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4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory
https://ddd.uab.cat/record/232600
The averaging theory of second order shows that for polynomial differential systems in ℝ4 with cubic homogeneous nonlinearities at least nine limit cycles can be born in a zero-Hopf bifurcation. Feddaoui, AminaMon, 28 Sep 2020 19:32:16 GMThttps://ddd.uab.cat/record/2326002020A new algorithm for finding rational first integrals of polynomial vector fields
https://ddd.uab.cat/record/232172
We present a new method to compute rational first integrals of planar polynomial vector fields. The algorithm is in general much faster than the usual methods and also allows to compute the remarkable curves associated to the rational first integral of the system. Ferragut, AntoniMon, 14 Sep 2020 15:33:32 GMThttps://ddd.uab.cat/record/2321722010N-dimensional zero-hopf bifurcation of polynomial differential systems via averaging theory of second order
https://ddd.uab.cat/record/232168
Using the averaging theory of second order, we study the limit cycles which bifurcate from a zero-Hopf equilibrium point of polynomial vector fields with cubic nonlinearities in ℝn. We prove that there are at least 3n-2 limit cycles bifurcating from such zero-Hopf equilibrium points. Moreover, we provide an example in dimension 6 showing that this number of limit cycles is reached. Kassa, SaraMon, 14 Sep 2020 15:33:32 GMThttps://ddd.uab.cat/record/2321682020Asymptotic dynamics of a difference equation with a parabolic equilibrium
https://ddd.uab.cat/record/232166
The aim of this work is the study of the asymptotic dynamical behaviour, of solutions that approach parabolic fixed points in difference equations. In one dimensional difference equations, we present the asymptotic development for positive solutions tending to the fixed point. For higher dimensions, through the study of two families of difference equations in the two and three dimensional case, we take a look at the asymptotic dynamic behaviour. To show the existence of solutions we rely on the parametrization method. Coll, BartomeuMon, 14 Sep 2020 15:33:32 GMThttps://ddd.uab.cat/record/2321662020Global dynamics and bifurcation of periodic orbits in a modified Nosé-Hoover oscillator
https://ddd.uab.cat/record/232160
We perform a global dynamical analysis of a modified Nosé-Hoover oscillator, obtained as the perturbation of an integrable differential system. Using this new approach for studying such an oscillator, in the integrable cases, we give a complete description of the solutions in the phase space, including the dynamics at infinity via the Poincaré compactification. Then using the averaging theory, we prove analytically the existence of a linearly stable periodic orbit which bifurcates from one of the infinite periodic orbits which exist in the integrable cases. Moreover, by a detailed numerical study, we show the existence of nested invariant tori around the bifurcating periodic orbit. Finally, starting with the integrable cases and increasing the parameter values, we show that chaotic dynamics may occur, due to the break of such an invariant tori, leading to the creation of chaotic seas surrounding regular regions in the phase space. Llibre, JaumeMon, 14 Sep 2020 15:33:31 GMThttps://ddd.uab.cat/record/2321602020Quadratic perturbations of a quadratic reversible Lotka-Volterra system
https://ddd.uab.cat/record/230979
We prove that perturbing the two periodic annuli of the quadratic polynomial reversible Lotka-Volterra differential system ̇x = -y + x2 - y2, ẏ = x(1 + 2y), inside the class of all quadratic polynomial differential systems we can obtain the following configurations of limit cycles (0,0), (1,0), (2,0), (1,1) and (1,2). Li, ChengzhiWed, 29 Jul 2020 09:22:18 GMThttps://ddd.uab.cat/record/2309792010Hilbert's sixteenth problem for polynomial Liénard equations
https://ddd.uab.cat/record/228095
This article reports on the survey talk 'Hilbert's Sixteenth Problem for Liénard equations,' given by the author at the Oberwolfach Mini-Workshop 'Algebraic and Analytic Techniques for Polynomial Vector Fields. ' It is written in a way that it is accessible to a public with heterogeneous mathematical background. The article reviews recent developments and techniques used in the study of Hilbert's 16th problem where the main focus is put on the subclass of polynomial vector fields derived from the Liérd equations. Caubergh, MagdalenaTue, 14 Jul 2020 13:29:36 GMThttps://ddd.uab.cat/record/2280952012Chini equations and isochronous centers in three-dimensional differential systems
