BZPEERReading, trust and collaboration for research
Menu

Discover

GraphFollow connections around a paper, topic or saved view.

Workspaces

HomePick up where your research flow left off.InboxHandle requests, replies and notifications from one place.CollectionsCurate reading lists, evidence sets and shareable dossiers.ResearchSee your private provenance graph, trail and imports.CollaborationMove from discovery into focused discussion rooms.PublishDraft, preview and publish full-text research articles.

Network

ExpertsFind specialists worth following or asking for help.CommunitiesJoin research circles organized around shared work.PartnersSee external organizations active around the same topics.

Opportunities

ListingsBrowse roles, calls and collaborations with clearer fit signals.

Account

Log inReturn to your reading and collaboration workspace.Create accountStart saving work, following people and joining teams.
Log inCreate account

Topic overview

math.DS

2975 works5216 researchers0 institutions
q-bio.MN

Topic snapshot

What this area looks like now

2975works
5216authors
0experts visible
0communities

Next steps

Move from topic reading into action

The graph preview below keeps the nearby papers, people and communities visible in the same reading flow.

Topic graph

See the topic as a live network

Open full explorer

Inspect nearby papers, researchers, institutions and communities without opening a separate graph page.

Building this graph slice

BZPEER is loading the nearby papers, people, topics and institutions for this page.

Papers in this area

24 featured work(s)

Showing a ranked slice from 2975 total work(s). Open graph or search to expand the full area.

preprint2012arXiv

Connections between the Sznajd Model with General Confidence Rules and graph theory

The Sznajd model is a sociophysics model, that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favour bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modelled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We present some graph theory concepts, together with examples, and comparisons between the mean-field and simulations in Barabási-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims. We also show that there is no qualitative difference in the mean-field results if we require that a group of size q>2, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean-field, this would coincide with the q-voter model).

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Zero-Hopf bifurcation in the FitzHugh-Nagumo system

We characterize the values of the parameters for which a zero--Hopf equilibrium point takes place at the singular points, namely, $O$ (the origin), $P_+$ and $P_-$ in the FitzHugh-Nagumo system. Thus we find two $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families we prove the existence of a periodic orbit bifurcating from the zero--Hopf equilibrium point $O$. We prove that exist three $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at $P_+$ and $P_-$ is a zero-Hopf equilibrium point. For one of these families we prove the existence of $1$, or $2$, or $3$ periodic orbits borning at $P_+$ and $P_-$.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

On Poincaré-Bendixson Theorem and Non-Trivial Minimal Sets in Planar Nonsmooth Vector Fields

In this paper some qualitative and geometric aspects of nonsmooth vector fields theory are discussed. In the class of nonsmooth systems, that do not present sliding regions, a Poincaré-Bendixson Theorem is presented. A minimal set in planar Filippov systems not predicted in classical Poincaré-Bendixson theory and whose interior is non-empty is exhibited. The concepts of limit sets, recurrence and minimal sets for nonsmooth systems are defined and compared with the classical ones. Moreover some differences between them are pointed out.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

On Multicorns and Unicorns II: Bifurcations in Spaces of Antiholomorphic Polynomials

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^d + c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio $2$. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Zero-Hopf bifurcation in a Chua system

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are $\pm ωi\neq 0$ and $0$. In general for a such equilibrium there is no theory for knowing when from it bifurcates some small-amplitude limit cycle moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on $6$ parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other differential system in dimension $3$ or higher. In this paper first we show that there are three $4$-parameter families of Chua systems exhibiting a zero-Hopf equilibrium. After, by using the averaging theory, we provide sufficient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that $1$ limit cycle bifurcate, and from the other two families we can prove that $1$, $2$ or $3$ limit cycles bifurcate simultaneously.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Periodic Solutions of the extended Duffing-Van der Pol Oscillator

In this paper some aspects on the periodic solutions of the extended Duffing-Van der Pol oscillator are discussed. Doing different rescaling of the variables and parameters of the system associated to the extended Duffing-Van der Pol oscillator we show that it can bifurcate one or three periodic solutions from a 2-dimensional manifold filled by periodic solutions of the referred system. For each rescaling we exhibit concrete values for which these bounds are reached. Beyond that we characterize the stability of some periodic solutions. Our approach is analytical and the results are obtained using averaging theory and some algebraic techniques.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion

We construct a $C^\infty$ area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is $\{(0,0)\}$, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Ergodicity and annular homeomorphisms of the torus

