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Bautin ideals and Taylor domination
https://ddd.uab.cat/record/118324
Yomdin, Y.Mon, 19 May 2014 17:50:50 GMThttps://ddd.uab.cat/record/1183242014Uniform methods to stablish Poincaré type linearization theorems
https://ddd.uab.cat/record/118322
We find a uniform method to establish Poincaré type linearization theorems for regular systems including classical autonomous, random and almost periodic ones via modified majorant norm methods. Wu, HaoMon, 19 May 2014 16:35:01 GMThttps://ddd.uab.cat/record/1183222014Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields
https://ddd.uab.cat/record/118321
We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress. Schlomiuk, DanaMon, 19 May 2014 16:31:48 GMThttps://ddd.uab.cat/record/1183212014Finding Hamiltonian isochronous centers by non-canonical transformations
https://ddd.uab.cat/record/118320
Starting from a class of isochronous Hamiltonian centers, we produce a new class of Hamiltonian isochronous centers by using non-canonical transformations. Sabatini, MarcoMon, 19 May 2014 16:28:04 GMThttps://ddd.uab.cat/record/1183202014A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations
https://ddd.uab.cat/record/118318
For stably dissipative Lotka{Volterra equations the dynamics on the attractor are Hamiltonian and we argue that complex dynamics can occur. We also present examples and properties of some infinite dimensional Volterra systems with applications related with stably dissipative Lotka-Volterra equations. We finish by mentioning recent contributions on the subject. Oliva, Waldyr M.Mon, 19 May 2014 16:21:26 GMThttps://ddd.uab.cat/record/1183182014Invariant tori in the lunar problem
https://ddd.uab.cat/record/118315
Theorems on the existence of invariant KAM tori are established for perturbations of Hamiltonian systems which are circle bundle ows. By averaging the perturbation over the bundle ow one obtains a Hamiltonian system on the orbit (quotient) space by a classical theorem of Reeb. A non-degenerate critical point of the system on the orbit space gives rise to a family of periodic solutions of the perturbed system. Conditions on the critical points are given which insure KAM tori for the perturbed ow. These general theorems are used to show that the near circular periodic solutions of the planar lunar problem are orbitally stable and are surrounded by KAM 2-tori. For the spatial case it is shown that there are periodic solutions of two types, either near circular equatorial, that is, the infinitesimal particle moves close to the plane of the primaries following near circular trajectories and the other family where the ifinitesimal particle moves along the axis perpendicular to the plane of the primaries following near rectilinear trajectories. We prove that the two solutions are elliptic and are surrounded by invariant 3-tori applying a recent theorem of Han, Li, and Yi. In the spatial case a second averaging is performed, and the corresponding or- bit space (called the twice-reduced space) is constructed. The flow of the averaged Hamiltonian on it is given and several families of invariant 3-tori are determined using Han, Li, and Yi Theorem. Meyer, Kenneth R.Mon, 19 May 2014 16:05:15 GMThttps://ddd.uab.cat/record/1183152014Integrable systems on S3
https://ddd.uab.cat/record/118314
We classify the links of basic periodic orbits of integrable vector fields on S3 generalizing results on two degree of freedom Hamiltonian systems. We also study the case of completely integrable systems and define invariants for the two classes of vector fields. Martínez-Alfaro, JoséMon, 19 May 2014 15:56:53 GMThttps://ddd.uab.cat/record/1183142014Delayed logistic population models revisited
https://ddd.uab.cat/record/118313
We discuss the global dynamics of some logistic models governed by delay-differential equations. We focus on models of exploited populations, and study the changes in the dynamics as the harvesting effort is increased. We get new results and highlight the link among different logistic equations usually employed in population models. Liz, EduardoMon, 19 May 2014 15:53:33 GMThttps://ddd.uab.cat/record/1183132014Complex length and persistence of limit cycles in a neighborhood of a hyperbolic polycycle
https://ddd.uab.cat/record/118310
Complex limit cycle located in a neighborhood of a hyperbolic polycycle can not vanish under a small deformation that preserves the characteristic values of the vertexes of the polycycle. The cycles either change holomorphically under the change of the parameter, or come to the boundary of the fixed neighborhood of the polycycle. The present paper makes these statements rigorous and proves them. Ilyashenko, YuMon, 19 May 2014 14:31:37 GMThttps://ddd.uab.cat/record/1183102014The discontinuous matching of two planar linear foci can have three nested crossing limit cycles
https://ddd.uab.cat/record/118308
The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincar'e maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found. In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S. -M. Huan and X. -S. Yang. Freire, EmilioMon, 19 May 2014 14:22:52 GMThttps://ddd.uab.cat/record/1183082014From Abel equations to Jacobian conjecture
https://ddd.uab.cat/record/118305
This is a survey relating several subjects bridging algebraic differential equations and their integrability, pertubative approach based on algebraic moments, the Jacobian conjecture and optimal transport of measures defined on the complement of an arrangement of hyperplanes. This text was written in relation with Jaume Llibre's 60th birthday and it touches several fields in which Jaume gave important contributions. It also contains some results wich have not yet been published before. Francoise, Jean-PierreMon, 19 May 2014 14:19:36 GMThttps://ddd.uab.cat/record/1183052014Outer billiard around a curvilinear triangle with a fixed diameter