https://ddd.uab.cat/record/226101
We study the number of limit cycles of T -periodic Chini equations and some generalized Abel equations and apply the results obtained to illustrate the existence of isochronous centers in three-dimensional autonomous differential systems. Chamberland, MarcFri, 03 Jul 2020 10:15:46 GMThttps://ddd.uab.cat/record/2261012010The secant map applied to a real polynomial with multiple roots
https://ddd.uab.cat/record/222644
We investigate the plane dynamical system given by the secant map applied to a polynomial p having at least one multiple root of multiplicity d > 1. We prove that the local dynamics around the ﬁxed points associated to the roots of p depend on the parity of d. Garijo, AntoniTue, 02 Jun 2020 18:48:51 GMThttps://ddd.uab.cat/record/2226442019Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones
https://ddd.uab.cat/record/221378
We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems ẋ = -y+x, y = x + xy, and ẋ = -y + xy, y = x + xy, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively. Itikawa, JacksonThu, 16 Apr 2020 05:59:19 GMThttps://ddd.uab.cat/record/2213782017Dynamical classification of a family of birational maps of C2 via algebraic entropy
https://ddd.uab.cat/record/221374
This work dynamically classifies a 9-parametric family of birational maps f: C→ C. From the sequence of the degrees d of the iterates of f, we find the dynamical degree δ(f) of f. We identify when d grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim (Mich Math J 54:647-670, 2006) as one of its subfamilies. Zafar, SundusThu, 16 Apr 2020 05:59:18 GMThttps://ddd.uab.cat/record/2213742019Phase portraits of Abel quadratic differential systems of second kind with symmetries
https://ddd.uab.cat/record/221366
We provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits. Ferragut, AntoniWed, 15 Apr 2020 15:23:10 GMThttps://ddd.uab.cat/record/2213662019Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
https://ddd.uab.cat/record/221365
In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y<0 and a virtual one for y>0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation. Cardoso, Joäo L.Wed, 15 Apr 2020 15:23:10 GMThttps://ddd.uab.cat/record/2213652020Dynamic rays of bounded-type transcendental self-maps of the punctured plane
https://ddd.uab.cat/record/221362
We study the escaping set of functions in the class B∗, that is, transcendental self-maps of ℂ∗ for which the set of singular values is contained in a compact annulus of ℂ∗ that separates zero from infinity. For functions in the class B∗, escaping points lie in their Julia set. If f is a composition of finite order transcendental self-maps of ℂ∗ (and hence, in the class B∗), then we show that every escaping point of f can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence e ∈ {0,∞}, we show that the escaping set of f contains a Cantor bouquet of curves that accumulate to the set {0,∞} according to e under iteration by f. Fagella Rabionet, NúriaWed, 15 Apr 2020 15:23:10 GMThttps://ddd.uab.cat/record/2213622017Rational parameterizations approach for solving equations in some dynamical systems problems
https://ddd.uab.cat/record/221348
We show how the use of rational parameterizations facilitates the study of the number of solutions of many systems of equations involving polynomials and square roots of polynomials. We illustrate the effectiveness of this approach, applying it to several problems appearing in the study of some dynamical systems. Our examples include Abelian integrals, Melnikov functions and a couple of questions in Celestial Mechanics: the computation of some relative equilibria and the study of some central configurations. Gasull i Embid, ArmengolWed, 15 Apr 2020 15:23:09 GMThttps://ddd.uab.cat/record/2213482019Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem
https://ddd.uab.cat/record/221345
We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piecewise linear planar vector field; a new counterexample of Kouchnirenko conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the (1 + 4)-body problem. Gasull i Embid, ArmengolWed, 15 Apr 2020 15:23:08 GMThttps://ddd.uab.cat/record/2213452020A note on the Lyapunov and period constants
https://ddd.uab.cat/record/221344
It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations. Cimà, AnnaWed, 15 Apr 2020 15:23:08 GMThttps://ddd.uab.cat/record/2213442020Cyclicity of (1,3)-switching FF type equilibria
