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Periodic solutions of linear, Riccati, and Abel dynamic equations
https://ddd.uab.cat/record/204401
We study the number of periodic solutions of linear, Riccati and Abel dynamic equations in the time scales setting. In this way, we recover known results for corresponding differential equations and obtain new results for associated difference equations. In particular, we prove that there is no upper bound for the number of isolated periodic solutions of Abel difference equations. One of the main tools introduced to get our results is a suitable Melnikov function. This is the first time that Melnikov functions are used for dynamic equations on time scales. Bohner, MartinThu, 16 May 2019 13:36:19 GMThttps://ddd.uab.cat/record/2044012019Final evolutions for simplified multistrain/two-stream model for tuberculosis and dengue fever
https://ddd.uab.cat/record/204397
The simplified multistrain/two-stream model for the tuberculosis and the Dengue fever here considered has three compartments, one susceptible and the other two infectious. We characterize all the final evolutions of this model under generic assumptions. Llibre, JaumeThu, 16 May 2019 13:36:19 GMThttps://ddd.uab.cat/record/2043972019Periodic orbits of the planar anisotropic generalized Kepler problem
https://ddd.uab.cat/record/204379
Many generalizations of the Kepler problem with homogeneous potential of degree -1/2 have been considered. Here, we deal with the generalized anisotropic Kepler problem with homogeneous potential of degree -1. We provide the explicit solutions of this problem on the zero energy level and show that all of them are periodic. Llibre, JaumeThu, 16 May 2019 13:36:17 GMThttps://ddd.uab.cat/record/2043792019Dynamics of the Higgins-Selkov and Selkov systems
https://ddd.uab.cat/record/199359
We describe the global dynamics in the Poincaré disc of the Higgins--Selkov model * x'= k₀-k₁xy², y'= k₂y+ k₁xy², * where k₀,k₁,k₂ are positive parameters, and of the Selkov model x'= -x+ay+x²y, y'= b-ay-x²y, * where a,b are positive parameters. Artés, Joan CarlesMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993592018Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers
https://ddd.uab.cat/record/199358
We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the x-axis having a nilpotent center at the origin. Dias, Fabio ScalcoMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993582018On Darboux integrability of Edelstein's reaction system in R³
https://ddd.uab.cat/record/199356
We consider Edelstein's dynamical system of three reversible reactions in R³ and show that it is not Darboux integrable. To do so we characterize its polynomial first integrals, Darboux polynomials and exponential factors. Ferragut, AntoniMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993562018Algebraic limit cycles on quadratic polynomial differential systems
https://ddd.uab.cat/record/199354
Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that for a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrally opposite singular point at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture. Llibre, JaumeMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993542018Limit cycles bifurcating from a zero-Hopf singularity in arbitrary dimension
https://ddd.uab.cat/record/199351
We study the limit cycles which can bifurcate from a zero--Hopf singularity of a C^m 1 differential system in \R^n, i. e. from a singularity with eigenvalues b i and n-2 zeros for n 3. If this singularity is at the origin of coordinates and the Taylor expansion of the differential system at the origin without taking into account the linear terms starts with terms of order m, from the origin it can bifurcate s limit cycles with s \ 0,1, 2^n-3\ if m=2 (see LZ), with s \ 0,1, 3^n-2\ if m=3, with s 6^n-2 if m=4, and with s 4 5^n-2 if m=5. Moreover, s \0,1,2\ if m=4 and n=3, and s \0,1,2,3,4,5\ if m=5 and n=3. Note that the maximum number of limit cycles bifurcating from this zero--Hopf singularity grows up exponentially with the dimension for m=2,3. Barreira, LuisMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993512018Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances
https://ddd.uab.cat/record/199347
Corbera Subirana, MontserratMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993472018Phase portraits for some symmetric cubic Riccati polynomial differential equations
https://ddd.uab.cat/record/199346
We classify the topological phase portraits in the Poincaré disc of two classes of symmetric Riccati cubic polynomial differential systems. Llibre, JaumeMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993462018Algebraic limit cycles for quadratic polynomial differential systems
https://ddd.uab.cat/record/199341
We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle. Llibre, JaumeMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993412018The cubic polynomial differential systems with two circles as algebraic limit cycles
https://ddd.uab.cat/record/199332
In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles. Giné, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993322018Global phase portraits of quadratic systems with a complex ellipse as invariant algebraic curve
https://ddd.uab.cat/record/199331