Let $f: \mathbb{T}^2 \to \mathbb{T}^2$ be a homeomorphism homotopic to the identity and $F: \mathbb{R}^2 \to \mathbb{R}^2$ a lift of $f$ such that the rotation set $ρ(F)$ is a line segment of rational slope containing a point in $\mathbb{Q}^2$. We prove that if $f$ is ergodic with respect to the Lebesgue measure on the torus and the average rotation vector (with respect to same measure) does not belong to $\mathbb{Q}^2$ then some power of $f$ is an annular homeomorphism.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Equivariant bifurcations in $4$-dimensional fixed point spaces

In this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach [14] and Lauterbach & Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Equivariant Bifurcation and Absolute Irreducibility in $\R^8$ A contribution to the Ize conjecture

M. Field [5] refers to an unpublished work by J. Ize for a result that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. The main ingredient of the proof is the hypotheses, that for an absolutely irreducible representation of a compact Lie group there exists a closed subgroup whose fixed point space is odd dimensional. Then, using Brouwer degree, one gets the result. We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the algebraic Ize conjecture (AIC). Lauterbach and Matthews [10] have shown that the (AIC) is in general not true. In fact they have constructed three infinite families of finite subgroups of SO{4} which act absolutely irreducibly on $\R^4$ and for each of them any isotropy subgroup has an even dimensional fixed point space. Moreover in [10] it is shown that in spite of this failure of the (AIC) the original conjecture is true at least for groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined in [10]. We go on and construct a family of groups acting absolutely irreducibly on $\R^8$ which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. We end this paper with a discussion on how to extend the results in [10] to larger sets of groups which act on $\R^{4}$.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Non-landing parameter rays of the multicorns

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn $\mathcal{M}_d^*$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of $\mathcal{M}_d^*$, and the Misiurewicz parameters are not dense on the boundary of $\mathcal{M}_d^*$.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Global results on reset-induced periodic trajectories of planar systems

We study the existence of asymptotically stable periodic trajectories induced by reset feedback. The analysis is developed for a planar system. Casting the problem into the hybrid setting, we show that a periodic orbit arises from the balance between the energy dissipated during flows and the energy restored by resets, at jumps. The stability of the periodic orbit is studied with hybrid Lyapunov tools. The satisfaction of the so-called hybrid basic conditions ensures the robustness of the asymptotic stability. Extensions of the approach to more general mechanical systems are discussed.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Strictly Toral Dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points $ine(f)$ is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points $ess(f)$ is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by Jäger. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line

In this paper we study the maximum number $N$ of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies $2\leq N \leq 3$ if one of the two linear differential systems has its equilibrium point on the straight line of discontinuity.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Dynamics of homeomorphisms of the torus homotopic to Dehn twists

In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under $f$. In other words, if we consider the lift $\hat{f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat{f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If $f$ is area preserving and has a lift $\hat{f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat{f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces

The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $Λ$ contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on $Λ$ is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold $Λ$, under the stronger assumption that the dynamics on $Λ$ is strongly chain transitive.

0 citations0 reviewsNo code linkedOpen access
preprint2007arXiv

Sobolev regularity of solutions of the cohomological equation

We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation surfaces our results are sharp and we find the expected relation between the regularity of the distributional obstructions and the Lyapunov exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we exactly determine the dimension of the space of obstructions in each Sobolev regularity class in terms of the Kontsevich-Zorich exponents. For a fixed arbitrary translation surface and a typical direction, our results are probably not optimal but are the best which can be achieved with the available harmonic analysis techniques we have introduced in an earlier paper.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder

Detect the birth of limit cycles in non-smooth vector fields is a very important matter into the recent theory of dynamical systems and applied sciences. The goal of this paper is to study the bifurcation of limit cycles from a continuum of periodic orbits filling up a two-dimensional isochronous cylinder of a vector field in $\mathbb{R}^{3}$. The approach involves the regularization process of non-smooth vector fields and a method based in the Malkin's bifurcation function for $C^{0}$ perturbations. The results provide sufficient conditions in order to obtain limit cycles emerging from the cylinder through smooth and non-smooth perturbations of it. To the best of our knowledge they also illustrate the implementation by the first time of a new method based in the Malkin's bifurcation function. In addition, some points concerning the number of limit cycles bifurcating from non-smooth perturbations compared with smooth ones are studied. In summary the results yield a better knowledge about limit cycles in non-smooth vector fields in $\mathbb{R}^{3}$ and explicit a manner to obtain them by performing non-smooth perturbations in codimension one Euclidean manifolds.

0 citations0 reviewsNo code linkedOpen access

People in this topic

12 visible researcher(s)

Ranked from 5216 total author signal(s) in this topic.