## About

154

Publications

8,975

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,785

Citations

Citations since 2016

Introduction

**Skills and Expertise**

## Publications

Publications (154)

Canard explosion is an appealing event occurring in singularly perturbed systems. In this phenomenon, upon variation of a parameter within an exponentially small range, the amplitude of a small limit cycle increases abruptly. In this letter we analyze the canard explosion in a limit cycle related to a degenerate center (with zero Jacobian matrix)....

We deal with analytic three-dimensional symmetric systems whose origin is a Hopf-zero singularity. Once it is not completely analytically integrable, we provide criteria on the existence of at least one functionally independent analytic first integral. In the generic case, we characterize the analytic partially integrable systems by using orbitally...

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degen...

We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerat...

In this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center probl...

In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first...

In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can b...

We consider the analytically integrable perturbations of cubic homogeneous differential systems whose origin is an isolated singularity. We prove that are orbitally equivalent to the cubic vector field associated. We also characterize the analytically integrable centers. We apply the results to two families of degenerate vector fields.

In the analysis of canard explosions, power series for estimating the critical value play a significant role for practical applications. In this chapter, an efficient analytical algorithm is developed to compute the series for a family of generic canard explosions in planar systems. In addition to the applicable formula for the first term, the prop...

In this work we consider an unfolding of a normal form of the Lorenz system near a triple-zero singularity. We are interested in the analysis of double-zero bifurcations emerging from that singularity. Their local study provide partial results that are extended by means of numerical continuation methods. Specifically, a curve of heteroclinic connec...

The goal of this work is twofold. On the one hand, we apply the nonlinear time transformation (NTT) method to the study of a homoclinic connection in a family of Rayleigh–Duffing mechanical oscillators. This efficient procedure provides high-order approximations for global connections in a planar system that can be written as a perturbation of a Ha...

We consider analytic perturbations of quadratic homogeneous differential systems having an isolated singularity at the origin. We characterize the systems with an analytic first integral at the origin. We apply the results to two families of degenerate vector fields.

The canard explosion is a significant phenomenon in singularly perturbed system which has attracted lots of attentions in the literature. Such a periodic behavior often appears near a Hopf bifurcation and variety of methods have been developed for studying it. In the present work, we introduce a degenerate canard explosion of which the canard cycle...

Based on the nonlinear time transformation method, in this paper we propose a recursive algorithm for arbitrary order approximation of heteroclinic orbits. This approach works fine for a wide class of systems that are perturbations of non-Hamiltonian integrable planar vector fields. Specifically, our method can provide an approximation up to any de...

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally re...

In this paper we define when a polynomial differential system is orbitally universal and we show the relevance of this notion in the classical center problem, i.e. in the problem of distinguishing between a focus and a center.

In this work we consider cuspidal loops, i.e., homoclinic orbits to cuspidal singular points. We develop an iterative procedure, founded on the nonlinear time transformation method, to estimate such codimension-three global bifurcations up to any wanted order, not only in the space of parameters but also in the phase plane. As far as we know, this...

A planar system has been proposed in the paper Rankin et al. (Nonlinear Dyn 66:681–688, 2011) to understand the canard explosion detected in a 6D aircraft ground dynamics model. A specific feature of this minimal 2D system is a critical manifold with a single fold and an asymptote. In this paper, we provide a high-order analytical prediction (in fa...

The aim of this paper is to obtain a high-order approximation of the canard explosion in the Brusselator equations. This classical chemical system has been extensively studied but, until now, only first-order approximation to the canard explosion has been provided. Here, with the help of the nonlinear time transformation method, we are able to obta...

Asymptotic expansions are of great interest and significance in the study of canard explosions in singularly perturbed systems. Several classical methods have been developed to compute such expansions. However, for the non-generic case considered in this letter, those methods fail to do so. There only exists an estimation on the first non-zero term...

In the present work, we investigate the canard explosion in a van der Pol electronic oscillator, a fast transition from a small amplitude periodic orbit to a relaxation oscillation. To this aim we develop a new effective procedure, based on the nonlinear time transformation method, that uses elementary trigonometric functions. In fact, it is able t...

A codimension-three Takens–Bogdanov bifurcation in reversible systems has been very recently analyzed in the literature. In this paper, we study with the help of the nonlinear time transformation method, the codimension-one and -two homoclinic and heteroclinic connections present in the corresponding unfolding. The algorithm developed allows to obt...

