk

**November 14, 2022, 15:00****Gian Italo Bischi,**University of Urbino (Italy)**Title:**Discrete dynamic models in social sciences: strategic interaction, rationality, evolution.

**Abstract.**Discrete time dynamical systems naturally arise in economic and social modelling, because changes in the state of a system occur as a consequence of decisions (event-driven time). Given a characteristic time interval, taken as a unit of time advancement, then the state at the next time period is obtained by the application of a map, i.e. a point transformation defined in a*n*-dimensional state space into itself.

In this lecture we consider two classical exemplary cases of dynamic models in economcs: the Cobweb model for price dynamics (Nicholas Kaldor, 1934) and the Cournot duopoly model (Augustine Cournot, 1838), as well some successive generalizations, such as a Cobweb with fading memory (represented by maps with a vanishing denominator, focal points and prefocal curves, leading to the creation of complex structures of the basins of attraction), nonlinear duopoly models with different boundedly rational strategies and adaptive dynamics (leading to complex attractors and basins’ boundaries) and duopolies with identical firms giving rise to chaos synchronization, riddling phenomena and on-off intermittency.

Recording of the talk Slides of the talk-
**October 24, 2022, 15:00****Jean-Pierre Françoise**(Sorbonne University)**Title:**Perturbation at infinite order of the Lotka-Volterra Double Center**Abstract:**We revisit the bifurcation theory of the Lotka-Volterra quadratic system $x' = − y − x^2 + y^2, y' = x − 2xy$, with respect to arbitrary quadratic deformations. The system has a double center, and we first compute an associated pair of Bautin ideals. We show that the deformed system can have at most two limit cycles on the finite plane, with possible distribution $(i, j)$, where $i + j \leq 2$. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two. This is a joint work with Lubomir Gavrilov.

Recording of the talk arXiv

**October 10, 2022, 15:00****Jaume Llibre**(Universitat Autònoma de Barcelona)**Title:**Continuous linear and quadratic polynomial differential systems on the $2$-dimensional torus.**Abstract:**We identify the $2$-dimensional torus $\mathbb{T}^2$ with $(\mathbb{R}/\mathbb{Z})^2$, and we study the dynamics of the continuous linear and quadratic polynomial differential systems on the torus $(\mathbb{R}/\mathbb{Z})^2$. The linear systems depend on two parameters, while the quadratic ones depend on five parameters. In particular we characterize all the local phase portraits of their equilibrium points, we study their limit cycles,... Our final objective is to obtain the global phase portraits of these differential systems. This is a joint work with Ali Bakhshalizadeh.

Recording of the talk Slides of the talk

**June 13, 2022, 15:00****Rafel Prohens**(UIB Palma)

T**itle:**Probability of occurrence of some planar random quasi-homogeneous vector fields**Abstract:**In this work we are concerned with the probability of occurrence of phase portraits in a family of planar quasi-homogeneous vector fields of quasi degree $q$, that is a natural extension of planar linear vector fields, which correspond to $q=1$. We obtain the exact values of the corresponding probabilities in terms of a simple one-variable definite integral that only depends on $q$. This integral is explicitly computable in the linear case, recovering known results, and it can be expressed in terms of either complete elliptic integrals or of generalized hypergeometric functions in the non-linear one. Moreover, it appears a remarkable phenomenon when $q$ is even: the probability to have a center is positive, in contrast with what happens in the linear case, or also when $q$ is odd, where this probability is zero.

Recording of the talk**May 30, 2022, 15:00****Lorena López Hernanz**(Universidad de Alcalá)

T**itle:**A flower theorem in dimension two**Abstract:**The local dynamics of a tangent to the identity biholomorphism in dimension one is described by Leau-Fatou flower theorem, that guarantees the existence of simply connected domains with 0 in the boundary, covering a punctured neighborhood of 0, in which the dynamics is either attracting or repelling, and where the biholomorphism is conjugated to the unit translation. We present a two-dimensional version of this result, valid when the fixed point is a non-degenerate singular point. This is a joint work with Rudy Rosas.**May 23, 2022, 15:00****André Zegeling**(Guangxi Normal University)

T**itle:**Time-to-return functions in a two-dimensional Hamiltonian system**Abstract:**In this talk I will discuss the relation between the solutions of the boundary value problem $\frac{d^2x(t)}{dt^2}+\lambda f(x(t))=0$, with $x(0)=x(1)=A\in\mathbb{R}$ and time-to-return functions $T_{n+\frac{1}{2}}(h)$, with $n\in\mathbb{N}$ which can be regarded as generalizations of the period function $T(h)$ for a period annulus in an autonomous system.

I will give a short historical overview of the methods used to study time-to-return functions. As an example to illustrate the concept, I will discuss the simplest possible case $f(u)=u(u+1)$. It is well-known that the period function $T(h)$ in this case is monotonically increasing. However, for the boundary value problem other solution types exist for which the corresponding time-to-return functions $T_{n+\frac{1}{2}}(h)$ are not monotonic. In some cases situations arise with at least one local maximum and one local minimum (in the literature referred to as S-shaped bifurcations). For these cases it is an open problem, even for the quadratic Hamiltonian, to prove that no other local maxima or minima occur.

At the end of the presentation I will give a list of open problems for other types of autonomous differential equations.

