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 heteroclinic loops at infinity for a class of polynomial vector fields

For polynomial vector fields in R3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

Societal metabolism of societies: the bifurcation between Spain and Ecuador

This paper presents an application of the Multiple-Scale Integrated Assessment of Societal Metabolism to the recent economic history of Ecuador and Spain. Understanding the relationship between the Gross Domestic Product (GDP) and the throughput of matter and energy over time in modern societies is crucial for understanding the sustainability predicament as it is linked to economic growth. When considering the dynamics of economic development, Spain was able to take a different path than Ecuador thanks to the different characteristics of its energy budget and other key variables. This and other changes are described using economic and biophysical variables (both extensive and intensive referring to different hierarchical levels). The representation of these parallel changes (on different levels and describable only using different variables) can be kept in coherence by adopting the frame provided by MSIASM.

Periodic orbits near a heteroclinic loop formed by one dimensional orbit and a 2-dimensional manifold: application to the charged collinear 3-body problem

The paper is devoted to the study of a type of differential systems which appear usually in the study of some Hamiltonian systems with 2 degrees of freedom. We prove the existence of infinitely many periodic orbits on each negative energy level. All these periodic orbits pass near the total collision. Finally we apply these results to study the existence of periodic orbits in the charged collinear 3–body problem.

Hopf bifurcation of a delay differential equation with two delays

We consider a delay differential equation with two delays. The Hopf bifurcation of this equation is investigated together with the stability of the bifurcated periodic solution, its period and the bifurcation direction. Finally, three applications are given.

Hopf bifurcation for degenerate singular points of multiplicity 2n-1 in dimension 3

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Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

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Hopf bifurcation in higher dimensional differential systems via the averaging method

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The Energy metabolism of China and India between 1971-2010: studying the bifurcation

This paper presents a comparison of the changes in the energetic metabolic pattern of China and India, the two most populated countries in the world, with two economies undergoing an important economic transition. The comparison of the changes in the energetic metabolic pattern has the scope to characterize and explain a bifurcation in their evolutionary path in the recent years, using the Multi-Scale Integrated Analysis of Societal and Ecosystem Metabolism (MuSIASEM) approach. The analysis shows an impressive transformation of China’s energy metabolism determined by the joining of the WTO in 2001. Since then, China became the largest factory of the world with a generalized capitalization of all sectors ―especially the industrial sector― boosting economic labor productivity as well as total energy consumption. India, on the contrary, lags behind when considering these factors. Looking at changes in the household sector (energy metabolism associated with final consumption) in the case of China, the energetic metabolic rate (EMR) soared in the last decade, also thanks to a reduced growth of population, whereas in India it remained stagnant for the last 40 years. This analysis indicates a big challenge for India for the next decade. In the light of the data analyzed both countries will continue to require strong injections of technical capital requiring a continuous increase in their total energy consumption. When considering the size of these economies it is easy to guess that this may induce a dramatic increase in the price of energy, an event that at the moment will penalize much more the chance of a quick economic development of India.

Loop bifurcation and magnetization rotation in exchange-biased Ni/FeF2

Exchange-biased Ni/FeF2 films have been investigated using vector coil vibrating-sample magnetometry as a function of the cooling field strength HFC . In films with epitaxial FeF2 , a loop bifurcation develops with increasing HFC as it divides into two sub-loops shifted oppositely from zero field by the same amount. The positively biased sub-loop grows in size with HFC until only a single positively shifted loop is found. Throughout this process, the negative and positive (sub)loop shifts maintain the same discrete value. This is in sharp contrast to films with twinned FeF2 where the exchange field gradually changes with increasing HFC . The transverse magnetization shows clear correlations with the longitudinal subloops. Interestingly, over 85% of the Ni reverses its magnetization by rotation, either in one step or through two successive rotations. These results are due to the single-crystal nature of the antiferromagnetic FeF2 , which breaks down into two opposite regions of large domains.

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.

Bifurcation of relative equilibria of the (1+3)-body problem

We study the relative equilibria of the limit case of the pla- nar Newtonian 4{body problem when three masses tend to zero, the so-called (1 + 3){body problem. Depending on the values of the in- nitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the oth- ers are concave. Each convex relative equilibrium of the (1 + 3){body problem can be continued to a unique family of relative equilibria of the general 4{body problem when three of the masses are su ciently small and every convex relative equilibrium for these masses belongs to one of these six families.

Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a two-parameter framework and the nonlinear phenomena associated with such collision are considered. By dealing with planar discontinuous (Filippov) systems, some of such phenomena are pointed out through specific representative cases. A methodology for obtaining the corresponding bi-parametric bifurcation sets is developed.