In this work we study the center conditions of a particular polynomial differential system with a nilpotent singularity using a new proposed algorithm. This problem was initially studied in [19] where some center conditions were found using the Cherkas' method and its full characterization was established as an open problem.

We consider a class of three-dimensional systems having an equilibrium point at the origin, whose principal part is of the form (−∂h∂y(x,y),∂h∂x(x,y),f(x,y))T. This principal part, which has zero divergence and does not depend on the third variable z, is the coupling of a planar Hamiltonian vector field Xh(x,y):=(−∂h∂y(x,y),∂h∂x(x,y))T with a one-d...

In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first...

The goal of this paper is to obtain a description of the global connections present in the Z2-symmetric Takens-Bogdanov normal form. The algorithm used, grounded on the nonlinear time transformation method, provides a perturbation solution up to any wanted order for the homoclinic and heteroclinic orbits, with the only restriction on the capabiliti...

In this paper, we present an algorithm based on the nonlinear time transformation method to approximate homoclinic orbits in planar autonomous nonlinear oscillators. With this approach, a unique perturbation solution up to any desired order can be obtained for them using trigonometric functions. To demonstrate its efficiency, the method is applied...

In this work it is characterized the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.

A family of planar nilpotent reversible systems with an equilibrium point located at the origin has been studied in the recent paper Algaba et al. (Nonlinear Dyn 87:835–849, 2017). The authors investigate the candidate for an universal unfolding of a codimension-three degenerate case which exhibits a rich bifurcation scenario. However, a codimensio...

In this paper we consider a two-parameter quadratic three-dimensional system with only six terms and two nonlinearities. First we analyze the Hopf bifurcation of its only equilibrium detecting several degeneracies. With this information we numerically obtain various bifurcation diagrams of periodic orbits in which saddle-node and period-doubling bi...

We consider the autonomous system of differential equations of the form x˙=P1(x,y)+P2(x,y),y˙=Q1(x,y)+Q3(x,y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{ali...

A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper (Phys. Lett. A 378 (2014) 1361–1363). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinit...

This work is about the analytic integrability problem around the origin in a family of degenerate nilpotent vector fields. The integrability problem for planar vector fields with first Hamiltonian component having simple factors in its factorization on C[x, y] is solved in Algaba et al. (Nonlinearity 22:395–420, 2009) [5]. Nevertheless, when the Ha...

We present the basic ideas of the Normal Form Theory by using quasi-homogeneous expansions of the vector field, where the structure of the normal form is determined by the principal part of the vector field. We focus on a class of tridimensional systems whose principal part is the coupling of a Hamiltonian planar system and an unidimensional system...

In this chapter, we review some bifurcations exhibited by the classical Lorenz system, where the parameters can have any real value. Analytical results on the pitchfork, Hopf and Takens–Bogdanov bifurcations of the origin, as well as the Hopf bifurcation of the nontrivial equilibria, are summarized. These results serve as a guide for the numerical...

In this work is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. The analytic integrability problem is characterized via the existence of a formal inverse integrating factor. The relation between the analytic integrability and the existence of an algebrai...

We solve, by using normal forms, the analytical integrability problem for differential systems in the plane whose first homogeneous component is a cubic Kolmogorov system whose origin is an isolated singularity. As an application, we give the analytically integrable systems of a class of systems x˙=x(P2+P3),y˙=y(Q2+Q3), with Pi, Qi homogeneous poly...

We give an expression of the irreducible invariant curves at the singular point. For analytically integrable systems, we provide an expression of its primitive first integral. This fact allows us to obtain necessary conditions of analytic integrability at degenerate singular points.

In this paper, we present different possibilities for the simplest orbital normal form of the Hopf bifurcation. Beyond the classical normal form we present other forms that have the structure of the Rayleigh and the van der Pol equations. The results obtained are applied to obtain orbital normal forms for Newtonian systems with one degree of freedo...

In this work, we study the structural stability of planar quasi-homogeneous vector fields under quasi-homogeneous perturbations, and provide a complete classification. This study, which has been the subject of previous works, is only complete in the homogeneous case. The main tool in our analysis is a splitting of planar quasi-homogeneous vector fi...

We say that a polynomial differential system x˙=P(x,y), y˙=Q(x,y) having the origin as a singular point is Z2-symmetric if P(−x,−y)=−P(x,y) and Q(−x,−y)=−Q(x,y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C∞ first integral. However these two kinds of nilpotent centers are not ch...

