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On 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/1824952017On the analytic integrability of the 5-dimensional Lorenz system for the gravitiy-wave activity
https://ddd.uab.cat/record/150747
The 5-dimensional Lorenz system for the coupled Rosby and gravity waves has exactly two independent analytic first integrals. Llibre, JaumeFri, 06 May 2016 08:59:53 GMThttps://ddd.uab.cat/record/1507472014Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems
https://ddd.uab.cat/record/150746
In this paper we study the limit cycles bifurcating from a nonisolated zero-Hopf equilibrium of a differential system in R3. The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using the averaging theory of the second order, explicit conditions are given for the existence of one or two limit cycles bifurcating from such a zero-Hopf equilibrium. To our knowledge, this is the first result on bifurcations from a non-isolated zero-Hopf equilibrium. This result is applied to study three-dimensional generalized Lotka-Volterra systems in [3]. The necessary and sufficient conditions for the existence of a non-isolated zero-Hopf equilibrium of this system are given, and it is shown that two limit cycles can be bifurcated from the non-isolated zero-Hopf equilibrium under a general small perturbation of three-dimensional generalized Lotka-Volterra systems. Llibre, JaumeFri, 06 May 2016 08:59:53 GMThttps://ddd.uab.cat/record/1507462014Algebraic invariant curves and first integrals for Riccati polynomial differential systems
https://ddd.uab.cat/record/150745
We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x′ = 1, y′ = a(x)y2 +b(x)y +c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals. Llibre, JaumeFri, 06 May 2016 08:59:53 GMThttps://ddd.uab.cat/record/1507452014On the connectivity of the escaping set for complex exponential Misiurewicz parameters.
https://ddd.uab.cat/record/150462
Jarque i Ribera, XavierFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504622011Skew product attractors and concavity
https://ddd.uab.cat/record/145369
We propose an approach to the attractors of skew products that tries to avoid unnecessary structures on the base space and rejects the assumption on the invariance of an attractor. When nonivertible maps in the base are allowed, one can encounter the mystery of the vanishing attractor. In the second part of the paper, we show that if the fiber maps are concave interval maps then contraction in the fibers does not depend on the map in the base. Alsedà i Soler, LluísTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453692015A note on the Dziobek central configurations
https://ddd.uab.cat/record/145368
For the Newtonian n-body problem in R^n−2 with n ≥ 3 we prove that the following two statements are equivalent. (a) Let x be a Dziobek central configuration having one mass located at the center of mass. (b) Let x be a central configuration formed by n-1 equal masses located at the vertices of a regular (n-2)-simplex together with an arbitrary mass located at its barycenter. Llibre, JaumeTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453682015