Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 in R3

In this paper we will find a continuous of periodic orbits passing near infinity for a class of polynomial vector fields in R3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane and that possess a “generalized heteroclinic loop” formed by two singular points e+ and e− at infinity and their invariant manifolds � and . � is an invariant manifold of dimension 1 formed by an orbit going from e− to e+, � is contained in R3 and is transversal to . is an invariant manifold of dimension 2 at infinity. In fact, is the 2–dimensional sphere at infinity in the Poincar´e compactification minus the singular points e+ and e−. The main tool for proving the existence of such periodic orbits is the construction of a Poincar´e map along the generalized heteroclinic loop together with the symmetry with respect to .

Symmetric periodic orbits near a heteroclinic loop in R3 formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper we consider C1 vector fields X in R3 having a “generalized heteroclinic loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere S2 and a diameter connecting the north with the south pole. The north pole is an attractor on S2 and a repeller on . The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S2. We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient Poincar´e map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3 satisfying this dynamics.

Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

In this paper we consider vector fields in R3 that are invariant under a suitable symmetry and that posses a “generalized heteroclinic loop” L formed by two singular points (e+ and e −) and their invariant manifolds: one of dimension 2 (a sphere minus the points e+ and e −) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e −). In particular, we analyze the dynamics of the vector field near the heteroclinic loop L by means of a convenient Poincar´e map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in R3, and the second one is the charged rhomboidal four body problem.

What determines the wavelength of self-organized shoreline sand waves?

Shoreline undulations extending into the bathymetric contours with a length scale larger than that of the rhythmic surf zone bars are referred to as shoreline sand waves. Many observed undulations along sandy coasts display a wavelength in the order 1-7 km. Several models that are based on the hypothesis that sand waves emerge from a morphodynamic instability in case of very oblique wave incidence predict this range of wavelengths. Here we investigate the physical reasons for the wavelength selection and the main parametric trends of the wavelength in case of sand waves arising from such instability. It is shown that the existence of a minimum wavelength depends on an interplay between three factors affecting littoral drift: (A) the angle of wave fronts relative to local shoreline, which tends to cause maximum transport at the downdrift flank of the sand wave, (B) the refractive energy spreading which tends to cause maximum transport at the updrift flank and (C) wave focusing (de-focusing) by the capes (bays), which tends to cause maximum transport at the crest or slightly downdrift of it. Processes A and C cause decay of the sand waves while process B causes their growth. For low incidence angles, B is very weak so that a rectilinear shoreline is stable. For large angles and long sand waves, B is dominant and causes the growth of sand waves. For large angles and short sand waves C is dominant and the sand waves decay. Thus, wavelength selection depends on process C, which essentially depends on shoreline curvature. The growth rate of very long sand waves is weak because the alongshore gradients in sediment transport decrease with the wavelength. This is why there is an optimum or dominant wavelength. It is found that sand wave wavelength scales with λ0/β where λ0 is the water wave wavelength in deep water and β is the mean bed slope from shore to the wave base.

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

The existence of FGFR1-5-HT1A receptor heterocomplexes in midbrain 5-HT neurons of the rat: relevance for neuroplasticity

The ascending midbrain 5-HT neurons to the forebrain may be dysregulated in depression and have a reduced trophic support. With in situ proximity ligation assay (PLA) and supported by coimmunoprecipitation and colocation of the FGFR1 and 5-HT1A immunoreactivities in the midbrain raphe cells, evidence for the existence of FGFR1-5-HT1A receptor heterocomplexes in the dorsal and median raphe nuclei of the Sprague Dawley rat as well as in the rat medullary raphe RN33B cells has been obtained. Especially after combined FGF-2 and 8-OH-DPAT treatment, a marked and significant increase in PLA clusters was found in the RN33B cells. Similar results were reached with the FRET technique in HEK293T cells, where TM-V of the 5HT1A receptor was found to be part of the receptor interface. The combined treatment with FGF-2 and the 5-HT1A agonist also synergistically increased FGFR1 and ERK1/2 phosphorylation in the raphe midline area of the midbrain and the RN33B cells as well as their differentiation, as seen from development of the increased number and length of extensions per cell and their increased 5-HT immunoreactivity. These signaling and differentiation events were dependent on the receptor interface since they were blocked by incubation with TM-V but not by TM-II. Together, the results indicate that the 5-HT1A autoreceptors by being part of a FGFR1-5-HT1A receptor heterocomplex in the midbrain raphe 5-HT nerve cells appear to have a trophic role in the central 5-HT neuron systems in addition to playing a key role in reducing the firing of these neurons

