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

**The International Online GSDUAB Seminari is closed until September.**

**Past talks**

**June 19, 2023, 15:00****Mattia Scomparin**(University of Padua, Italy)**Title:**Conserved currents from non local constants in relativistic scalar field theories.

**Abstract.**We call*nonlocal constants*some functions which have the property of being constant along solutions of Euler-Lagrange equations, but whose value depends on the past history of the motion itself. Nonlocal constants are part of a research line that began in 2014, when they were introduced to study classical mechanical systems providing first integrals in particular cases, Lyapunov functions, and non standard separations of variables. This talk is devoted to the problem of extending the nonlocal theory to the framework of Lagrangian*scalar field theories*. Driven by a symmetry-oriented approach, we provide a set of useful theorems giving locally-conserved currents from nonlocal constants and we prove the consistency of our results by recovering some standard Noetherian results. Applications include the real/complex nonlinear interacting theory and the real dissipative*Klein-Gordon*theory.

Recording of the talk

**May 15, 2023, 15:00****Jun Zhang**(Chengdu University of Technology, China)**Title:**Global center of polynomial Newton's system.

**Abstract.**Using toroidal compactification and desingularization, we obtain a complete characterization of the monodromy of polynomial Newton's system at infinity. This leads to a necessary and sufficient condition on global centers. We further prove that the period of orbit near infinity approaches zero monotonically, which shows the nonexistence of isochronous global centers. This Joint work with Colin Christopher and Weinian Zhang.

Recording of the talk

**April 24, 2023, 15:00****Dmitry Sinelshchikov**(**HSE University (Russia) and Instituto Biofisika (UPV/EHU, CSIC))****Title:**Integrability criteria for autonomous and non-autonomous second-order differential equations**.****Abstract.**In this talk, we consider a family of cubic, with respect to the first derivative, second-order differential equations. Particular members of the considered family often appear in various applications in mechanics, physics, biology, and so on. In addition, this family of equations is a projection of two-dimensional geodesics flow equations. We study equivalence problems for the considered family of equations and its integrable subcases, including Painleve-type equations. As equivalence transformations, we use generalized nonlocal transformations. It is demonstrated that solutions to these equivalence problems lead to new integrability criteria for the considered class of equations. For each member of the constructed equivalence classes, it is possible to obtain the general solution in the parametric form and in the case of autonomous transformations an autonomous first integral in the parametric form. Furthermore, we study the possibility of the existence of a Lax representation for equations from the constructed equivalence classes. In particular, we demonstrate that an equation from this family admits a certain quadratic rational first integral if and only if it possesses a Lax representation with the L matrix from the sl(2,C) algebra. Moreover, we show that the existence of this Lax representation, and, hence, a quadratic first integral, is equivalent to linearizability of the corresponding equation via certain nonlocal transformations. Intrinsic characterization of this subfamily of the considered family can be obtained by calculating compatibility conditions for an overdetermined system of equations on the elements of the L matrix. We explicitly find these conditions in some particular cases and also illustrate our results with several examples. Finally, we demonstrate that these results can be used for constructing integrable and superintegrable two-dimensional Riemannian metrics. In particular, we present a first example of a superintegrable metric with a linear and a transcendential first integrals.

**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**March 20, 2023, 15:00****Changjian Liu**(Sun Yat-Sen University, China)**Title: The number of limit cycles of the 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 the detection of nonzero Lyapunov coefficients**The goal of my talk is to introduce a new fractal formula for the 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**(HSE University, Russian Federation)**M. V. Demina****Title:**Invariant algebraic manifolds of ordinary differential equations**Abstract.**The talk is devoted to the problem of finding polynomials that produce invariant algebraic manifolds 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, 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 studied by 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 the 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 on 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 behavioral 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 of whether this type of response 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****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 are 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 issued from the set related to a center bifurcation of the fixed point. Moreover, several regimes of multistability are evidenced.

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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, Urbino, Italy)**Title:**Discrete dynamic models in social sciences: strategic interaction, rationality, evolution.

**Abstract.**Discrete-time dynamical systems naturally arise in economic and social modeling, 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 economics: 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, France)**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 with 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**(Universitat de les Illes Balears, Spain)

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 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, Spain)

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 the Leau-Fatou flower theorem, which 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, China)

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, 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 partial 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, France)

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, Netherlands)

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 the absence of the predator, and 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 shows richer dynamics than their analytical counterparts. As examples of this more complicated behavior, 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, Italy)

T**itle:**Fixed points and periodic points for maps that 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 that 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, Spain)**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 $k, l, m, n$ nonnegative 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, Belgium)*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, and also indicate the huge difficulties that lie ahead. We then 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 in 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, Brazil)**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 diseases 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