Category: Seminar at UAB

Date: 09.03.20

Time: 15:30

Additional Information:

**Abstract:**

We investigate the root finding algorithm given by the secant method applied to a real polynomial $p$ as a discrete dynamical system defined in $\mathbb{R}^2$. We study the main dynamical properties associated to the basins of attraction of the roots of $p$.

]]>Category: Seminar at UAB

Date: 02.03.20

Time: 15:30

Additional Information:

**Abstract:**

In 1960 Markus and Yamabe made the following conjecture: If a $C^1$ differential system $\dot {\bf x}=F({\bf x})$ in $\R^n$ has a unique equilibrium point and the Jacobian matrix of $F({\bf x})$ for all ${\bf x}\in \R^n$ has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. Until 1997 we do not have the complete answer to this conjecture. It is true in $\R^2$, but it is false in $\R^n$ for all $n>2$.

Here we extend the conjecture of Markus and Yamabe to continuous and discontinuous piecewise linear differential systems in $\R^n$ separated by a hyperplane, and we prove that for the continuous piecewise linear differential systems it is true in $\R^2$, but it is false in $\R^n$ for all $n>2$. But for discontinuous piecewise linear differential systems it is false in $\R^n$ for all $n\ge 2$.

This is a joint work with Xiang Zhang.

]]>Category: Seminar at UAB

Date: 10.02.20

Time: 15:30

Additional Information:

During this week there will be no seminar because Jaume Llibre and Pablo Pedregal organize at Centre de Recerca Matemàtica (Auditori) a working seminar entitled "On the proof of 16th Hilbert problem"

]]>Category: Seminar at UAB

Date: 03.02.20

Time: 15:30

Additional Information:

During this week there will be no seminar because at Centre de Recerca Matemàtica it will be the advanced course "Recent Trends in Nonlinear Science 2020" organized by CRM and the Spanish Dynamical Systems Network.

]]>Category: Seminar at UAB

Date: 27.01.20

Time: 15:30

Additional Information:

**Abstract.**

Let $\mathcal{P}_n$ the class of polynomial differential systems of degree $n.$ In this class, we are interested in the isolated periodic orbits, the so called limit cycles, surrounding only one equilibrium point of monodromic type. For the unperturbed system, the origin is always an equilibrium point of nondegenerate center-focus type. We define $M(n)$ as the maximum number of small limit cycles bifurcating from the origin via a degenerate Hopf bifurcation. We will prove that $M(5)\geq 33$. We will also consider this problem in the class of piecewise polynomial systems defined in two zones. Here, we are interested in the small crossing limit cycles surrounding only one equilibrium point or an sliding segment. When the separation curve is a straight line, we provide a piecewise cubic system exhibiting at least $26$ small crossing limit cycles. All of them nested surrounding only one equilibrium point, in fact an sliding segment. The computations use a parallelization algorithm.

This is a joint work with Joan Torregrosa.

This seminar will be presented in Spanish.

]]>Category: Seminar at UAB

Date: 20.01.20

Time: 15:30

Additional Information:

**Abstract:**

This talk is dedicated to study the subclass of linear type centers which we call the {\it weak centers}. We say that the linear type center is a weak center if the Poincare-Liapunov first integral can be written as $H=(x^2 +y^2)/2(1+h.o.t.).$

We have characterized the expression of an analytic (polynomial) differential systems having a weak center. We prove that the uniform and holomorphic centers are weak centers. Moreover we give the conjecture that all the weak centers are *quasi Darboux integrable*. Finally we established the relations between a particular case of weak centers and reversibility.

Category: Seminar at UAB

Date: 14.01.20

Time: 12:00

Additional Information:

Defense of his PhD.

The advisor is Joan Torregrosa (UAB).

]]>Category: Seminar at UAB

Date: 16.12.19

Time: 15:30

Additional Information:

**Abstract:**

We consider the chaos which appears in real problems assuming:1) determinism; 2) a mathematical model; 3) good agreement between physical experiments and predictions coming from the model.

After looking at some examples coming from a simple nonlinear model, with or without dissipation and forcing, we introduce the requirements that we consider for chaos: Sensitive Dependence on Initial Conditions and Topological Transitivity.

Then we study the reasons why chaos appears, looking at the objects that appear in the phase space. Next we discuss where these objects appear and which facts lead to creation/destruction/coupling of chaos.

Finally we comment on different ways to measure the amount of chaos, some additional facts and dynamical consequences and a few of the many open problems.

Category: Seminar at UAB

Date: 09.12.19

Time: 15:30

Additional Information:

**Abstract:**

The plane dynamical system generated by the secant method applied to real polynomials ia a particular example to a large family of differentiable plane dynamical systems given by rational components. In this seminar we will present some results we have found about the shape and distribution of the atrracting basins, as well as other considerations. This is a joint work with Antonio Garijo.

]]>Category: Seminar at UAB

Date: 26.11.19

Time: 12:00

Additional Information:

Defense of his PhD

Advisor Jaume Llibre (UAB)

]]>Category: Seminar at UAB

Date: 25.11.19

Time: 15:30

Additional Information:

**Abstract**

We will present a robust algorithm to compute the rotation number of a degree one map that has a constant section. The inputs are the interval in which the function is constant and the maximum number of iterates.

We will show how the algorithm performs against some other algorithms used for maps that fall under the same hypothesis. Finally we will plot the Arnol'd Tongues and rotation intervals for a variety of functions.

