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.

Resonances and subharmonic bifurcations of large amplitude periodic orbits of planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restriction to any compact set. The global study involving infinity is performed via the Poincare Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves C-(m) of subharmonic bifurcations, for which the periodically perturbed system has subharmonics of order m, for any integer m.

Periodic perturbations of quadratic planar polynomial vector fields

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Heteroclinic cycles in lattice dynamical systems

A criterion of spatial chaos occurring in lattice dynamical systems-heteroclinic cycle-is discussed. It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos.

Discussion on the geometric structure of strange attractor

We discuss the transversal heteroclinic cycle formed by hyperbolic periodic pointes of diffeomorphism on the differential manifold. We point out that every possible kind of transversal heteroclinic cycle has the Smalehorse property and the unstable manifolds of hyperbolic periodic points have the closure relation mutually. Therefore the strange attractor may be the closure of unstable manifolds of a countable number of hyperbolic periodic points. The Henon mapping is used as an example to show that the conclusion is reasonable.

The strange attractor of a kind of 2-dimensional map and dynamical properties on it

On the basis of previous works, the strange attractor in real physical systems is discussed. Louwerier attractor is used as an example to illustrate the geometric structure and dynamical properties of strange attractor. Then the strange attractor of a kind of two-dimensional map is analysed. Based on some conditions, it is proved that the closure of the unstable manifolds of hyberbolic fixed point of map is a strange attractor in real physical systems.

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

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Effects of cooling time on a closed LWR fuel cycle

In this study, the effects of cooling time prior to reprocessing spent LWR fuel has on the reactor physics characteristics of a PWR fully loaded with homogeneously mixed U-Pu or U-TRU oxide (MOX) fuel is examined. A reactor physics analysis was completed using the CASM04e code. A void reactivity feedback coefficient analysis was also completed for an infinite lattice of fresh fuel assemblies. Some useful conclusions can be made regarding the effect that cooling time prior to reprocessing spent LWR fuel has on a closed homogeneous MOX fuel cycle. The computational analysis shows that it is more neutronically efficient to reprocess cooled spent fuel into homogeneous MOX fuel rods earlier rather than later as the fissile fuel content decreases with time. Also, the number of spent fuel rods needed to fabricate one MOX fuel rod increases as cooling time increases. In the case of TRU MOX fuel, with time, there is an economic tradeoff between fuel handling difficulty and higher throughput of fuel to be reprocessed. The void coefficient analysis shows that the void coefficient becomes progressively more restrictive on fuel Pu content with increasing spent fuel cooling time before reprocessing.

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.

Net power optimization of an irreversible Otto cycle using ECOP and ecological function

This paper presents an analysis of an irreversible Otto cycle aiming to optimize the net power through ECOP and ecological function. The studied cycle operates between two thermal reservoirs of infinite thermal capacity, with internal irreversibilities derived from non-isentropic behavior of compression and expansion processes, irreversibilities from thermal resistance in heat exchangers and heat leakage from the high temperature reservoir to the low temperature reservoir. Analytical expressions are applied for the power outputs optimized by the ECOP, by the ecological function and by the maximum power criteria, in conjunction with a graphic analysis, in which some cycle operation parameters are analyzed for an increased comprehension of the effects of the irreversibilities in the optimized power.