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Very strong El Niño events occur sporadically every 10-20 years. The origin of this bursting behavior still remains elusive. Using a simplified 3-dimensional dynamical model of the tropical Pacific climate system, which captures the El Niño-Southern Oscillation (ENSO) combined with recently developed mathematical tools for fast-slow systems we show that decadal ENSO bursting behavior can be explained as a Mixed Mode Oscillation (MMO), which also predicts a critical threshold for rapid amplitude growth. It is hypothesized that the MMO dynamics of the low-dimensional climate model can be linked to a saddle-focus equilibrium point, which mimics a tropical Pacific Ocean state without ocean circulation.

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A large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25,7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

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BACKGROUND: Maternal postpartum hypertensive emergency is a major cause of maternal mortality and maternal readmission, yet prediction of women who require readmission is limited with false negatives and false positives. OBJECTIVE: This study aimed to develop and validate a predictive algorithm for maternal postpartum readmission from complications of hypertensive disorders of pregnancy using machine learning. STUDY DESIGN: We performed a cohort study of pregnant women delivering at a single institution using prospectively collected clinical information available from the electronic medical record at the time of discharge. Our primary outcome was readmission within 42 days of delivery for complications of hypertensive disorders of pregnancy. The data set was divided into a derivation and a separate validation cohort. In the derivation cohort, 10 independent data sets were created by randomly suppressing 10% of the population, and then clinical features predictive of complications of hypertensive disorders of pregnancy were analyzed using machine learning to optimize the area under the curve. Once an optimal model was determined, this model was then validated using a second independent validation cohort. RESULTS: A total of 20,032 delivering women with 238 readmissions for complications of hypertensive disorders of pregnancy (1.2%) were included in the derivation cohort. The validation cohort consisted of 5823 women with 82 readmissions for complications of hypertensive disorders of pregnancy (1.4%). The demographics were similar between the 2 populations as was the test performance characteristics (area under the curve, 0.85 in the derivation cohort vs 0.81 in the validation cohort). Both the derivation and validation algorithms used 31 clinical features that were found to be comparably predictive in both models. CONCLUSION: In this evaluation of a machine learning algorithm, readmission for complications of hypertensive disorders of pregnancy can be predicted with reasonable accuracy using clinical data at the time of discharge. Validation of this model in other care settings is necessary to validate its utility.

We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard phenomena in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards "with heads." The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation-whether leading to canards or super-explosion-can be subcritical.