Past Seminars at UABHome page of "Grup de Sistemes Dinàmics de la UAB"http://www.gsd.uab.es/index.php2020-02-23T20:19:21ZJoomla! 1.5 - Open Source Content ManagementOn the proof of the 16th Hilbert problem 2019-12-18T12:12:16Z2019-12-18T12:12:16Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1620%3A16thhilbertproofannouncement&option=com_simplecalendar&Itemid=43&lang=enEvent name: On the proof of the 16th Hilbert problem <br />Place: Centre de Recerca Matemàtica<br />Category: Seminar at UAB<br />Date: 10.02.20<br />Time: <br />Additional Information: <p>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"</p>Event name: On the proof of the 16th Hilbert problem <br />Place: Centre de Recerca Matemàtica<br />Category: Seminar at UAB<br />Date: 10.02.20<br />Time: <br />Additional Information: <p>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"</p>Recent Trends in Nonlinear Science 20202019-12-18T12:06:53Z2019-12-18T12:06:53Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1619%3Artns2020&option=com_simplecalendar&Itemid=43&lang=enEvent name: Recent Trends in Nonlinear Science 2020<br />Place: Centre de Recerca Matemàtica<br />Category: Seminar at UAB<br />Date: 03.02.20<br />Time: <br />Additional Information: <p>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.</p>Event name: Recent Trends in Nonlinear Science 2020<br />Place: Centre de Recerca Matemàtica<br />Category: Seminar at UAB<br />Date: 03.02.20<br />Time: <br />Additional Information: <p>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.</p>New lower bounds for the local Hilbert number for cubics systems and piecewise systems2019-12-19T08:37:53Z2019-12-19T08:37:53Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1624%3Alocal-cyclicity-in-lower-degree-piecewise-polynomial-vector-fiel&option=com_simplecalendar&Itemid=43&lang=enEvent name: New lower bounds for the local Hilbert number for cubics systems and piecewise systems<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 27.01.20<br />Time: 15:30<br />Additional Information: <p><strong>Abstract.</strong></p>
<p>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.</p>
<p> </p>
<p style="text-indent: 0px; margin: 0px;">This is a joint work with Joan Torregrosa.</p>
<p style="text-indent: 0px; margin: 0px;"> </p>
<p style="text-indent: 0px; margin: 0px;">This seminar will be presented in Spanish.</p>Event name: New lower bounds for the local Hilbert number for cubics systems and piecewise systems<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 27.01.20<br />Time: 15:30<br />Additional Information: <p><strong>Abstract.</strong></p>
<p>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.</p>
<p> </p>
<p style="text-indent: 0px; margin: 0px;">This is a joint work with Joan Torregrosa.</p>
<p style="text-indent: 0px; margin: 0px;"> </p>
<p style="text-indent: 0px; margin: 0px;">This seminar will be presented in Spanish.</p>The theory of the weak centers2019-12-19T09:58:10Z2019-12-19T09:58:10Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1626%3Athe-theory-of-the-weak-centers&option=com_simplecalendar&Itemid=43&lang=enEvent name: The theory of the weak centers<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 20.01.20<br />Time: 15:30<br />Additional Information: <p><strong>Abstract:</strong></p>
<p><strong></strong><br />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 $H=(x^2 +y^2)/2(1+h.o.t.).$</p>
<p>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 <em>quasi Darboux integrable</em>. Finally we established the relations between a particular case of weak centers and reversibility.</p>Event name: The theory of the weak centers<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 20.01.20<br />Time: 15:30<br />Additional Information: <p><strong>Abstract:</strong></p>
<p><strong></strong><br />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 $H=(x^2 +y^2)/2(1+h.o.t.).$</p>
<p>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 <em>quasi Darboux integrable</em>. Finally we established the relations between a particular case of weak centers and reversibility.</p>Limit cycles of small amplitude in polynomial and piecewise polynomial planar vector fields 2019-12-18T11:52:30Z2019-12-18T11:52:30Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1618%3Aphddefenseluizfernandogouveia&option=com_simplecalendar&Itemid=43&lang=enEvent name: Limit cycles of small amplitude in polynomial and piecewise polynomial planar vector fields <br />Place: UAB - Sala de Graus (C1/070-1)<br />Category: Seminar at UAB<br />Date: 14.01.20<br />Time: 12:00<br />Additional Information: <p>Defense of his PhD.</p>
<p>The advisor is Joan Torregrosa (UAB).</p>Event name: Limit cycles of small amplitude in polynomial and piecewise polynomial planar vector fields <br />Place: UAB - Sala de Graus (C1/070-1)<br />Category: Seminar at UAB<br />Date: 14.01.20<br />Time: 12:00<br />Additional Information: <p>Defense of his PhD.</p>
<p>The advisor is Joan Torregrosa (UAB).</p>Chaos: why, where and how much2019-11-20T14:48:07Z2019-11-20T14:48:07Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1613%3Atba&option=com_simplecalendar&Itemid=43&lang=enEvent name: Chaos: why, where and how much<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 16.12.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract:</strong></p>
<p>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.<br /> <br /> 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.<br /> <br /> 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.<br /> <br /> 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.</p>Event name: Chaos: why, where and how much<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 16.12.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract:</strong></p>
