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.

]]>Category: Seminar at UAB

Date: 13.05.19

Time: 15:30

Additional Information:

**Abstract**

Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations $x" + f(x)x'+ x = 0$, being $f(x)$ a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.

This work has been done in collaboration with C. Pessoa and J. Donizetti

]]>Category: Seminar at UAB

Date: 18.03.19

Time: 15:30

Additional Information:

**Abstract**

The minimum entropy degree one circle maps depending on the rotation interval are known since 1988 [ALMM]. In this very technical seminar I will obtain the intertwinning of the the two extremal twist orbits of these maps when the endpoints of the interval are rational and I will show that this information characterizes completely the dynamics of such maps. In a second part I will show how to extend these maps to an arbitrary graph with a single circuit by essentially keeping the dynamics. This is a factory of particular examples with (a limited) dynamics analogous to circle dynamics of the the minimum entropy degree one circle maps.

[ALMM] Lluı́s Alsedà, Jaume Llibre, Francesc Mañosas, and Michal Misiurewicz. Lower bounds of the topological entropy for continuous maps of the circle of degree one. Nonlinearity, 1(3):463–479, 1988.

]]>Category: Seminar at UAB

Date: 11.03.19

Time: 15:30

Additional Information:

**Abstract**

We consider dynamical systems that depend on some coefficients which vary randomly following certain probability distributions. The starting point is the right election of the probability space and the distribution of the coefficients. After that we give a statistical measure of:

- The linear stability index, which is the dimension of the stable manifold associated to an equilibrium.
- The phase portraits of planar homogeneous vector fields of degrees 2, 3 and 4.

Category: Seminar at UAB

Date: 25.02.19

Time: 15:30

Additional Information:

**Abstract**

We present a brief survey of recent results on the second part of the 16th Hilbert problem for algebraic limit cycles. We put special emphasis in the circular limit cycles.

]]>Category: Seminar at UAB

Date: 18.02.19

Time: 00:00

Additional Information:

**Abstract**

A diffeomorphism of the plane is Anosov if it has a hyperbolic splitting at every point of the plane. The two known topological conjugacy classes ofsuch diffeomorphisms are linear hyperbolic automorphisms and translations (the existence of Anosov structures for plane translations was originally shown by W. White). P. Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. We prove that this claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time one map.

]]>Category: Seminar at UAB

Date: 11.02.19

Time: 15:30

Additional Information:

**Abstract**

Experimental study of a uni junction transistor (UJT) has enabled to show that this electronic component has the same features as the so-called “memristor”. So, we have used the memristor’s direct current (DC) vM–iM characteristic for modeling the UJT’s DC current–voltage characteristic. This has led us to confirm on the one hand, that the UJT is a memristor and, on the other hand, to propose a new four-dimensional autonomous dynamical system allowing to describe experimentally observed phenomena such as the transition from a limit cycle to torus breakdown.

]]>Category: Seminar at UAB

Date: 04.02.19

Time: 15:30

Additional Information:

**Abstract**

We consider polynomial generalized Liénard systems that come from second order polynomial perturbations of a linear center. For these systems we found a generic (open and dense) subset in the space of perturbations of degree $2n$, with $n\geq 1$, such that each associated perturbed generalized Liénard system has at most $2n-1$ medium limit cycles.

]]>Category: Seminar at UAB

Date: 29.01.19

Time: 12:00

Additional Information: PhD defense.]]>

Category: Seminar at UAB

Date: 28.01.19

Time: 15:30

Additional Information:

**Abstrat**

In this talk we present the new results concerning the structurally unstable quadratic differential systems of codimension one and two. In 1998, Artés, Kooij and Llibre proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincaré disc of quadratic vector fields modulo limit cycles, and, in 2018, Artés, Llibre and Rezende showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. Now, we begin to study the codimension-two quadratic systems. Combining the groups of codimension-one quadratic vector fields one to each other, we obtain ten new groups. We first consider group $AA$ obtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of group $AA$ and prove their realization. We got 34 new topologically distinct phase portraits in the Poincaré disc modulo limit cycles.

]]>Category: Seminar at UAB

Date: 21.01.19

Time: 15:30

Additional Information:

We investigate the topological attractors of some quasi-periodically forced one-dimensional maps, which are in form of\[F(\theta,x) = (\, \theta + \omega \mbox{\ mod(1)}, \, \psi(\theta,x) \,),\]where $(\theta, x) \in \mathbb{S}^1 \times \mathbb{R} $, $\omega$ is a fixed irrational real number and the function $\psi(\theta,x)$ is continuous on both $x$ and $\theta$. We present some simple but essential properties of the topological structure of their attractors, and elaborate the dynamics of two types of specific systems. The first type consists of two quasi-periodically forced increasing real maps, whose dynamics displays clearly the basic distinction between the pinched and non-pinched cases. The other type is forced S-unimodal maps. We propose the mechanism for the change of periodicity of their attractors according to the forced terms. This is based on our analysis of the block structure of topological attractors of unforced S-unimodal maps, and is substantiated by numerical evidence.

