DEGENERATE HOPF BIFURCATIONS IN CHUA'S SYSTEM

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system

In this paper, by using the Poincare compactification in R(3) we make a global analysis of the Lorenz system, including the complete description of its dynamic behavior on the sphere at infinity. Combining analytical and numerical techniques we show that for the parameter value b = 0 the system presents an infinite set of singularly degenerate heteroclinic cycles, which consist of invariant sets formed by a line of equilibria together with heteroclinic orbits connecting two of the equilibria. The dynamical consequences related to the existence of such cycles are discussed. In particular a possibly new mechanism behind the creation of Lorenz-like chaotic attractors, consisting of the change in the stability index of the saddle at the origin as the parameter b crosses the null value, is proposed. Based on the knowledge of this mechanism we have numerically found chaotic attractors for the Lorenz system in the case of small b > 0, so nearby the singularly degenerate heteroclinic cycles.

Development and validation of a stability-indicative agar diffusion assay to determine the potency of linezolid in tablets in the presence of photodegradation products

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

Measurement of oxygen atom diffusion in Nb and Ta by anelastic spectroscopy

Impurity interstitial atoms present in metals with BCC structure can diffuse in the metallic matrix by jumps to energetically equivalent crystallographic sites. Anelastic spectroscopy (internal friction) is based on the measurement of mechanical loss or internal friction as a function of temperature. Due to its selective and nondestructive nature, anelastic spectroscopy is well suited for the study of diffusion of interstitial elements in metals. Internal friction measurements were made using the torsion pendulum technique with oscillation frequency of a few Hz, temperature interval from 300 to 700 K, heating rate of about 1 K/min, and vacuum better than 10-5 mbar. The polycrystalline Nb and Ta samples used were supplied by Aldrich Inc. The results obtained showed thermally activated relaxation structures due to stress-induced ordering of oxygen atoms around the Nb (or Ta) atoms of the metallic matrix. The results were interpreted by three methods and led to activation enthalpy values for the diffusion of oxygen in Nb and Ta of 1.15 eV and 1.10 eV, respectively.

Determination of the oxygen and nitrogen interstitial diffusion coeficient in niobium by mechanical spectroscopy

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

Diffusion of Interstitial Elements in Ti Alloys used as Biomaterials

Titanium alloys are excellent implant materials for orthopedic applications due to their desirable properties, such as good corrosion resistance, low elasticity modulus, and excellent biocompatibility. The presence of interstitial elements (such as oxygen and nitrogen) causes strong changes in the material's mechanical properties, mainly in its elastic properties. Study of the interaction among interstitial elements present in metals began with Snoek's postulate, that a stress-induced ordering of interstitials gives rise to a peak in the mechanical relaxation (internal friction) spectra. In the mechanical relaxation spectra, each species of interstitial solute atom gives rise to a distinct Snoek's peak, whose temperature and position depend on the measurement frequency. This effect is very interesting because its peculiar parameters are directly related to the diffusion coefficient (D) for the interstitial solute. This paper presents a study of diffusion of heavy interstitial elements in Ti-35Nb-7Zr-5Ta alloys using mechanical spectroscopy. Pre-exponential factors and activation energies are calculated for oxygen and nitrogen in theses alloys.

Oxygen diffusion in superconducting oxides

Since the discovery of high-temperature superconductivity of cuprate oxides, it has been clear that it is strongly affected by the oxygen content, which is also a crucial factor to determine other physical properties of high T-c superconductors. Non-stoichiometric (interstitial) oxygen strongly influences the physical properties of various superconducting oxides, in particular by creating conducting holes. It is now ascertained that the amount of holes injected depends not only on the content of interstitial oxygen, but also on its ordering. Rearrangement of the oxygen ordering may occur even below room temperature due to the unusual high mobility of these atoms. This way, mechanical spectroscopy is one of the most adequate techniques for the study of the mobility (diffusion) of oxygen atoms. This technique allows the determination of the jump frequency of an atomic species precisely, regardless of the model or the different possible types of jumps. In order to evaluate the mobility and the effect of oxygen content on these oxides, ceramic samples we prepared and submitted to several oxygen removal cycles alternately with mechanical relaxation measurements. As for SBCO, it was assumed that the peak was due to O(1)-O(5) jumps of oxygen atoms at the chain terminals or in chain fragments in the orthorhombic phase. In the case of BSCCO, the results showed complex anelastic relaxation structures, which were attributed to interstitial oxygen atom jumps between two adjacent CuO planes.

Diffusion of Nitrogen in Ti-13V-11Cr-3Al Alloys

Metals with a bcc crystalline structure such as Ti-13V-11Cr-3Al alloys have their physical properties significantly changed through the addition of interstitial elements such as oxygen and nitrogen. These metals can dissolve substantial amounts of interstitial elements forming solid solutions. Mechanical spectroscopy measurements constitute a powerful tool for studying interactions of these interstitial elements with other elements that make up the alloy. From these measurements, it is possible to obtain information regarding diffusion, interstitial concentration, interaction between interstitials, and other imperfections of the crystalline lattice, In this paper, Ti-13V-11Cr-3Al alloys with several amount of nitrogen, in a solid solution, were studied using mechanical spectroscopy (internal friction) measurements. The results presented complex internal friction spectra which were resolved in a series of constituent Debye peaks corresponding to different interactions and interstitial diffusion coefficients. Pre-exponential factors and activation energies were calculated for nitrogen in theses alloys.