We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically integrable systems of a class of systems $\dot{x}= x(P1+P2)$; $\dot{y}= y(Q1+Q2)$; being Pi,Qi homogeneous polyno...

In this work it is solved the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.

We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasihomogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbati...

We characterize the center conditions for a cubic Kolmogorov differential system. We also study the number of small limit cycles that can bifurcate from the singular point of focus type for such systems.

In the commented paper, the authors consider a three-dimensional system and analyze the presence of Shilnikov chaos as well as a Hopf bifurcation. On the one hand, they state that the existence of a chaotic attractor is verified via the homoclinic Shilnikov theorem. The homoclinic orbit of this system is determined by using the undetermined coeffic...

En el trabajo comentado , los autores presentan diez sistemas autónomos no lineales caóticos , de los que afirman que no tienen caos en el sentido de Shilnikov . Desgraciadamente, esta afirmación carece de fundamento pues utilizan un teorema erróneo de la literatura .

The study of the local bifurcations of equilibria in the Lorenz system, when the parameters are allowed to take any real value, has been successfully completed in the case of the pitchfork, Hopf and Takens–Bogdanov bifurcations. However, the Hopf-pitchfork and the triple-zero bifurcations of the origin cannot be analyzed with the standard procedure...

Le us consider the well-known Lorenz system [4] ˙ x = σ(y − x), ˙ y = ρx − y − xz, ˙ z = −bx + xy, where σ, ρ and b are real parameters. This system is invariant under the change of variables (x, y, z) → (−x, −y, z). Moreover , it has an equilibrium E 0 = (0, 0, 0) and a pair of non-trivial equilibria E ± = (± b(ρ − 1), ± b(ρ − 1), ρ − 1), fo...

In this paper, we present a bifurcation analysis for planar nilpotent reversible systems with an equilibrium point located at the origin. We study candidates for the universal unfoldings of the codimension-one non-degenerate cases, as well as a pair of codimension-two degenerate cases, and a codimension-three degenerate case, where a rich bifurcati...

In this paper, we use a geometric criterium based on the classical method of the construction of Lyapunov functions to determine if a differential system has a focus or a center at a singular point. This criterium is proved to be useful for several examples studied in previous works with other more specific methods.

In this paper we study the analytic integrability around the origin inside a family of degenerate centers or perturbations of them. For this family analytic integrability does not imply formal orbital equivalence to a Hamiltonian system. It is shown how difficult is the integrability problem even inside this simple family of degenerate centers or p...

The paper, “Study on the reliable computation time of the numerical model using the sliding temporal correlation method,” was published in Theoretical and Applied Climatology. In that work, the sliding temporal correlation analysis is employed to investigate the predictable time of two typical chaotic numerical models, namely the Lorenz system and...

Usually, the study of differential systems with linear part null is done using quasi-homogeneous expansions of vector fields. Here, we use this technique for analyzing the existence of an inverse integrating factor for generalized nilpotent systems, in general non-integrable, whose lowest-degree quasi-homogeneous term is the Hamiltonian system \(y^...

Usually, the physical interest of the Lorenz system is restricted to the region where its three parameters are positive. However, this famous system appears, when \(\sigma <0\), in the study of a thermosolutal convection model and in the analysis of traveling-wave solutions of the Maxwell–Bloch equations. In this context, a Takens–Bogdanov bifurcat...

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not suffic...

In this work we present, for the Lorenz system, analytical and numerical results on the existence of periodic orbits with unbounded amplitude and whose period tends to zero. Since a particle moving on these periodic orbits would be faster-than-light, we call them superluminal periodic orbits. To achieve this goal, we first find analytical expressio...

Newton diagram of a planar vector field allows to determine whether a singular point of an analytic system is a monodromic singular point. We solve the monodromy problem for the nilpotent systems and we apply our method to a wide family of systems with a degenerate singular point, so-called generalized nilpotent cubic systems.

In this work, we analyze some aspects of the center problem from the perspective of the normal form theory. We provide alternative proofs of some well known results in the case of non-vanishing linear part (nondegenerate and nilpotent centers). Moreover, some new results are also derived. In particular, we show that the unique characterization for...

In this work we study Takens–Bogdanov bifurcations of equilibria and periodic orbits in the
classical Lorenz system, allowing the parameters to take any real value. First, by computing the
corresponding normal form we determine where the Takens–Bogdanov bifurcation of equi-
libria is non-degenerate, namely of homoclinic or of heteroclinic type. The...