On the existence and function of galanin receptor heteromers in the central nervous system

Galanin receptor (GalR) subtypes 1-3 linked to central galanin neurons may form heteromers with each other and other types of G protein-coupled receptors in the central nervous system (CNS). These heteromers may be one molecular mechanism for galanin peptides and their N-terminal fragments (gal 1-15) to modulate the function of different types of glia-neuronal networks in the CNS, especially the emotional and the cardiovascular networks. GalR-5-HT1A heteromers likely exist with antagonistic GalR-5-HT1A receptor-receptor interactions in the ascending midbrain raphe 5-HT neuron systems and their target regions. They represent a novel target for antidepressant drugs. Evidence is given for the existence of GalR1-5-HT1A heteromers in cellular models with trans-inhibition of the protomer signaling. A GalR1-GalR2 heteromer is proposed to be a galanin N-terminal fragment preferring receptor (1-15) in the CNS. Furthermore, a GalR1-GalR2-5-HT1A heterotrimer is postulated to explain why only galanin (1-15) but not galanin (1-29) can antagonistically modulate the 5-HT1A receptors in the dorsal hippocampus rich in gal fragment binding sites. The results underline a putative role of different types of GalR-5-HT1A heteroreceptor complexes in depression. GalR antagonists may also have therapeutic actions in depression by blocking the antagonistic GalR-NPYY1 receptor interactions in putative GalR-NPYY1 receptor heteromers in the CNS resulting in increases in NPYY1 transmission and antidepressant effects. In contrast the galanin fragment receptor (a postulated GalR1-GalR2 heteromer) appears to be linked to the NPYY2 receptor enhancing the affinity of the NPYY2 binding sites in a putative GalR1-GalR2-NPYY2 heterotrimer. Finally, putative GalR-α2-adrenoreceptor heteromers with antagonistic receptor-receptor interactions may be a widespread mechanism in the CNS for integration of galanin and noradrenaline signals also of likely relevance for depression

Correlation of the EPR properties of perchlorotriphenylmethyl radical and their efficiency as DNP polarizers

Water soluble perchlorinated trityl (PTM) radicals were found to be effective 95 GHz DNP (dynamic nuclear polarization) polarizers in ex situ (dissolution) 13C DNP (Gabellieri et al., Angew Chem., Int. Ed. 2010, 49, 3360). The degree of the nuclear polarization obtained was reported to be dependent on the position of the chlorine substituents on the trityl skeleton. In addition, on the basis of the DNP frequency sweeps it was suggested that the 13C NMR signal enhancement is mediated by the Cl nuclei. To understand the DNP mechanism of the PTM radicals we have explored the 95 GHz EPR characteristics of these radicals that are relevant to their performance as DNP polarizers. The EPR spectra of the radicals revealed axially symmetric g-tensors. A comparison of the spectra with the 13C DNP frequency sweeps showed that although the solid effect mechanism is operational the DNP frequency sweeps reveal some extra width suggesting that contributions from EPR forbidden transitions involving 35,37Cl nuclear flips are likely. This was substantiated experimentally by ELDOR (electron-electron double resonance) detected NMR measurements, which map the EPR forbidden transitions, and ELDOR experiments that follow the depolarization of the electron spin upon irradiation of the forbidden EPR transitions. DFT (density functional theory) calculations helped to assign the observed transitions and provided the relevant spin Hamiltonian parameters. These results show that the 35,37Cl hyperfine and nuclear quadrupolar interactions cause a considerable nuclear state mixing at 95 GHz thus facilitating the polarization of the Cl nuclei upon microwave irradiation. Overlap of Cl nuclear frequencies and the 13C Larmor frequency further facilitates the polarization of the 13C nuclei by spin diffusion. Calculation of the 13C DNP frequency sweep based on the Cl nuclear polarization showed that it does lead to an increase in the width of the spectra, improving the agreement with the experimental sweeps, thus supporting the existence of a new heteronuclear assisted DNP mechanism.