Lyapunov coefficients for Hopf bifurcations in systems with piecewise smooth nonlinearity

Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching (super- versus sub-criticality), but in general this differs from any fixed smoothening of the vector field. We focus on non-smooth nonlinearities of the form $u_i|u_j|$, but our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part. In addition, we discuss some codimension-one degeneracies and apply the results to a model of a shimmying wheel.Comment: 39 pages, 10 figure

Behavior change training for pregnant women's communication during birth: A randomized controlled trial.

Applying health psychological theories can improve communication interventions to empower pregnant women and ensure safe births. The aim was to test a short digital communication intervention based on the health action process approach. A randomized-controlled trial was conducted with pregnant women at two German university hospitals. The intervention group (NT1  = 225; NT2  = 142) received a 2.5 h online training focusing on communication planning, self-efficacy and communicating personal needs and preferences under difficult circumstances. This group was compared with a passive control group (NT1  = 199; NT2  = 144). Data from the N = 286 women with complete datasets were used for multilevel analyses. Data from all recruited N = 424 women were used for intention-to-treat analyses with multiple imputation. Both groups improved regarding communication behavior, quality of birth, action planning, coping planning and coping self-efficacy after birth, which was more pronounced in the intervention group. The intention-to-treat analyses confirmed the higher improvement for communication behavior, perceived quality of birth and coping planning. The intervention was related to improvements in pregnant women's communication behavior and quality of birth. Hence, future research and practice should apply and evaluate health psychological theories when targeting communication and empowerment