https://ddd.uab.cat/record/221339
Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of (1, 2)-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of (1, 3)-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3. Chen, XingwuWed, 15 Apr 2020 15:23:07 GMThttps://ddd.uab.cat/record/2213392019Centers of discontinuous piecewise smooth quasi-homogeneous polynomial differential systems
https://ddd.uab.cat/record/221337
In this paper we investigate the center problem for the discontinuous piecewise smooth quasi-homogeneous but non-homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi-homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi-homogeneous cubic and quartic polynomial differential systems. Chen, HebaiWed, 15 Apr 2020 15:23:07 GMThttps://ddd.uab.cat/record/2213372019On the dynamics of the Euler equations on so(4)
https://ddd.uab.cat/record/221336
This paper deals with the Euler equations on the Lie Algebra so(4). These equations are given by a polynomial differential system in R6. We prove that this differential system has four 3-dimensional invariant manifolds and we give a complete description of its dynamics on these invariant manifolds. In particular, each of these invariant manifolds are fulfilled by periodic orbits except in a zero Lebesgue measure set. Buzzi, ClaudioWed, 15 Apr 2020 15:23:07 GMThttps://ddd.uab.cat/record/2213362019Conservation laws in biochemical reaction networks
https://ddd.uab.cat/record/221331
We study the existence of linear and nonlinear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The nonlinear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using the Darboux theory of integrability, we provide necessary structural (i. e. , parameter-independent) conditions on a reaction network to guarantee the existence of nonlinear conservation laws of a certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that nonlinear first integrals only exist under the same structural condition (as in the case of the Lotka-Volterra system). We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples. Mahdi, AdamWed, 15 Apr 2020 15:23:07 GMThttps://ddd.uab.cat/record/2213312017Bifurcation of relative equilibria generated by a circular vortex path in a circular domain
https://ddd.uab.cat/record/221321
We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number. Rojas, DavidWed, 15 Apr 2020 15:23:06 GMThttps://ddd.uab.cat/record/2213212020Classification of linear skew-products of the complex plane and an affine route to fractalization
https://ddd.uab.cat/record/221317
Linear skew products of the complex plane, θ↦θ+ω,z↦a(θ)z,} where θ∈T, z∈C, ω/2π is irrational, and [θ↦a(θ)∈C∖{0} is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of θ↦a(θ). We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an affine variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present. Fagella Rabionet, NúriaWed, 15 Apr 2020 15:23:06 GMThttps://ddd.uab.cat/record/2213172019Phase portraits of Abel quadratic differential systems of the second kind
https://ddd.uab.cat/record/221315
We provide normal forms and the global phase portraits on the Poincaré disk of some Abel quadratic differential equations of the second kind. Moreover, we also provide the bifurcation diagrams for these global phase portraits. Ferragut, AntoniWed, 15 Apr 2020 15:23:06 GMThttps://ddd.uab.cat/record/2213152018Generalized rings around the McMullen domain
https://ddd.uab.cat/record/221310
We consider the family of rational maps given by F (z) = z + λ/ z where n, d∈ N with 1 / n+ 1 / d< 1, the variable z∈ C^ and the parameter λ∈ C. It is known that when n= d≥ 3 there are infinitely many rings S with k∈ N, around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter λ-plane where the Julia sets of F are Cantor sets of simple closed curves. The rings S converge to the boundary of the McMullen domain as k→ ∞ and contain parameter values that lie at the center of Sierpiński holes, i. e. , open simply connected subsets of the parameter space for which the Julia sets of F are Sierpiński curves. The rings also contain the same number of superstable parameter values, i. e. , parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1 / n+ 1 / d< 1 where n is not necessarily equal to d. The number of Sierpiński holes and superstable parameters on S is τ1n,d=n-1, and on S for k> 1 is given by τkn,d=dnk-2(n-1)-nk-1+1. Garijo, AntoniWed, 15 Apr 2020 15:23:05 GMThttps://ddd.uab.cat/record/2213102019