]]>Category: Seminar at UAB

Date: 18.11.19

Time: 15:30

Additional Information:

**Abstract**

We will perform a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate $\Sigma$-center). Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from the nondegenerate $\Sigma$-center that have k hyperbolic limit cycles bifurcating from it. Moreover, we also exhibit families of piecewise smooth vector fields of codimension k also emerging from the nondegenerate $\Sigma$-center. As a consequence we prove that it has infinite codimension. It is a joint work with Tiago de Carvalho and Marco Antonio Teixeira.

]]>Category: Seminar at UAB

Date: 11.11.19

Time: 15:30

Additional Information:

**Abstract**

The configuration of the singular points together with their topological indices for a planar quadratic and cubic polynomial differential system when this system has the maximum number of finite singular points can be studied using the Euler-Jacobi formula.

First we recall the result for the quadratic polynomial differential systems, i.e. we recall the classical Berlinskii’s Theorem, and after we will present the classification of the mentioned configurations for all the planar cubic polynomial differential systems.

This is a joint work with Claudia Valls.

Category: Seminar at UAB

Date: 04.11.19

Time: 15:30

Additional Information:

**Abstract**

**En esta sesión introduciremos la noción de función de período para un sistema de ecuaciones diferenciales en el plano que presenta un centro en el origen, y presentaremos el concepto de período crítico. En analogía con el 16º Problema de Hilbert, consideramos el problema de hallar el máximo número de períodos críticos que bifurcan de un centro isócrono al añadir una perturbación que mantiene la propiedad de centro. Con el trabajo que presentaremos hemos hallado n^2/2+n/2-2 períodos críticos para sistemas de grado n con 2<n<17. Para los casos cúbico y cuártico (n=3,4), usamos una técnica que mejora este resultado y, por lo que nosotros sabemos, obtiene las mejores cotas conocidas hasta la fecha.**

Category: Seminar at UAB

Date: 28.10.19

Time: 15:30

Additional Information:

**Abstract**

In this talk we will introduce the notion of rigid centers on center manifolds of vector fields in the three-dimensional space and classify some families of polynomial vector fields that have this kind of center.

** **

Category: Seminar at UAB

Date: 21.10.19

Time: 15:30

Additional Information:

**Abstract**

Category: Seminar at UAB

Date: 26.09.19

Time: 15:30

Additional Information:

**Abstract**

In 1979 R. Bowen and C. Series gave a beautiful construction of a dynamical system associated to a Fuschian group. They constructed a particular map that was a piecewise diffeomorphism of the circle from the action of the group on the hyperbolic 2 space. More recently I generalized this idea to construct some piecewise homeomorphisms of the circle, for surface groups given by some particular group presentations.In this talk I’ll try to explain the reverse the question: How to construct a group from a piecewise homeomorphism on the circle and verify it is indeed a surface group? This is a geometrization problem and I will describe a construction for a class of such piecewise homeomorphisms of the circle.

]]>Category: Seminar at UAB

Date: 01.07.19

Time: 15:30

Additional Information:

**Abstract**

For smooth autonomous systems the problem of precise non-local estimation of the limit cycles number in a simply-connected domain of a real phase plane containing one or three equilibrium points with a total Poincaré index +1 is considered. To solve this problem, we present new approaches that are based on a sequential two-step construction of the Dulac or Dulac-Cherkas which provide the closed transversal curves decomposing the simply-connected domain in simply-connected subdomains, doubly-connected subdomains, and possibly a three-connected subdomain. As an additional approach, we consider a generalization of the Dulac-Cherkas method, where the traditional requirement can be weakened and replaced by the condition of the transversality of the curves on which the divergence vanishes. The efficiency of the developed approaches is demonstrated by several examples of some classes of the polynomial systems, for which it is proved that there exists a limit cycle in each of the doubly-connected subdomains and two limit cycles in the three-connected subdomain.

]]>Category: Seminar at UAB

Date: 27.05.19

Time: 15:30

Additional Information:

**Abstract**

In the theory of autonomous ordinary differential equations invariant sets play an important role. In particular, we are interested in local analytic invariant sets near stationary points. Invariant sets of a differential equation correspond to invariant ideals of the associated derivation in the power series algebra. Poincaré-Dulac normal forms are very useful in studying semi-invariants and invariant ideals. We prove that an invariant ideal with respect to a vector field, given in normal form, is already invariant with respect to the semi-simple part of its Jacobian at the stationary point. This generalizes a known result about semi-invariants, that is invariantsets of codimension 1. Moreover, we give a characterization of all ideals which areinvariant with respect to the semi-simple part of the Jacobian. As an application, we consider polynomial systems and we provide a sharp bound of the total degree of possible polynomial semi-invariants under some generic conditions

]]>Category: Seminar at UAB

Date: 20.05.19

Time: 15:30

Additional Information:

**Abstract**

We consider polynomially nonlinear, input-affine control systems $\dot x= f(x)+g(x)u$ with a focus on invariant and controlled invariant algebraic varieties. Specifically we introduce and discuss natural controlled invariant varieties (NCIV) with respect to a given input matrix g, i.e. varieties which are controlled invariant sets of the control problem for any choice of the drift vector f. We use tools from commutative algebra and algebraic geometry in order to characterize NCIV's, provide some constructions and and present an algorithmic method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.

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