Ducks on the torus: existence and uniqueness

We show that there exist generic slow-fast systems with only one (time-scaling) parameter on the two-torus, which have canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. Here we treat systems with a convex slow curve. In this case there is a set of parameter values accumulating to zero for which the system has exactly one attracting and one repelling canard cycle. The basin of the attracting cycle is almost the whole torus.Comment: To appear in Journal of Dynamical and Control Systems, presumably Vol. 16 (2010), No. 2; The final publication is available at www.springerlink.co

The counterphobic defense in children

The clinical data for this study were derived from the case histories of five children who consistently used the counterphobic defense either alone or in combination with phobic attitudes. The children's manifestations of this defense appeared in both verbal and nonverbal behavioral patterns. The choice of defensive style was found related to at least three factors: an early history of trauma, especially separation, parental encouragement of “toughness,” and essentially a counterphobic family style.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43947/1/10578_2005_Article_BF01433642.pd

Measuring Program Outcome

The Progress Evaluation Scales (PES) provide an efficient measuring devicefor evaluating current functioning, setting treatment goals, and assessing change over time in clinically relevant aspects of personal, social, and community adjustment. The PES can be completed by patients, significant others, and therapists, making it possible to obtain various points of view of the outcome of mental health services. This article describes the seven domains measured by the PES and the underlying dimensions they were designed to tap, and presents the generalizability, validity, and usefulness of the scales as applied to an adult mental health center population.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67322/2/10.1177_0193841X8100500402.pd

The regularized visible fold revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law

Arthroscopic removal of an osteoid osteoma of the acetabulum

In this case report, we describe the arthroscopic removal of an osteoid osteoma from the acetabulum in a young adolescent. After identifying the osteoid osteoma close to the cartilage with MRI and CT investigations, we decided that in this case, arthroscopic removal was the best treatment. In the case of an osteoid osteoma in the acetabulum close to the cartilage, arthroscopic removal should be considered as one can treat the associated osteochondritic lesion during this procedure

A mathematical framework for critical transitions: normal forms, variance and applications

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio

Moment Closure - A Brief Review

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog

Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow

Equation-free methods make possible an analysis of the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behavior depends on parameter values including an understanding of transitions between different types of qualitative behavior. These methods are introduced and explained for traffic jam formation and emergence of oscillatory pedestrian counter flow in a corridor with a narrow door

An Application of the Concept of the Therapeutic Alliance To Sadomasochistic Pathology

This paper traces the history of the therapeutic alliance concept, examining how it has been used and misused, at times elevated to a central position and at others rejected altogether. The loss of this concept created a vacuum in classical psychoanalysis that has been filled by rival theories. The continuing usefulness of looking at the treatment process through the lens of the therapeutic alliance, particularly in relation to the manifold difficulties of working with sadomasochistic pathology, is suggested. To this end, revisions of the theory of the therapeutic alliance are suggested to address some of the difficulties that have arisen in conceptualizing this aspect of the therapeutic relationship, and to provide an integrated dynamic model for working with patients at each phase of treatment. This revised model acknowledges the complexity of the domain and encompasses the multiple tasks, functions, partners, and treatment phases involved. The utility of the revised theory is illustrated in application to understanding the sadomasochistic, omnipotent resistances of a female patient through the phases of her analysis.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/66889/2/10.1177_00030651980460031301.pd