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.

Relationships between gender, age, family conditions, physical and mental health, and social isolation of elderly caregivers

Background: In an aging population an increasing number of elderly caregivers will be called upon to provide care over a long period, during which time they will be burdened both by caregiving and by the physiological effects of their own aging. Among them there will be more aged male caregivers, who will probably be less prepared than women to become caregivers. The aim of this study was to investigate the relationship between caregivers' gender, age, family income, living arrangements and social support as independent variables, and depressive symptoms, comorbidities, level of frailty, grip strength, walking speed and social isolation, as dependent variables. Methods: 176 elderly people (123 women) were selected from a sample of a population-based study on frailty (n = 900), who had cared for a spouse (79.3%) and/or parents (31.4%) in the past five years (mean age = 71.8 +/- 4.86 years; mean monthly family income in minimum wages = 4.64 +/- 5.14). The study used questionnaires and self-report scales, grip strength and walking speed tests. Results: 65% of participants evaluated caregiving as being very stressful. Univariate analyses of regression showed low family income as a risk factor for depression; being female and low perceived social support as a risk for comorbidities; being 80 years of age and above for low grip strength; and being male for social isolation indicated by discontinuity of activities and social roles. In multivariate analyses of regression, poverty arose as a risk factor for depression and being female for comorbidities. Conclusions: Gender roles, age, income and social support interacted with physical and emotional health, and with the continuity of social participation of elderly caregivers. Special attention must be given to male caregivers.