A reversible bifurcation analysis of the inverted pendulum

The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincaré map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.

Quasi-periodic stability of normally resonant tori

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.

Pacer cell response to periodic Zeitgebers

Almost all organisms show some kind of time periodicity in their behavior. Especially in mammals the neurons of the suprachiasmatic nucleus form a biological clock regulating the activity-inactivity cycle of the animal. This clock is stimulated by the natural 24-hour light-dark cycle. In our model of this system we consider each neuron as a so called phase oscillator, coupled to other neurons for which the light-dark cycle is a Zeitgeber. To simplify the model we first take an externally stimulated single phase oscillator. The first part of the phase interval is called the active state and the remaining part is the inactive state. Without external stimulus the oscillator oscillates with its intrinsic period. An external stimulus, be it from activity of neighboring cells or the periodic daylight cycle, acts twofold, it may delay the change form active to inactive and it may advance the return to the active state. The amount of delay and advance depends on the strength of the stimulus. We use a circle map as a mathematical model for this system. This map depends on several parameters, among which the intrinsic period and phase delay and advance. In parameter space we find Arnol'd tongues where the system is in resonance with the Zeitgeber. Thus already in this simplified system we find entrainment and synchronization. Also some other phenomena from biological experiments and observations can be related to the dynamical behavior of the circle map

The Paraprofessional Conundrum: Why We Need Alternative Support Strategies

This four-page article describes a conundrum facing schools utilizing paraprofessionals to support students receiving special education. It considers three factors: (a) asked to engage in teacher-type instructional roles, (b) trained and supervised for teacher-type instructional roles, and (c) compensation commensurate with teacher-type instructional roles across six combinations. Regardless of whether or not the factors are present or not it often leads to series of undesirable outcomes. The article suggest a series of alternatives

Guidelines for Selecting Alternatives to Overreliance on Paraprofessionals

The Guidelines for Selecting Alternatives to Overreliance on Paraprofessionals is a field-tested schhol-based planning process. Support for the preparation of this article was provided by the United States Department of Education, Office of Special Education and Rehabilitative Services under the funding category, Model Demonstration Projects for Children and Youth with Disabilities, CFDA 84.324M (H324M02007), awarded to the Center on Disability and Community Inclusion at the University of Vermont. The contents of this document reflect the ideas and positions of the authors and do not necessarily reflect the ideas or positions of the U.S. Department of Education; therefore, no official endorsement should be inferred