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.

PERIODIC PERTURBATION of QUADRATIC SYSTEMS WITH TWO INFINITE HETEROCLINIC CYCLES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the existence of infinite heteroclinic cycles in polynomial systems and its dynamic consequences

In this work we consider the dynamic consequences of the existence of infinite heteroclinic cycle in planar polynomial vector fields, which is a trajectory connecting two saddle points at infinity. It is stated that, although the saddles which form the cycle belong to infinity, for certain types of nonautonomous perturbations the perturbed system may present a complex dynamic behavior of the solutions in a finite part of the phase plane, due to the existence of tangencies and transversal intersections of their stable and unstable manifolds. This phenomenon might be called the chaos arising from infinity. The global study at infinity is made via the Poincare Compactification and the argument used to prove the statement is the Birkhoff-Smale Theorem. (c) 2004 WILEY-NCH Verlag GmbH & Co. KGaA, Weinheim.

On the global dynamics of the Rabinovich system

In this paper by using the Poincare compactification in R(3) make a global analysis of the Rabinovich system(x) over dot = hy - v(1)x + yz, (y) over dot = hx - v(2)y - xz, (z) over dot = -v(3)z + xy,with (x, y, z) is an element of R(3) and ( h, v(1), v(2), v(3)) is an element of R(4). We give the complete description of its dynamics on the sphere at infinity. For ten sets of the parameter values the system has either first integrals or invariants. For these ten sets we provide the global phase portrait of the Rabinovich system in the Poincare ball (i.e. in the compactification of R(3) with the sphere S(2) of the infinity). We prove that for convenient values of the parameters the system has two families of singularly degenerate heteroclinic cycles. Then changing slightly the parameters we numerically found a four wings butterfly shaped strange attractor.

Periodic perturbations of quadratic planar polynomial vector fields

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Dynamics at infinity and other global dynamical aspects of Shimizu-Morioka equations

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

GLOBAL DYNAMICS IN THE POINCARE BALL of THE CHEN SYSTEM HAVING INVARIANT ALGEBRAIC SURFACES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

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.

Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes

In this paper, we propose a model for the destruction of three-dimensional horseshoes via heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call generating. © 2008 Cambridge University Press.

Severity of banana leaf spot in an intercropping system in two cycles of banana Prata Ana

Prata Ana is the most planted banana cultivar in northern Minas Gerais, Brazil. It is however susceptible to several pathogens. This study was carried out to evaluate the disease severity of banana leaf spot in the Prata Ana cv. in the first and second cycle under six different planting systems. The randomized block experimental design was used with six treatments and four replications. lit an evaluation of the severity of banana leaf spot, no disease symptoms were found on Thap Maeo and Caipira. The evolution curve of the disease indicated seasonal effects in the first and second cycles. The severity, of banana leaf spot was highest soon after the regional rainy period from November to March. A comparison of the means of the evaluations indicated a reduction in disease severity from the first to the second cycle.

Monitoring ovarian cycles and pregnancy in brown brocket deer (Mazama gouazoubira) by measurement of fecal progesterone metabolites

In the present study, pregnancy and the estrous cycle were monitored in captive brown brocket deer (Mazama gouazoubira) by measuring fecal progestagens with a commercial enzyme immunoassay (EIA), along with behavioral data. Fecal samples were collected twice a week during pregnancy and daily during the estrous cycle and post-partum period. It was possible to distinguish between inter-luteal and luteal phases of the estrous cycle. Behavioral estrus corresponded with low concentrations of fecal progestagens. Samples from two consecutive cycles were available from five hinds, and the mean estrous cycle (n = 10) was 26.9 +/- 1.7 d (mean +/- S.E.M.). However, when two extreme cycles (34 and 37 d) were deleted, the mean estrous cycle was 24.7 +/- 1.2 d. Three animals became pregnant (gestation ranged from 208 to 215 d). After fertile breeding, progestagen concentration in these hinds remained among luteal phase concentrations throughout pregnancy, with the exception of a few peaks. Within 4 d post-partum, two hinds reached interluteal phase values, while one hind maintained luteal concentrations for at least 1 week. (C) 2005 Elsevier B.V. All rights reserved.

LIMIT CYCLES of DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS

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

The effect of thermal cycles on the mechanical properties of fiber-metal laminates

The effect of thermal-shock cycles on the mechanical properties of fiber-metal laminates (FMLs) has been evaluated. FML plates were composed by two AA2024 Al sheets (1.6 mm thick) and one composite ply formed by two layers of unidirectional glass fiber epoxy prepreg and two layers of epoxy adhesive tape of glass fiber reinforced epoxy adhesive. The set was manufactured by hand layup and typical vacuum bag technique. The curing cycle was in autoclave at 125 +/- 5 degrees C for 90 min and an autoclave pressure of 400 kPa. FML coupons taken from the manufactured plate were submitted to temperature variations between -50 and +80 degrees C, with a fast transition between these temperatures. Tensile and interlaminar shear strength were evaluated on samples after 1000 and 2000 cycles, and compared to nonexposed samples. 2000 Cycles corresponds to typical C Check interval for commercial aircraft maintenance programs. It was observed that the thermal-shock cycles did not result in significant microstructural changes on the FML, particularly on the composite ply. Similarly, no appreciable effect on the mechanical properties of FML was observed by the thermal-shock cycles. (c) 2012 Elsevier Ltd. All rights reserved.

Bifurcation of limit cycles from an n-dimensional linear center inside a class of piecewise linear differential systems

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