https://ddd.uab.cat/record/118302
We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes. Dobbs, NeilMon, 19 May 2014 14:11:48 GMThttps://ddd.uab.cat/record/1183022014Lyapunov exponent and almost sure asymptotic stability of a stochastic SIRS model
https://ddd.uab.cat/record/118299
Epidemiological models with bilinear incidence rate usually have an asymptotically stable trivial equilibrium corresponding to the disease-free state, or an asymptotically stable nontrivial equilibrium (i. e. interior equilibrium) corresponding to the endemic state. In this paper, we consider an epidemiological model, which is a SIRS (susceptible-infected-removed-susceptible) model in uenced by random perturbations. We prove that the solutions of the system are positive for all positive initial conditions and that the solutions are global, that is, there is no finite explosion time. We present necessary and suficient condition for the almost sure asymptotic stability of the steady state of the stochastic system. Chen, GuotingMon, 19 May 2014 14:04:31 GMThttps://ddd.uab.cat/record/1182992014Briot-Bouquet's Theorem in high dimension
https://ddd.uab.cat/record/118298
Let X be a germ of holomorphic vector field at 0 2 Cn and let E be a linear subspace of Cn which is invariant for the linear part of X at 0. We give a suficient condition that imply the existence of a non-singular invariant manifold tangent to E at 0. It generalizes to higher dimensions the conditions in the classical Briot{Bouquet's Theorem: roughly speaking, we impose that the convex hull of the eigenvalues ui corresponding to E does not contain 0 and there are no resonances between the ui and the complementary eigenvalues. As an application, we propose an elementary proof of the analyticity of the local stable and unstable manifolds of a real analytic vector field at a singular point. Carrillo Torres, Sergio AlejandroMon, 19 May 2014 14:00:52 GMThttps://ddd.uab.cat/record/1182982014Geometric singular perturbation theory for non-smooth dynamical systems
https://ddd.uab.cat/record/118297
Cardin, Pedro T.Mon, 19 May 2014 13:54:30 GMThttps://ddd.uab.cat/record/1182972014Half-Reeb components, Palais-Smale condition and global injectivity of local diffeomorphisms in R3
https://ddd.uab.cat/record/118294
Let F = (F1, F2, F3): R3 → R3 be a C∞ local diffeomorphism. We prove that each of the following conditions are sufficient to the global injectivity of F: A) The foliations FFi made up by the connected components of the level surfaces Fi = constant, consist of leaves without half-Reeb components induced by Fj , j ∈ {1, 2, 3} \ {i}, for i ∈ {1, 2, 3}. B) For each i 6= j ∈ {1, 2, 3}, Fi. Braun, FranciscoMon, 19 May 2014 13:46:50 GMThttps://ddd.uab.cat/record/1182942014Irregular sets for ratios of Birkhoff averages are residual
https://ddd.uab.cat/record/118293
It follows from Birkhoff's Ergodic Theorem that the irregular set of points for which the Birkhff averages of a given continuous function diverge has zero measure with respect to any finite invariant measure. In strong contrast, for systems with the weak specification property, we show here that if the irregular set is nonempty, then it is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with the specfication property. We consider also the more general case of ratios of Birkhoff averages of continuous functions and the case when the set of accumulation points of the ratios of Birkhoff averages is a prescribed closed interval. Finally, we give an application of our work to the pointwise dimension of a Gibbs measure on a repeller of a conformal map. Barreira, LuisMon, 19 May 2014 13:42:56 GMThttps://ddd.uab.cat/record/1182932014On the integrability of some three-dimensional Lotka-Volterra equations with rank-1 resonances
https://ddd.uab.cat/record/118291
We investigate the local integrability in C3 of some three-dimensional Lotka–Volterra equations at the origin with (p : q : r)-resonance, x˙ = P = x(p + ax + by + cz), y˙ = Q = y(q + dx + ey + fz), z˙ = R = z(r + gx + hy + kz). Recent work on this problem has centered on the case where the resonance is of "rank-2". That is, there are two independent linear dependencies of p, q and r over Q. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of (i, −i, λ)-resonance with λ /∈ iR (after a scaling, this is just the case p + q = 0 with q/r /∈ R), and also the case of (i − 1, −i − 1, 2)-resonance (a subcase of p + q + r = 0) under the additional assumption that a = k = 0. Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form x αy βz γ(1 + O(x, y, z). However, a new feature in the case of rank-1 resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural. Aziz, WaleedMon, 19 May 2014 13:38:32 GMThttps://ddd.uab.cat/record/1182912014Participants List
https://ddd.uab.cat/record/118222
Wed, 14 May 2014 18:43:07 GMThttps://ddd.uab.cat/record/1182222014Introduction
https://ddd.uab.cat/record/118221
In this volume we present the proceedings of the conference "New Trends in Dynamical Systems" held in Salou, Catalonia, from October 1st to 5th, 2012. This international event was devoted to celebrate Jaume Llibre's 60th birthday. The meeting was organized by the Dynamical Systems Group of the Universitat Autònoma de Barcelona, that is leaded by Jaume Llibre and has members distributed over most of the Catalonian universities. Alsedà i Soler, LluísWed, 14 May 2014 18:38:09 GMThttps://ddd.uab.cat/record/1182212014