Nitrogen diffusion in the Nb-1.0wt%Zr measured by mechanical spectroscopy

The scientific and technological development in the area of new materials contributed to several applications of niobium and its alloys in nuclear power plants as well as in aerospace, aeronautics, automobile and naval industries. This paper presents the interstitial diffusion coefficients of nitrogen in solid solution in the Nb-1.0wt%Zr alloy using internal friction measurements obtained by mechanical spectroscopy, which uses a torsion pendulum operating at an oscillation frequency between 1.0 Hz and 10.0 Hz. The temperature range varies from 300K to 700K, at a heating rate of 1 K/min and vacuum better than 2 x 10(-6) Torr. The results showed an increase of the interstitial diffusion coefficient of nitrogen that was correlated with configurational considerations for the octahedral interstitials.

Diffusion of Interstitial Solutes in Nb-46wt% Ti Alloys Measured by Mechanical Spectroscopy

The mechanical properties of metals with a body-centered cubic (bcc) structure, such as Nb, Ta, V, and their alloys, are modified with the introduction of interstitial impurities, such as O, N, C, or H. These metals can dissolve great amounts of O and N, for example, to form solid solutions. The interstitial solute atoms (ISA) in metals with a bcc structure occupy octahedral sites and cause local distortion with tetragonal symmetry. So ISA in these metals forms an elastic dipole that can align along one of the three cubic axis of the crystal. In the present paper, the torsion pendulum technique was employed for the investigation of various interactions among the metallic matrix and different interstitial solutes in the Nb-46wt%Ti alloy. From the relaxation spectra, we obtained the diffusion coefficients, pre-exponential factors, and activation energies for nitrogen in the Nb-46wt%Ti alloy.

Oxygen diffusion in Ti-10Mo alloys measured by mechanical spectroscopy

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

Oxygen Diffusion in an Nb-Ta Alloy Measured by Mechanical Spectroscopy

When metals that present bcc crystalline structure receive the addition of interstitial atoms as oxygen, nitrogen, hydrogen and carbon, they undergo significant changes in their physical properties because they are able to dissolve great amounts of those interstitial elements, and thus form solid solutions. Niobium and most of its alloys possess a bcc crystalline structure and, because Brazil is the largest world exporter of this metal, it is fundamental to understand the interaction mechanisms between interstitial elements and niobium or its alloys. In this study, mechanical spectroscopy (internal friction) measurements were performed on Nb-8.9wt%Ta alloys containing oxygen in solid solution. The experimental results presented complex internal friction spectra. With the addition of substitutional solute, interactions between the two types of solutes (substitutional and interstitial) were observed, considering that the random distribution of the interstitial atoms was affected by the presence of substitutional atoms. Interstitial diffusion coefficients, pre-exponential factors and activation energies were calculated for oxygen in this alloy.

Oxygen Diffusion in Ti-20Mo Alloys, used as Biomaterial, Measured by Mechanical Spectroscopy

Titanium alloys are favorable implant materials for orthopedic applications, due to their desirable properties such as good corrosion resistance, low elasticity modulus, and excellent biocornpatibility. The research on titanium alloys is concentrated in the beta type, as the Ti-20Mo alloys and the addition of interstitial elements in these metals cause changes in their mechanical properties. The mechanical spectroscopy measurements have been frequently used in order to verify the behavior of these interstitials atoms in metallic alloys. This paper presents the study of oxygen diffusion in Ti-20Mo alloys using mechanical spectroscopy measurements. A thermally activated relaxation structure was observed in the sample after oxygen doping. It was associated with the interstitial diffusion of oxygen atoms in a solid solution in the alloy. The diffusion coefficient for the oxygen diffusion in the alloy was obtained by the frequency dependence of the peak temperature and by using a simple mathematical treatment of the relaxation structure and the Arrhenius law.

Diffusion of Oxygen and Nitrogen in the Ti-15Mo Alloy used for Biomedical Applications

The Ti-15Mo alloy is a promising material for use as a biomaterial because of its excellent corrosion resistance and its good combination of mechanical properties, such as fatigue, hardness, and wears resistance. This alloy has a body-centered predominantly cubic crystalline structure and the addition of interstitial atoms, such as oxygen and nitrogen, strongly alters its mechanical properties. Mechanical spectroscopy is a powerful tool to study the interaction of interstitial elements with the matrix metal or substitutional solutes, providing information such as the distribution and the concentration of interstitial elements. The objective of this paper is to study of the effects of heavy interstitial elements, such as oxygen and nitrogen, on the anelastic properties of the Ti-15Mo alloy by using mechanical spectroscopy measurements. In this study, the diffusion coefficients, pre-exponential factors, and activation energies were calculated for the oxygen in the Ti-15Mo alloy.

Nitrogen diffusion in the Nb-2.0wt%Ti measured by mechanical spectroscopy

Metals that present bcc crystalline structure, when receiving addition of interstitial atoms as oxygen, nitrogen, hydrogen and carbon, undergo significant changes in their physical properties, being able to dissolve great amounts of those interstitial elements, thus forming solid solutions. Niobium and most of its alloys possess bcc crystalline structure and, as Brazil is the largest world exporter of this metal, it is fundamental to understand the interaction mechanisms between interstitial elements and niobium or its alloys. In this paper, mechanical spectroscopy (internal friction) measurements were performed in Nb-2.0wt%Ti alloys containing nitrogen in solid solution. The experimental results presented complex internal friction spectra and with the addition of substitutional solute, it was observed interactions between the two types of solutes (substitutional and interstitial), considering that the random distribution of the interstitial atoms was affected by the presence of substitutional atoms. Interstitial diffusion coefficients, pre-exponential factors and activation energies were calculated for nitrogen in the Nb-2.0wt%Ti alloys.