In this Letter we consider a three parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point. Using blow-up techniques we obtain a system where an exact homoclinic connection is determined. The numerical continuation of this global connection shows that it exhibits three different kinds of codimens...

The Newton diagram and, in particular, the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine the existence of characteristic orbits and separatrices of an isolated singular point. We give an easy algorithm for obtaining the local phase portrait near the origin of a bi-dimensional differential system and...

In this paper we show numerically the existence of a T-point-Hopf bifurcation in the Lorenz system. This codimension-three degeneracy occurs when the nontrivial equilibria involved in the T-point heteroclinic loop undergo a subcritical Hopf bifurcation. Shil’nikov-Hopf bifurcations of the heteroclinic and the homoclinic orbits of the nontrivial equ...

We give necessary conditions for the orbital-reversibility for a class of planar dynamical systems, based on properties of some invariant curves. From these necessary conditions we formulate a suitable algorithm, to detect orbital-reversibility, which is applied to a family of nilpotent systems and to a family of degenerate systems.

This paper is devoted to the classification of analytic integrable cases of two families of degenerate planar vector fields with a monodromic singular point at the origin. This study falls in the still open degenerate center problem. This classification can be done using the formal normal form theory and knowing a suitable normal form of any differ...

We study the existence of an inverse integrating factor for a class of systems, in general non-integrable, whose lowest-degree quasi-homogeneous term is a Hamiltonian system and its Hamiltonian function only has simple factors over C[x,y]C[x,y].

In the commented paper, the authors claim to have proved the existence of heteroclinic and homoclinic orbits of Šilnikov type in two-Lorenz like systems, the so-called Lü and Zhou systems. According to them, they have analytically demonstrated that both systems exhibit Smale horseshoe chaos. Unfortunately, we show that the results they obtain are i...

In the present paper we characterize the analytic integrability around the origin of a family of degenerate differential systems. Moreover, we study the analytic integrability of some degenerate systems through the orbital reversibility and from the existence of a Lie's symmetry for these systems. The results obtained for this family are similar to...

We give a new algorithmic criterium that determines wether an isolated degenerate singular point of a system of differential equations on the plane is monodromic. This criterium involves the conservative and dissipative parts associated to the edges and vertices of the Newton diagram of the vector field.

In a very recent paper by Deng (Z Angew Math Phys 64:1443–1449, 2013), the author claims to have successfully found all the invariant algebraic surfaces of the generalized Lorenz system, \({\dot{x} = a(y - x), \ \dot{y} = bx + cy - xz, \ \dot{z} = dz + xy}\) . He provides six invariant algebraic surfaces, found according to the idea of the weight o...

We provide in a very straightforward manner a proof for the existence of centers on center manifolds, for the generalized Lorenz system, ẋ=a(y-x),ẏ=bx+cy-xz,ż=dz+xy. From this result, the presence of this Hopf bifurcation of codimension infinity is trivially deduced for the Lorenz, Chen and Lü systems. Our outcomes are novel for the Lorenz and Chen...

The characterization of all Darboux polynomials and rational first integrals of the generalized Lorenz system, x˙=a(y−x), y˙=bx+cy−xz, z˙=dz+xy, was published very recently in [K. Wu, X. Zhang, Bull. Sci. Math. 136 (2012) 291–308]. In this paper we improve that work in two aspects. On the one hand, we obtain the same results in a much more straight...

In the commented paper the authors study some aspects of boundedness in the general Lorenz family considering that it contains four independent parameters. However, as we show here by means of a linear scaling in time and coordinates, they are dealing with a system homothetically equivalent to the Lorenz system. Consequently, the novel and interest...

In the recent paper ‘Global dynamics of the generalized Lorenz systems having invariant algebraic
̇
̇
surfaces’ published in Physica D, the authors study the generalized Lorenz system, x = a(y − x), y = bx +
̇
cy − xz , z = dz + xy, and consider the subclass of these systems which have an invariant algebraic surface.
Within this subclass they prese...

In this paper, we perform a complete study of the Hopf bifurcations in the three-parameter Lorenz system, \(\dot{x} = \sigma (y-x),\,\dot{y} = \rho x - y - xz,\,\dot{z} = -bz + xy\), with \(\sigma , \rho , b \in \mathbb {R}\). On the one hand, we reobtain the results found in the literature for the Lorenz model when the three parameters are positiv...