Two-parametric unfoldings for planar invisible double-tangency singularities.

We consider the family of planar discontinuous piecewise linear systems in two regions with a straight line as boundary and one tangency point in each region. First, we recall a seven-parametric canonical form for such a family. Next, we select two parameters as bifurcation parameters. One of them represents the distance between the tangency points. It is known that this bifurcation parameter unfolds the called pseudo-Hopf bifurcation. The coefficient that determines the stability of the crossing limit cycle that emerges from this bifurcation mechanism, called the first Lyapunov coefficient, is considered our second bifurcation parameter. Finally, with these two bifurcation parameters, we give a two-parametric unfolding for the invisible fold-fold and focus-fold singularities. The bifurcation diagram for each unfolding consists of two curves of bifurcation points: a curve of pseudo-Hopf bifurcation points and a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin (pB) bifurcation because of the dynamical behavior that is same as the Bautin bifurcation for smooth dynamical systems.

Short-term outcomes from a pilot randomized controlled trial evaluating a virtual culturally adapted parenting intervention among Filipino parents of school-age children.

BACKGROUND: Filipino youth in the United States have significant behavioral health problems, including high rates of depression and suicidal behavior. Evidence-based parenting groups promote positive parenting practices and improve child behavior, yet few have been implemented online. OBJECTIVES: This study tested the short-term effects of a culturally adapted hybrid version of the Incredible Years School Age Basic and Advance Programs when delivered online among groups of parents of Filipino children and estimated intervention effect sizes. METHOD: Forty-nine parents of children aged 8-12 years recruited from schools, clinics, community organizations, and social media were randomly assigned to intervention or a wait-list control group that received literature from the American Academy of Pediatrics' Bright Futures program. The intervention consisted of 12 weekly 2-hr sessions. Parent perceptions of child behavior, parenting practices, and parenting stress as well as child surveys of anxiety and depression symptoms using validated assessments were obtained at baseline and 3-month postintervention follow-up. RESULTS: Forty parents completed both baseline and follow-up surveys with a mean attendance of 9.35 out of 12 sessions (n = 18). Analysis of covariance comparing 3-month (pre-/postintervention) changes revealed that the program had a statistically significant positive impact on parenting practices (positive verbal discipline, praise and incentives, and clear expectations); parent perceptions of their child's internalizing symptoms; and child-reported anxiety and depression symptoms. CONCLUSIONS: Results support the feasibility and potential effectiveness of offering an online evidence-based parenting program to promote positive parenting and decrease child anxiety and depression. This multigenerational approach to mental health prevention could potentially help address the growing mental health epidemic among youth. (PsycInfo Database Record (c) 2023 APA, all rights reserved).