On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

The effects of parental illness and other ill family members on the adjustment of children

Background This study addresses limitations of prior research that have used group comparison designs to test the effects of parental illness on youth. Purpose This study examined differences in adjustment between children of a parent with illness and peers from ‘healthy’ families controlling for the effects of whether a parent or non-parent family member is ill, illness type, demographics and caregiving. Methods Based on questionnaire data, groups were derived from a community sample of 2,474 youth (‘healthy’ family, n = 1768; parental illness, n = 336; other family member illness, n = 254; both parental and other family illness, n = 116). Results The presence of any family member with an illness is associated with greater risk of mental health difficulties for youth relative to peers from healthy families. This risk is elevated if the ill family member is a parent and has mental illness or substance misuse. Conclusions Serious health problems within a household adversely impact youth adjustment.