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 .

Equilibrium points and central configurations for the Lennard-Jones 2-and 3-body problems

Abstract. In this paper we study the relative equilibria and their stability for a system of three point particles moving under the action of a Lennard{Jones potential. A central con guration is a special position of the particles where the position and acceleration vectors of each particle are proportional, and the constant of proportionality is the same for all particles. Since the Lennard{Jones potential depends only on the mutual distances among the particles, it is invariant under rotations. In a rotating frame the orbits coming from central con gurations become equilibrium points, the relative equilibria. Due to the form of the potential, the relative equilibria depend on the size of the system, that is, depend strongly of the momentum of inertia I. In this work we characterize the relative equilibria, we nd the bifurcation values of I for which the number of relative equilibria is changing, we also analyze the stability of the relative equilibria.

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.

Myofascial pain syndrome associated with trigger points: A literature review. (I): Epidemiology, clinical treatment and etiopathogeny

Over the last few decades, advances have been made in the understanding of myofascial pain syndrome epidemiology, clinical characteristics and aetiopathogenesis, but many unknowns remain. An integrated hypothesis has provided a greater understanding of the physiopathology of trigger points, which may allow the development of new diagnostic, and above all, therapeutic methods, as well as the establishment of prevention policies and protocols by the health profession. Nevertheless, randomized studies are needed to provide a better understanding and detection of the different factors involved in the origin of trigger points.

Leverage and beliefs: Personal experience and risk taking in margin lending

What determines risk-bearing capacity and the amount of leverage in financial markets? Thispaper uses unique micro-data on collateralized lending contracts during a period of financialdistress to address this question. An investor syndicate speculating in English stocks wentbankrupt in 1772. Using hand-collected information from Dutch notarial archives, we examinechanges in lenders' behavior following exposure to potential (but not actual) losses. Before thedistress episode, financiers that lent to the ill-fated syndicate were indistinguishable from therest. Afterwards, they behaved differently: they lent with much higher haircuts. Only lendersexposed to the failed syndicate altered their behavior. The differential change is remarkable sincethe distress was public knowledge, and because none of the lenders suffered actual losses ? allfinanciers were repaid in full. Interest rates were also unaffected; the market balanced solelythrough changes in collateral requirements. Our findings are consistent with a heterogeneousbeliefs-interpretation of leverage. They also suggest that individual experience can modify thelevel of leverage in a market quickly.

Density, climate and varying return points: an analysis of long-term population fluctuations in the threatened European tree frog.

Experimental research has identified many putative agents of amphibian decline, yet the population-level consequences of these agents remain unknown, owing to lack of information on compensatory density dependence in natural populations. Here, we investigate the relative importance of intrinsic (density-dependent) and extrinsic (climatic) factors impacting the dynamics of a tree frog (Hyla arborea) population over 22 years. A combination of log-linear density dependence and rainfall (with a 2-year time lag corresponding to development time) explain 75% of the variance in the rate of increase. Such fluctuations around a variable return point might be responsible for the seemingly erratic demography and disequilibrium dynamics of many amphibian populations.

Improving T2 -weighted imaging at high field through the use of kT -points.

PURPOSE: At high magnetic field strengths (B0 ≥ 3 T), the shorter radiofrequency wavelength produces an inhomogeneous distribution of the transmit magnetic field. This can lead to variable contrast across the brain which is particularly pronounced in T2 -weighted imaging that requires multiple radiofrequency pulses. To obtain T2 -weighted images with uniform contrast throughout the whole brain at 7 T, short (2-3 ms) 3D tailored radiofrequency pulses (kT -points) were integrated into a 3D variable flip angle turbo spin echo sequence. METHODS: The excitation and refocusing "hard" pulses of a variable flip angle turbo spin echo sequence were replaced with kT -point pulses. Spatially resolved extended phase graph simulations and in vivo acquisitions at 7 T, utilizing both single channel and parallel-transmit systems, were used to test different kT -point configurations. RESULTS: Simulations indicated that an extended optimized k-space trajectory ensured a more homogeneous signal throughout images. In vivo experiments showed that high quality T2 -weighted brain images with uniform signal and contrast were obtained at 7 T by using the proposed methodology. CONCLUSION: This work demonstrates that T2 -weighted images devoid of artifacts resulting from B1 (+) inhomogeneity can be obtained at high field through the optimization of extended kT -point pulses. Magn Reson Med 71:1478-1488, 2014. © 2013 Wiley Periodicals, Inc.

Equidistribution of Fekete Points on the Sphere

Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of Fekete points in the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz-Zygmund arrays and interpolating arrays, that have been studied recently.