<p>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.<br /> <br /> 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.<br /> <br /> 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.<br /> <br /> 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.</p>On the secant map as a plane dynamical system2019-11-20T14:47:03Z2019-11-20T14:47:03Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1612%3Atba&option=com_simplecalendar&Itemid=43&lang=enEvent name: On the secant map as a plane dynamical system<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 09.12.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract:</strong></p>
<p>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.</p>Event name: On the secant map as a plane dynamical system<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 09.12.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract:</strong></p>
<p>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.</p>Qualitative theory of differential equations in the plane and in the space, with emphasis on the center-focus problema and on the Lotka-Volterra systems2019-11-07T17:35:20Z2019-11-07T17:35:20Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1608%3Aqualitative-theory-of-differential-equations-in-the-plane-and-in&option=com_simplecalendar&Itemid=43&lang=enEvent name: Qualitative theory of differential equations in the plane and in the space, with emphasis on the center-focus problema and on the Lotka-Volterra systems<br />Place: UAB - Sala de Graus (C1/070-1)<br />Category: Seminar at UAB<br />Date: 26.11.19<br />Time: 12:00<br />Additional Information: <p>Defense of his PhD</p>
<p>Advisor Jaume Llibre (UAB)</p>Event name: Qualitative theory of differential equations in the plane and in the space, with emphasis on the center-focus problema and on the Lotka-Volterra systems<br />Place: UAB - Sala de Graus (C1/070-1)<br />Category: Seminar at UAB<br />Date: 26.11.19<br />Time: 12:00<br />Additional Information: <p>Defense of his PhD</p>
<p>Advisor Jaume Llibre (UAB)</p>An algorithm for rotation interval computation2019-11-06T10:59:22Z2019-11-06T10:59:22Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1607%3Aan-algorithm-for-rotation-interval-computation&option=com_simplecalendar&Itemid=43&lang=enEvent name: An algorithm for rotation interval computation<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 25.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>
<p>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.</p>Event name: An algorithm for rotation interval computation<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 25.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>
<p>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.</p>Birth of limit cycles bifurcating from a nonsmooth center2019-09-16T08:39:05Z2019-09-16T08:39:05Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1593%3Atba&option=com_simplecalendar&Itemid=43&lang=enEvent name: Birth of limit cycles bifurcating from a nonsmooth center<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 18.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>Event name: Birth of limit cycles bifurcating from a nonsmooth center<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 18.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>The quadratic and cubic polynomial differential systems in R^2 and the Euler Jacobi formula2019-11-05T08:02:32Z2019-11-05T08:02:32Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1606%3Athe-quadratic-and-cubic-polynomial-differential-system-in-r2-and&option=com_simplecalendar&Itemid=43&lang=enEvent name: The quadratic and cubic polynomial differential systems in R^2 and the Euler Jacobi formula<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 11.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.<br />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.</p>
<p><br />This is a joint work with Claudia Valls.</p>Event name: The quadratic and cubic polynomial differential systems in R^2 and the Euler Jacobi formula<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 11.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.<br />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.</p>
<p><br />This is a joint work with Claudia Valls.</p>New lower bounds of the number of critical periods in reversible centers2019-11-03T19:57:21Z2019-11-03T19:57:21Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1604%3Anew-lower-bounds-of-the-number-of-critical-periods-in-reversible&option=com_simplecalendar&Itemid=43&lang=enEvent name: New lower bounds of the number of critical periods in reversible centers<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 04.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p><strong>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.</strong></p>Event name: New lower bounds of the number of critical periods in reversible centers<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 04.11.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p><strong>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.</strong></p>Rigid centers on center manifolds of vector fields in the R32019-10-23T09:19:01Z2019-10-23T09:19:01Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1602%3Arigid-centers-on-center-manifolds-of-vector-fields-in-the-r3&option=com_simplecalendar&Itemid=43&lang=enEvent name: Rigid centers on center manifolds of vector fields in the R3<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 28.10.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<div>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.</div>
<p><strong> </strong></p>Event name: Rigid centers on center manifolds of vector fields in the R3<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 28.10.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<div>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.</div>
<p><strong> </strong></p>Tongues in degree 4 Blaschke products2019-10-14T08:36:49Z2019-10-14T08:36:49Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1600%3Atongues-for-degree-2-covering-maps-of-the-cercle&option=com_simplecalendar&Itemid=43&lang=enEvent name: Tongues in degree 4 Blaschke products<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 21.10.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
