Continuity of Lyapunov functions and of energy level for generalized gradient semigroup

The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions)-defined on the whole phase space- which proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results.

Analysis of energy dissipation in stirred suspension polymerisation reactors using computational fluid dynamics

This work evaluates the spatial distribution of normalised rates of droplet breakage and droplet coalescence in liquidliquid dispersions maintained in agitated tanks at operation conditions normally used to perform suspension polymerisation reactions. Particularly, simulations are performed with multiphase computational fluid dynamics (CFD) models to represent the flow field in liquidliquid styrene suspension polymerisation reactors for the first time. CFD tools are used first to compute the spatial distribution of the turbulent energy dissipation rates (e) inside the reaction vessel; afterwards, normalised rates of droplet breakage and particle coalescence are computed as functions of e. Surprisingly, multiphase simulations showed that the rates of energy dissipation can be very high near the free vortex surfaces, which has been completely neglected in previous works. The obtained results indicate the existence of extremely large energy dissipation gradients inside the vessel, so that particle breakage occurs primarily in very small regions that surround the impeller and the free vortex surface, while particle coalescence takes place in the liquid bulk. As a consequence, particle breakage should be regarded as an independent source term or a boundary phenomenon. Based on the obtained results, it can be very difficult to justify the use of isotropic assumptions to formulate particle population balances in similar systems, even when multiple compartment models are used to describe the fluid dynamic behaviour of the agitated vessel. (C) 2011 Canadian Society for Chemical Engineering

Decay of energy and suppression of Fermi acceleration in a dissipative driven stadium-like billiard

The behavior of the average energy for an ensemble of non-interacting particles is studied using scaling arguments in a dissipative time-dependent stadium-like billiard. The dynamics of the system is described by a four dimensional nonlinear mapping. The dissipation is introduced via inelastic collisions between the particles and the moving boundary. For different combinations of initial velocities and damping coefficients, the long time dynamics of the particles leads them to reach different states of final energy and to visit different attractors, which change as the dissipation is varied. The decay of the average energy of the particles, which is observed for a large range of restitution coefficients and different initial velocities, is described using scaling arguments. Since this system exhibits unlimited energy growth in the absence of dissipation, our results for the dissipative case give support to the principle that Fermi acceleration seems not to be a robust phenomenon. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3699465]

Maternal Melatonin Programs the Daily Pattern of Energy Metabolism in Adult Offspring

Background: Shift work was recently described as a factor that increases the risk of Type 2 diabetes mellitus. In addition, rats born to mothers subjected to a phase shift throughout pregnancy are glucose intolerant. However, the mechanism by which a phase shift transmits metabolic information to the offspring has not been determined. Among several endocrine secretions, phase shifts in the light/dark cycle were described as altering the circadian profile of melatonin production by the pineal gland. The present study addresses the importance of maternal melatonin for the metabolic programming of the offspring. Methodology/Principal Findings: Female Wistar rats were submitted to SHAM surgery or pinealectomy (PINX). The PINX rats were divided into two groups and received either melatonin (PM) or vehicle. The SHAM, the PINX vehicle and the PM females were housed with male Wistar rats. Rats were allowed to mate and after weaning, the male and female offspring were subjected to a glucose tolerance test (GTT), a pyruvate tolerance test (PTT) and an insulin tolerance test (ITT). Pancreatic islets were isolated for insulin secretion, and insulin signaling was assessed in the liver and in the skeletal muscle by western blots. We found that male and female rats born to PINX mothers display glucose intolerance at the end of the light phase of the light/dark cycle, but not at the beginning. We further demonstrate that impaired glucose-stimulated insulin secretion and hepatic insulin resistance are mechanisms that may contribute to glucose intolerance in the offspring of PINX mothers. The metabolic programming described here occurs due to an absence of maternal melatonin because the offspring born to PINX mothers treated with melatonin were not glucose intolerant. Conclusions/Significance: The present results support the novel concept that maternal melatonin is responsible for the programming of the daily pattern of energy metabolism in their offspring.