Fold, transcritical and pitchfork singularities for time-reversible systems

This paper deals with a class of singularly perturbed reversible planar vector fields around the origin where the normal hyperbolicity assumption is not assumed. We exhibit conditions for the existence of infinitely many periodic orbits and hetero-clinic cycles converging to singular orbits with respect to the Hausdorf distance. In addition, generic normal forms of such singularities are presented.

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in R-n without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

Path formulation for multiparameter D3-equivariant bifurcation problems

We implement a singularity theory approach, the path formulation, to classify D3-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a Ba-miniversal unfolding f0 of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of F0 onto its unfolding parameter space. We apply our results to degenerate bifurcation of period-3 subharmonics in reversible systems, in particular in the 1:1-resonance.

Campos de vetores lineares reversíveis equivariantes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Homoclinic bifurcations in reversible Hamiltonian systems

We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u(iv) + au - u +f(u, b) = 0 as a model, where fis an analytic function and a, b real parameters. These equations are important in several physical situations such as solitons and in the existence of finite energy stationary states of partial differential equations, but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others Hamiltonian systems and successfully employed in situations where standard methods fail. We reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations. We also plot the homoclinic values configuration in parameters space, giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations. (c) 2005 Published by Elsevier B.V.

Carbon dioxide emissions after application of tillage systems for a dark red latosol in southern Brazil

Soil tillage may influence CO2 emissions in agricultural systems. Agricultural soils are managed in several ways in Brazil, ranging from no tillage to intensive land preparation. The objective of this study was to determine the effect of common soil tillage treatments (disk harrow, reversible disk plow, rotary tiller and chisel plow tillage systems) on the intermediate CO2 emissions of a dark red latosol, located in southern Brazil. Different tillage systems produced significant differences in the CO2 emissions, and the results indicate that the chisel plow produced the highest soil carbon loss during the 15 days period after tillage treatments were performed. Emissions to the atmosphere increased as much as 74 g CO2 m(-2), at the end of a 2-week period, in the plot where the chisel plow treatment was applied, in comparison to the non-disturbed plot. The results indicate that the total increase on the intermediate term soil CO2 emissions due to tillage treatments in southern Brazil is comparable to that reported for the more humid and cooler regions. (C) 2001 Elsevier B.V. B.V All rights reserved.

Thermo-reversible sol-gel transition of TiO2 nanoparticles with surface modified by p-toluene sulfonic acid

In this paper an unprecedent thermo-reversible sol-gel transition for titania nanoparticles dispersed in a solution of p-toluene sulfonic acid (PTSH) in isopropanol is reported. The sol formed by the thermo-hydrolysis at 60 degrees C of titanium tetraisopropoxide (Ti((OPr)-Pr-i)(4)) reversibly changes into a turbid gel upon cooling to room temperature. Turbidimetric measurements performed for samples containing different nominal acidity ratios (A = [PTSH]/[Ti]) have evidenced that the gel transformation temperature increases from 20 to 35 degrees C as the [PTSH]/[Ti] ratio increases from 0.2 to 2.0. SAXS results indicate that the thermo-reversible gelation is associated to a reversible aggregation of a monodisperse set of titania nanoparticles with average gyration radius of approximate to 2 nm. From the different PTSH species evidenced by Raman spectroscopy and TG/DTA of dried gels we proposed that the then-no-reversible gelation in this systems is induced by the formation of a supramolecular network, in which the protonated surface of nanoparticles is interconnected through cooperative hydrogen bonds between -SO3 groups of p-toluene sulfonic acid. (C) 2009 Elsevier Ltd. All rights reserved.

On the number of critical periods for planar polynomial systems

In this paper we get some lower bounds for the number of critical periods of families of centers which are perturbations of the linear one. We give a method which lets us prove that there are planar polynomial centers of degree l with at least 2[(l - 2)/2] critical periods as well as study concrete families of potential, reversible and Lienard centers. This last case is studied in more detail and we prove that the number of critical periods obtained with our approach does not. increases with the order of the perturbation. (C) 2007 Elsevier Ltd. All rights reserved.

Reversible Hamiltonian Liapunov center theorem

We study the existence of periodic solutions in the neighbourhood of symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector fields. These are Hamiltonian vector fields with an involutory reversing symmetry R. We contrast the cases where R acts symplectically and anti-symplectically. In case R acts anti-symplectically, generically purely imaginary eigenvalues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing symmetric periodic solutions encircling the equilibrium point. In case R acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds containing nonsymmetric periodic solutions encircling the equilibrium point. In addition, there exists a three-dimensional invariant surface containing a two-parameter family of symmetric periodic solutions.