In this talk we will present the concept of tongue for degree 2 covering maps of the circle. This concept may be applied to maps such as the double standard family and generalized Blaschke products. We will analyze the properties of the tongues of a family of degree 4 generalized Blaschke products and describe how bifurcations take place near their tips. Afterwards, we will study how they may be extended beyond their natural range of definition.Event name: Tongues in degree 4 Blaschke products<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 21.10.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
In this talk we will present the concept of tongue for degree 2 covering maps of the circle. This concept may be applied to maps such as the double standard family and generalized Blaschke products. We will analyze the properties of the tongues of a family of degree 4 generalized Blaschke products and describe how bifurcations take place near their tips. Afterwards, we will study how they may be extended beyond their natural range of definition.Geometrization of piecewise homeomorphisms of the circle2019-09-16T08:31:38Z2019-09-16T08:31:38Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1592%3Ageometrization-of-piecewise-homeomorphisms-of-the-circle&option=com_simplecalendar&Itemid=43&lang=enEvent name: Geometrization of piecewise homeomorphisms of the circle<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 26.09.19<br />Time: 15:30<br />Additional Information: <p><strong><span style="color: #000000;">Abstract</span></strong></p>
<p>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.</p>Event name: Geometrization of piecewise homeomorphisms of the circle<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 26.09.19<br />Time: 15:30<br />Additional Information: <p><strong><span style="color: #000000;">Abstract</span></strong></p>
<p>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.</p>Dulac - Cherkas method for detecting the exact number of limit cycles surrounding one or three equilibrium points of planar autonomous systems2019-06-25T04:37:56Z2019-06-25T04:37:56Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1589%3Adulac-cherkas-method-for-detecting-the-exact-number-of-limit-cyc&option=com_simplecalendar&Itemid=43&lang=enEvent name: Dulac - Cherkas method for detecting the exact number of limit cycles surrounding one or three equilibrium points of planar autonomous systems<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 01.07.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>Event name: Dulac - Cherkas method for detecting the exact number of limit cycles surrounding one or three equilibrium points of planar autonomous systems<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 01.07.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>Local invariant sets of analytic vector fields2019-05-06T13:47:31Z2019-05-06T13:47:31Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1581%3Alocal-invariant-sets-of-analytic-vector-fields&option=com_simplecalendar&Itemid=43&lang=enEvent name: Local invariant sets of analytic vector fields<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 27.05.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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</p>Event name: Local invariant sets of analytic vector fields<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 27.05.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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</p>Natural controlled invariant varieties for polynomial control problems2019-05-14T05:27:31Z2019-05-14T05:27:31Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1582%3Anatural-controlled-invariant-varieties-for-polynomial-control-pr&option=com_simplecalendar&Itemid=43&lang=enEvent name: Natural controlled invariant varieties for polynomial control problems<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 20.05.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>Event name: Natural controlled invariant varieties for polynomial control problems<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 20.05.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations2019-05-06T13:34:12Z2019-05-06T13:34:12Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1580&option=com_simplecalendar&Itemid=43&lang=enEvent name: Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 13.05.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>
<p>This work has been done in collaboration with C. Pessoa and J. Donizetti</p>Event name: Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 13.05.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>
<p>This work has been done in collaboration with C. Pessoa and J. Donizetti</p>Combinatorial dynamics of the minimum entropy degree one circle maps depending on the rotation interval and its use as examples factory for graph maps2019-03-14T06:39:07Z2019-03-14T06:39:07Zhttp://www.gsd.uab.es/index.php?view=detail&catid=7%3Aseminar-at-uab&id=1571%3Acombinatorial-dynamics-of-the-minimum-entropy-degree-one-circle-&option=com_simplecalendar&Itemid=43&lang=enEvent name: Combinatorial dynamics of the minimum entropy degree one circle maps depending on the rotation interval and its use as examples factory for graph maps<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 18.03.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>
<p>[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.</p>Event name: Combinatorial dynamics of the minimum entropy degree one circle maps depending on the rotation interval and its use as examples factory for graph maps<br />Place: UAB - Dept. Matemàtiques (C1/-128)<br />Category: Seminar at UAB<br />Date: 18.03.19<br />Time: 15:30<br />Additional Information: <p><strong>Abstract</strong></p>
<p>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.</p>
<p>[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.</p>