Polynomial first integrals of quadratic vector fields

We classify all quadratic polynomial differential systems having a polynomial first integral, and provide explicit normal forms for such systems and for their first integrals.     

On some open problems in planar differential systems and Hilbert’s 16th problem

This review paper contains a brief summary of topics and concepts related with some open problems of planar differential systems. Most of them are related with 16th Hilbert problem which refers to the existence of a uniform upper bound on the number of limit cycles of a polynomial system in function of its degree. These open problems are proposed as open questions throughout the text. Finally, an extensive bibliography, which does not intend to be exhaustive, is also given.     

Characterization of transversal homothetic solutions in the n-body problem

Homothetic solutions of the n-body problem can be seen as heteroclinic orbits when the dynamical variables are changed via the McGehee blow-up and the time is suitably scaled. Transversality of the invariant asymptotic manifolds which contain the heteroclinic orbits is related to some structural stability. We fully characterize the cases in which such transversality is obtained for the n-body problem in any dimension.     

Three-dimensional open-frameworks based on Ln(III) ions and open-/closed-shell PTM ligands: synthesis, structure, luminescence, and magnetic properties

A series of isostructural open-framework coordination polymers formulated as [Ln(dmf)(3)(ptmtc)] (Ln = Sm (1), Eu (2), Gd (3), Tb (4), Dy (5); PTMTC = polychlorotriphenylmethyl tricarboxylate) and [Ln(dmf)(2)H(2)O(αH-ptmtc)] (Ln = Sm (1'), Eu (2'), Gd (3'), Tb (4'), Dy (5')) have been obtained by treating Ln(III) ions with PTMTC ligands with a radical (PTMTC(3-)) or a closed-shell character (αH-PTMTC(3-)). X-ray diffraction analyses reveal that these coordination polymers possess 3D architectures that combine large channels and fairly rare lattice complex T connectivity. In addition, these compounds show selective framework dynamic sorption properties. For both classes of ligands, the ability to act as an antenna in Ln sensitization processes has been investigated. No luminescence was observed for compounds 1-5, and 3' because of the PTMTC(3-) ligand and/or Gd(III) ion characteristics. Conversely, photoluminescence measurements show that 1', 2', 4', and 5' emit dark orange, red, green, and dark cyan metal-centered luminescence. The magnetic properties of all of these compounds have been investigated. The nature of the {Ln-radical} exchange interaction in these compounds has been assessed by comparing the behavior of the radical-based coordination polymers 1-5 with those of the compounds with the diamagnetic ligand set. While antiferromagnetic {Sm-radical} interactions are found in 1, ferromagnetic {Ln-radical} interactions propagate in the 3D architectures of 3, 4, and 5 (Ln = Gd, Tb, and Dy, respectively). This procedure also provided access to information on the {Ln-Ln} exchange existing in these magnetic systems.     

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: CPn the n-dimensional complex projective space, HPn the n-dimensional quaternion projective space, Sn the n-dimensional sphere and Sp×Sq the product space of the p-dimensional with the q-dimensional spheres.     

Limit cycles for generalized Kukles polynomial differential systems

We study the limit cycles of generalized Kukles polynomial differential systems using averaging theory of first and second order.     

Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ)=U(γ ₁,γ ₂,γ ₃). This motion subject to the constraint 〈ν,ω〉=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω ₁,ω ₂,ω ₃) is the angular velocity and 〈?,?〉 is the inner product of $\mathbb{R}^{3}$ . We provide the following new integrable cases. (i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that $$U=-\frac{1}{2I_1I_2}\bigl(I_1\mu^2_1+I_2 \mu^2_2\bigr), $$ where I ₁,I ₂, and I ₃ are the principal moments of inertia of the body, μ ₁ and μ ₂ are solutions of the first-order partial differential equation $$\gamma_3 \biggl(\frac{\partial\mu_1}{\partial\gamma_2}- \frac{\partial\mu_2}{\partial \gamma_1} \biggr)- \gamma_2\frac{\partial \mu_1}{\partial\gamma_3}+\gamma_1\frac{\partial\mu_2}{\partial \gamma_3}=0. $$ (ii) The Veselova problem is integrable for the potential $$U=-\frac{\varPsi^2_1+\varPsi^2_2}{2(I_1\gamma^2_2+I_2\gamma^2_1)}, $$ where Ψ ₁ and Ψ ₂ are the solutions of the first-order partial differential equation where $p=\sqrt{I_{1}I_{2}I_{3} (\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}} )}$ . Also it is integrable when the potential U is a solution of the second-order partial differential equation where $\tau_{2}=I_{1}\gamma^{2}_{1}+I_{2}\gamma^{2}_{2}+I_{3}\gamma^{2}_{3}$ and $\tau_{3}=\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}}$ . Moreover, we show that these integrable cases contain as a particular case the previous known results.