In this paper we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 y^2 1=0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc. Llibre, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993312018Normal forms and hyperbolic algebraic limit cycles for a class of polynomial differential systems
https://ddd.uab.cat/record/199329
We study the normal forms of polynomial systems having a set of invariant algebraic curves with singular points. We provide sufficient conditions for the existence of hyperbolic algebraic limit cycles. Llibre, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993292018On the Darboux integrability of the Hindmarsh-Rose burster
https://ddd.uab.cat/record/199325
We study the HindmarshRose burster which can be described by the differential system x?=y?x3+bx2+I?z,y?=1?5x2?y,z?=?(s(x?x0)?z), where b, I, ?, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist. Llibre, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993252018Centers for the Kukles homogeneous systems with even degree
https://ddd.uab.cat/record/182729
Giné, JaumeThu, 07 Dec 2017 14:53:38 GMThttps://ddd.uab.cat/record/1827292017On the global dynamics of a finance model
https://ddd.uab.cat/record/182540
Recently several works have studied the following model of finance \[ x= z (y-a) x, y= 1-b y -x^2, z= -x -c z, \] where a, b and c are positive real parameters. We study the global dynamics of this polynomial differential system, and in particular for a one--dimensional parametric subfamily we show that there is an equilibrium point which is a global attractor. Llibre, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825402018Centers for generalized quintic polynominal differential systems
https://ddd.uab.cat/record/182538
Giné, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825382017Proper rational and analytic first integrals for asymmetric 3-dimensional Lotka-Volterra systems
https://ddd.uab.cat/record/182536
We go beyond in the study of the integrability of the classical model of competition between three species studied by May and Leonard [19], by considering a more realistic asymmetric model. Our results show that there are no global analytic first integrals and we provide all proper rational first integrals of this extended model by classifying its invariant algebraic surfaces. Llibre, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825362017On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity
https://ddd.uab.cat/record/182533
Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that for a quadratic polynomial differential system having two pairs of diametrally opposite equilibrium points at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture. Llibre, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825332017On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers
https://ddd.uab.cat/record/182528
We study the number of limit cycles bifurcating from the peri- od annulus of a real planar polynomial Hamiltonian ordinary differential system with a center at the origin when it is perturbed in the class of polynomial vector fields of a given degree. Colak, IlkerTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825282017Center Problem for trigonometric Liénard systems
https://ddd.uab.cat/record/182524
We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Liénard systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials. Our results are also applied to some particular subfamilies of planar trigonometric Liénard systems. The results obtained are reminiscent of the ones for planar polynomial Liénard systems but the proofs are different. Gasull i Embid, ArmengolTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825242017Limit cycles for a variant of a generalized Riccati equation
https://ddd.uab.cat/record/182520
In this paper we provide a lower bound for the maximum number of limit cycles surrounding the origin of systems (x, y = x) given by a variant of the generalized Riccati equation \[ x x^2n 1 x b x^4n 3=0, \] where b>0, b \R, n is a non--negative integer and is a small parameter. The tool for proving this result uses Abelian integrals. Llibre, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825202017On the integrability of the 5-dimensional Lorenz system for the gravity-wave activity
https://ddd.uab.cat/record/182504
We consider the 5-dimensional Lorenz system \[ U' &= -V W b V Z, \\ V' &= UW-b UZ, \\ W'&= -U V,\\ X' &= -Z, \\ Z'&=b UV X \] where b \R \0\ and the derivative is with respect to T. This system describes coupled Rosby waves and gravity waves. First we prove that the number of functionally independent global analytic first integrals of this differential system is two. This solves an open question in the paper On the analytic integrability of the 5-dimensional Lorenz system for the gravity-wave activity, Proc. Amer. Math. Soc. 142 (2014), 531--537, where it was proved that this number was two or three. Moreover, we characterize all the invariant algebraic surfaces of the system, and additionally we show that it has only two functionally independent Darboux first integrals. Llibre, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1825042017Centers of weight-homogeneous polynomial vector fields on the plane
https://ddd.uab.cat/record/182495
We characterize all centers of a planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of a planar weight-homogeneous polynomial vector fields of degrees 6 and 7. Giné, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1824952017