]]>Category: Seminar at UAB

Date: 14.01.19

Time: 15:30

Additional Information:

**Resum**

En aquesta xerrada explicaré alguns resultats d’un treball (encara inacabat) que estem duent a terme el David Marín i jo sobre el càlcul dels coeficients del desenvolupament asimptòtic de la funció temps de Dulac. Sense entrar massa en detalls sobre el problema general que estem abordant, vull centrar-me en dos aspectes concrets. En primer lloc, en el càlcul de l’expressió general d’aquests coeficients, on usem una mena de generalització de la transformada de Mellin. En segon lloc, en l’obtenció d’aquests coeficients pels sistemes Loud, on usem resultats de prolongació analítica de la funció hipergeomètrica. La intenció és que aquesta sigui una xerrada de pissarra, entrant (si hi ha temps) en els detalls de les proves.

]]>Category: Seminar at UAB

Date: 17.12.18

Time: 15:30

Additional Information:

**Abstract**

We consider a discrete dynamical system in the plane defined by an analytic map having a parabolic fixed point with nilpotent component. We will prove the existence of an analytic stable curve associated to such a fixed point using the parameterization method for invariant manifolds. The proof that we will present brings some advantages to the previous ones: it allows to give the result of existence as an "a posteriori theorem" and also it allows to determine an explicit algorithm to compute an approximation of the invariant curve. The talk is based on the results of the master thesis "The parameterization method for invariant curves associated to parabolic points" performed under the supervision of Ernest Fontich

]]>Category: Seminar at UAB

Date: 03.12.18

Time: 15:30

Additional Information:

**Abstract**

We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: (i) a unique limit cycle, (ii) a unique one-parameter family of periodic orbits, (iii) scrolls, (iv) invariant cylinders foliated by orbits which can be periodic or no.

]]>Category: Seminar at UAB

Date: 26.11.18

Time: 15:30

Additional Information:

**Asbract**

We will present some results for the understanding of the dynamics of non-smooth systems. In this work our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. Our approach combines several techniques such as regularization process, blowing-up method and singularperturbation theory. Some of the results generalize the regularization process introduced by Sotomayor and Teixeira. This is a joint work with Marco A. Teixeira and Paulo R. da Silva.

]]>Category: Seminar at UAB

Date: 19.11.18

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 $M(3)\geq 12$. 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 $24$ 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.

The seminar will be presented in Spanish.

]]>Category: Seminar at UAB

Date: 12.11.18

Time: 15:30

Additional Information:

**Abstract**

Ecological systems are complex dynamical systems. Modelling efforts on the dynamical stability of ecosystems have revealed that population dynamics, being highly nonlinear, can be governed by complex fluctuations. Experimental and field research has provided mounting evidence for the existence of chaotic fluctuations in species' abundances. Especially for discrete-time food chain systems. Discrete-time dynamics, mainly arising in temperate ecosystems for species with non-overlapping generations, have been largely studied to understand the dynamical outcome due to changes in relevant ecological parameters. The behaviour of many of these models is difficult to investigate in the parameter space, and most of these models have been studied mainly numerically when the dimension of the phase space is large. In this article we investigate a discrete-time food chain system considering two predators predating a single prey species. We provide a full description of the local behaviour of equilibria and stability within a volume of the parameter space containing relevant dynamics. The parameter space is build with three parameters including prey's growth rate and the predation rates. We discuss the behaviour of this model, paying special attention to the pressure of predators on the prey. The dynamics undergo the Ruelle-Takens-Newhouse route to chaos at increasing predation rates. Interestingly, we find that increasing predation directly on preys can shift the extinction of top predators to their survival, allowing an unstable persistence of the three species by means of periodic and strange chaotic attractors.

]]>Category: Seminar at UAB

Date: 05.11.18

Time: 15:30

Additional Information:

Given a field of forces in the Euclidean space which is central and attractive there always exist circular periodic solutions. In 1873 Bertrand proved the following result: among all central fields of forces in the Euclidean space there are only two exceptional cases (the harmonic oscillator and the Newtonian potential) in which all solutions close to the circular motions are also periodic. Besides the original proof, nowadays there are several methods of proofs.

In this talk we present a connection between Bertrand's theorem and the theory of planar isochronous potential centers. In particular, we are interested in the families of potentials of the type $V_{\lambda}(x)=\frac{1}{2}x^2+\lambda\Phi(x),\,x>0,$ where $\Phi\in C^2(0,+\infty)$ is a given function. We prove that there are only two isochronous families of this type and we derive a new proof of Bertrand's theorem from it. The proof is more or less direct in the analytic scenario but there are some subtleties when the central field of forces is only $C^1$.

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