A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

Families of periodic horseshoe orbits in the restricted three-body problem

We compute families of symmetric periodic horseshoe orbits in the restricted three-body problem. Both the planar and three-dimensional cases are considered and several families are found.We describe how these families are organized as well as the behavior along and among the families of parameters such as the Jacobi constant or the eccentricity. We also determine the stability properties of individual orbits along the families. Interestingly, we find stable horseshoe-shaped orbit up to the quite high inclination of 17◦

A Note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits

Bifurcations of homoclinic tangency in two-dimensional area-preserving and reversible maps

Report for the scientific sojourn at the Research Institute for Applied Mathematics and Cybernetics, Nizhny Novgorod, Russia, from July to September 2006. Within the project, bifurcations of orbit behavior in area-preserving and reversible maps with a homoclinic tangency were studied. Finitely smooth normal forms for such maps near saddle fixed points were constructed and it was shown that they coincide in the main order with the analytical Birkhoff-Moser normal form. Bifurcations of single-round periodic orbits for two-dimensional symplectic maps close to a map with a quadratic homoclinic tangency were studied. The existence of one- and two-parameter cascades of elliptic periodic orbits was proved.

L'albedo lunar i els satèl·lits: Recerca sobre els efectes de l'albedo lunar en les plaques solars d'un satèl·lit artificial que orbita al voltant de la Lluna

Treball de recerca realitzat per una alumna d'ensenyament secundari i guardonat amb un Premi CIRIT per fomentar l'esperit científic del Jovent l'any 2009. L’albedo lunar i els satèl•lits és un treball que relaciona l’enginyeria aeroespacial amb l’astronomia. El seu objectiu principal investigar si l’albedo lunar, els rajos de sol reflectits a la superfície lunar, pot modificar significativament la temperatura de les plaques solars d’un satèl•lit artificial que orbiti la Lluna i, en conseqüència, afectar-ne el rendiment. El segon objectiu del treball és calcular si seria possible fer un mapa d’albedo lunar, a partir de la temperatura d’un satèl•lit en òrbita al voltant de la Lluna, que permetria conèixer amb més precisió la composició de la superfície lunar. Després d’adquirir els fonaments teòrics necessaris per a realitzar el treball, del procés per a trobar la manera de dur a terme els càlculs i d’efectuar els càlculs en si, les conclusions del treball són que l’albedo lunar causa un augment de temperatura en els satèl•lits prou significatiu per afectar-ne el rendiment; i que amb les temperatures enregistrades per un satèl•lit en òrbita al voltant de la Lluna es podria crear un mapa d’albedo. Aquesta recerca ha estat feta per suggeriment i sota la supervisió del CTAE (Centre de Tecnologia Aeroespacial) per analitzar si els resultats són aplicables al satèl•lit que s’enviarà a la Lluna, Lunar Mission BW1.

Exponentially small splitting of separatrices in the perturbed McMillan map

The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [MSS08].

Generic Nekhoroshev theory without small divisors

In this article, we present a new approach of Nekhoroshev theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak which combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem.Amer.Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

Emissió ràdio a baixa freqüència en fenòmens eruptius i en fonts esteses de raigs gamma d'alta i molt alta energia

La tasca investigadora presentada en aquesta memòria s'ha centrat en les fonts galàctiques de raigs gamma de molt alta energia LS I +61 303, HESS J1708-410 i HESS J1858+020. La primera és una binària de raigs gamma molt estudiada, formada per una estrella massiva i un objecte compacte. S'ha proposat un escenari on l'objecte compacte seria un púlsar jove, i la interacció del seu vent amb el vent de l'estrella generaria els raigs gamma. De totes formes, no s'ha detectat polsos procedents d'aquest putatiu púlsar. L'investigador va realitzar observacions en fase a 1280 MHz amb el radiotelescopi GMRT, sense trobar-hi polsos, cosa que implica un estricte límit superior de 0,38 mJy a la densitat mitjana de flux polsat en un putatiu púlsar amb un període major que 2 mil•lisegons en el sistema binari LS I +61 303. Per altra banda, HESS J1708-410 i HESS J1858+020 són dues fonts esteses de raigs gamma de molt alta energia de les quals no es coneix cap contrapart a d'altres longituds d'ona. L'investigador les va observar amb el GMRT, quatre vegades HESS J1708-410 (dues a 610 MHz i dues a 1400 MHz) i dues vegades HESS J1858+020 (una a cada freqüència). En les imatges realitzades amb aquestes dades no hi ha emissió estesa coincident amb les regions d'emissió de raigs gamma. HESS J1858+020 se solapa parcialment amb una font estesa que podria ser un SNR. De confirmar-se la falta de contrapartida ràdio de HESS J1708-410, estaríem parlant d'un accelerador hadrònic extraordinàriament eficient, d'una classe desconeguda fins ara.