**May 2, 2022, 15:00****Álvaro Castañeda**(Universidad de Chile)

T**itle:**Global stability and injectivities in a nonautonomous framework**Abstract:**In this talk, we present a problem of nonautonomous global stability for $x' = f(t,x)$ in terms of the spectrum associated with nonuniform exponential dichotomy. Also, we introduce, for a family of maps parametrized $F_t=f(t,x)$, the concepts of partially injectivity and $\sigma$-uniformly injectivity. Finally, we show that partial injectivity follows if the above global stability problem is satisfied.**April 25, 2022, 15:00****Xavier Buff**(Institut de Mathématiques de Toulouse)

T**itle:**Spiraling domains in dimension 2**Abstract:**I will present a work in progress with Jasmin Raissy. We are studying the complex dynamics of the map $(x,y) \rightarrow (x+y^2+2x^2y,y+x^2+2y^2x)$. This polynomial map fixes the origin in $C^2$ and is tangent to the identity at the origin. We are trying to prove that the interior of the basin of attraction of the origin contains infinitely many fixed connected components. Those fixed components are closely related to the periodic trajectories in an equilateral triangular billiard.Recording of the talk**April 4, 2022, 15:00****Robert Kooij**(Delft University of Technology)

T**itle:**Limit cycles in two-dimensional predator-prey systems**Abstract:**In this talk we discuss two-dimensional predator-prey systems, in particular the generalized Gause model. This model assumes a logistic growth rate for the prey in absence of the predator, a constant death rate for the predator. Often it is assumed that the functional response, i.e. the capture rate of prey per predator, is an analytical function, such as the functional response of Holling Type II and III. The aim of this talk is to discuss the generalized Gause model for several classes of non-analytical functional response. Our main interest is the number of closed orbits of the systems under consideration. We will show that the system with a non-analytical functional response show richer dynamics than their analytical counterparts. As examples of this more complicated behaviour we mention: the co-existence of a stable equilibrium with a stable limit cycle and the co-existence of a family of closed orbits and a limit cycle.Slides of the talk**March 28, 2022, 15:00****Fabio Zanolin**(Università degli Studi di Udine)

T**itle:**Fixed points and periodic points for maps which are expansive in one direction, with applications.**Abstract:**In the first part of the talk, I will describe a geometric approach, based on the theory of topological horseshoes, in order to prove the existence of fixed points, periodic points and complex dynamics for maps which are defined in rectangular regions of the plane and which are expansive in one direction.

In the second part of the talk, I will present some applications to periodically perturbed planar systems in which the unperturbed system has a center (local or global) with an associated monotone period map. In this case, the results can be seen as a natural variant of the classical Poincaré-Birkhoff fixed point theorem, in which the usual twist condition on the angular coordinate is paired with a compression/expansion condition on the radial component.Recording of the talk**March 21, 2022, 15:00****María Jesús Álvarez**(Universitat de les Illes Balears)**T****itle:**Uniqueness of limit cycles of complex differential equations with two monomials.**Abstract:**We prove that the family of complex differential equations with two monomials, $z' = az^k \bar{z}^l + bz^m z\bar{z}^n,$ with $k, l, m, n$ non negative integers and $a, b \in C$, has at most one limit cycle. Moreover, we characterize when it exists and prove that it is hyperbolic. For this family we also solve the center-focus problem.Recording of the talk**March 7, 2022, 15:00**(Hasselt University)*Peter De Maesschalk*

T**itle:**Finite cyclicity of singular transitory canards and application to Smale's 13th problem**Abstract:**Smale's 13th problem is about finding a uniform upper bound on the number of limit cycles of classical Lienard systems $(x',y')=(y-F(x),-x)$. We present an overview of the method of dealing with this problem by means of geometric singular perturbation theory, show the progress that was already made but also indicate the huge difficulties that lie ahead. We thnn proceed to talk about work in progress aimed at providing a slow-fast version of a 2008 result by Dumortier and Caubergh, i.e. we target uniform finite cyclicity of unbounded slow-fast cycles.

At the moment it is joint work with my PhD student Melvin Yeung.

Recording of the talk-
**February 28, 2022, 15:00**(Centre de Recerca Matemàtica)**Josep Sardanyés****Title:**How intrinsic noise shapes transients and scaling laws close to saddle-node bifurcations**Abstract:**In this talk we will investigate the role of demographic (intrinsic) noise in delayed transitions occurring at the vicinity of saddle-node bifurcations. We have addressed this question by means of analytical work on simple stochastic models, extensive numerical simulations, and a Hamiltonian approach explaining the changes in the shape of the scaling functions for transient times.

Recording of the talk **February 21, 2022, 15:00**(Universidade de São Paulo)**Tiago Carvalho**Studying models of HIV, cancer and SarsCov-19 using piecewise smooth vector fields.

Title:**Abstract:**We will study, briefly, the qualitative behavior and the dynamics of recent models of piecewise smooth vector fields used to model disease like HIV, cancer and SarsCov-19. When it is possible, the sliding vector field will be presented, limit cycles will be exhibited and tangential sliding vector fields will be defined at the simultaneous occurrence of tangencies of both vector fields along a subset of the switching manifold.Recording of the talk**February 14, 2022, 15:00**(Universitat de Barcelona)**Xavier Jarque**Beyond the secant method on the plane.

Title:**Abstract:**In this talk, we will study the secant method as a plane dynamical system. After a brief introduction, we will present new results on the secant map "at infinity" (using homogeneous coordinates in $RP^2$), and on a model which allows us to explain some interesting invariant objects of the dynamical plane.

Recording of the talk