Limit cycles bifurcating from a zero–Hopf singularity in arbitrary dimension

Nonlinear Dynamics - For a $$C^{m+1}$$ differential system on $$\mathbb {R}^n$$ , we study the limit cycles that can bifurcate from a zero–Hopf singularity, i.e., from a singularity with...     

The symmetric periodic orbits for the two-electron atom

We analyse the existence of symmetric periodic orbits of the two-electron atom. The results obtained show that there exist six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré.     

Exploring the interaction of mercury(II) by N(2)S(2) and NS(3) anthracene-containing macrocyclic ligands: photophysical, analytical, and structural studies

The complexation properties toward Hg(II) of six macrocyclic ligands, 3,11-dithia-7,17-diazabicyclo[11.3.1]heptadeca-1(17),13,15-triene (L1), 7-(9-anthracenylmethyl)-3,11-dithia-7,17-diazabicyclo[11.3.1]heptadeca-1(17),13,15-triene (L2), 7-(10-methyl-9-anthracenylmethyl)-3,11-dithia-7,17-diazabicyclo[11.3.1]heptadeca-1(17),13,15-triene (L3), 7,7'-[9,10-anthracenediylbis(methylene)]bis-3,11-dithia-7,17-diazabicyclo[11.3.1]heptadeca-1(17),13,15-triene (L4), 1,4,7-trithia-11-azacyclotetradecane (L5), and 11,-(anthracen-9-ylmethyl)-1,4,7-trithia-11-azacyclotetradecane (L6), were studied. The stoichiometries of the formed species were determined from absorption and fluorescence titrations. In these anthracene-containing macrocycles, a fluorescent quenching of the emission was found upon Hg(II) addition. The X-ray crystal structure of [HgCl2(L2)] x 1/2CH2Cl2 was determined. The asymmetric unit contains two independent [HgCl2(L2)] molecules and one dichloromethane molecule. Each Hg(II) ion is coordinated by the pyridine nitrogen, the two sulfur atoms of one L2 molecule, and two chloride ions. Analytical studies using solvent extraction separation of Hg(II) from aqueous solutions were performed to determine the Hg(II) extraction capability of ligands L1, L2, and L5.     

First-row transition-metal complexes based on a carboxylate polychlorotriphenylmethyl radical: trends in metal-radical exchange interactions

We report the synthesis, crystal structures, and magnetic properties of a series of mononuclear, metal-radical complexes with first-row transition-metal ions using a new class of radical-based ligands, the polychlorinated triphenylmethyl (PTM) radicals. Crystal structures of three new PTM-based complexes of general formula M(PTMMC)2(py)4-x(H2O)x [PTMMC = PTM radical functionalized at the para position with one carboxylic group; M = Zn(II), x = 2 (1); M = Ni(II), x = 1 (2); M = Co(II), x = 1 (3)] show similar molecular structures in which mononuclear complexes are formed by an octahedral metal ion coordinated by two monodentated PTMMC units. From a magnetic point of view, these similar configurations describe a quasilinear, trimeric magnetic model (PTMMC-M(II)-PTMMC), in which the metal [Ni(II) or Co(II)]-radical magnetic-exchange coupling constants have been determined for the first time. In all of these complexes, the temperature dependence of the magnetic susceptibility reveals moderate antiferromagnetic-exchange coupling constants between the PTMMC radicals and Ni(II) (2J/kB = -47.1 K) and Co(II) ions (2J/kB = -15.2 K) based on the exchange Hamiltonian H = -2JSM(Srad1 + Srad2).     

Old materials with new tricks: multifunctional open-framework materials

The literature on open-framework materials has shown numerous examples of porous solids with additional structural, chemical, or physical properties. These materials show promise for applications ranging from sensing, catalysis and separation to multifunctional materials. This critical review provides an up-to-date survey to this new generation of multifunctional open-framework solids. For this, a detailed revision of the different examples so far reported will be presented, classified into five different sections: magnetic, chiral, conducting, optical, and labile open-frameworks for sensing applications. (413 references.)     

Intramolecular electron transfer and charge delocalization in bistable donor–acceptor systems based on perchlorotriphenylmethyl radicals linked to ferrocene and tetrathiafulvalene units

A comparative analysis of the physicochemical properties of different donor–acceptor dyads, based on a perchlorotriphenylmethyl radical acceptor subunit linked through a vinylene π‐bridge to ferrocene derivatives (1–3) or a tetrathiafulvalene (4) donor subunit, is presented. Intramolecular electron transfer and charge delocalization of these donor–acceptor dyads are discussed on the basis of data obtained in solution by cyclic voltammetry, UV–Vis near‐infrared, and electron spin resonance spectroscopies. Bistability in the crystalline phase is also discussed on the basis of the results provided by X‐ray diffraction analysis and temperature‐dependent Mössbauer spectra. Copyright © 2014 John Wiley & Sons, Ltd.     

On the polynomial limit cycles of polynomial differential equations

In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If φ(x) is a polynomial, then φ(x) is called a polynomial solution.     

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.