Compositional evolution with mass transfer in closed systems

Evolution of compositions in time, space, temperature or other covariates is frequentin practice. For instance, the radioactive decomposition of a sample changes its composition with time. Some of the involved isotopes decompose into other isotopes of thesample, thus producing a transfer of mass from some components to other ones, butpreserving the total mass present in the system. This evolution is traditionally modelledas a system of ordinary di erential equations of the mass of each component. However,this kind of evolution can be decomposed into a compositional change, expressed interms of simplicial derivatives, and a mass evolution (constant in this example). A rst result is that the simplicial system of di erential equations is non-linear, despiteof some subcompositions behaving linearly.The goal is to study the characteristics of such simplicial systems of di erential equa-tions such as linearity and stability. This is performed extracting the compositional differential equations from the mass equations. Then, simplicial derivatives are expressedin coordinates of the simplex, thus reducing the problem to the standard theory ofsystems of di erential equations, including stability. The characterisation of stabilityof these non-linear systems relays on the linearisation of the system of di erential equations at the stationary point, if any. The eigenvelues of the linearised matrix and theassociated behaviour of the orbits are the main tools. For a three component system,these orbits can be plotted both in coordinates of the simplex or in a ternary diagram.A characterisation of processes with transfer of mass in closed systems in terms of stability is thus concluded. Two examples are presented for illustration, one of them is aradioactive decay

A total order in [0,1] defined through a 'next' operator

A `next' operator, s, is built on the set R1=(0,1]-{ 1-1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. Besides, the orbits {sn(a)}nare all dense in R1 and are constituted by elements of the samearithmetical character: if a is an algebraic irrational of degreek all the elements in a's orbit are algebraic of degree k; if a istranscendental, all are transcendental. Moreover, the asymptoticdistribution function of the sequence formed by the elements in anyof the half-orbits is a continuous, strictly increasing, singularfunction very similar to the well-known Minkowski's ?(×) function.

On the binary nature of the γ-ray sources AGL J2241+4454 (= MWC 656) and HESS J0632+057 (= MWC 148)

We present optical spectroscopy of MWC 656 and MWC 148, the proposed optical counterparts of the gamma-ray sources AGL J2241+4454 and HESS J0632+0 57, respectively. The main parameters of the Halpha emission line (EW, FWHM and centroid velocity) in these stars are modulated on the proposed orbital periods of 60.37 and 321 days, respectively. These modulations are likely produced by the resonant interaction of the Be discs with compact stars in eccentric orbits. We also present radial velocity curves of the optical stars folded on the above periods and obtain the first orbital elements of the two gamma-ray sources thus confirming their binary nature. Our orbital solution support eccentricities e~0.4 and 0.83+-0.08 for MWC 656 and MWC 148, respectively. Further, our orbital elements imply that the X-ray outbursts in HESS J0632+057/MWC 148 are delayed ~0.3 orbital phases after periastron passage, similarly to the case of LS I +61 303. In addition, the optical photometric light curve maxima in AGL J2241+4454/MWC 656 occur ~0.25 phases passed periastron, similar to what is seen in LS I +61 303. We also find that the orbital eccentricity is correlated with orbital period for the known gamma-ray binaries. This is explained by the fact that small stellar separations are required for the efficient triggering of VHE radiation. Another correlation between the EW of Halpha and orbital period is also observed, similarly to the case of Be/X-ray binaries. These correlations are useful to provide estimates of the key orbital parameters Porb and e from the Halpha line in future Be gamma-ray binary candidates.

Classical cross sections for relativistic spinning particles

We propose a definition of classical differential cross sections for particles with essentially nonplanar orbits, such as spinning ones. We give also a method for its computation. The calculations are carried out explicitly for electromagnetic, gravitational, and short-range scalar interactions up to the linear terms in the slow-motion approximation. The contribution of the spin-spin terms is found to be at best 10-6 times the post-Newtonian ones for the gravitational interaction.

New mechanisms for lack of equipartition of energy

We describe several mechanisms that prevent equipartition of energy in mechanical systems. In certain regimes, we present a quantitative prediction of the relative abundance of orbits exhibiting these mechanisms. This quantitative prediction is confirmed in numerical experiments.

Periodic points of holomorphic maps via Lefschetz numbers

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension n and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of CP(n) of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.