Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

The comfortable roller coaster -- on the shape of tracks with constant normal force

A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is: what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.Comment: 10 pages, 4 figure

Chatter, sticking and chaotic impacting motion in a two-degree of freedom impact oscillator

We consider the dynamics of a two-degree of freedom impact oscillator subject to a motion limiting constraint. These systems exhibit a range of periodic and nonperiodic impact motions. For a particular set of parameters, we consider the bifurcations which occur between differing regimes of impacting motion and in particular those which occur due to a grazing bifurcation. Unexpected resonant behavior is also observed, due to the complexity of the dynamics. We consider both periodic and chaotic chatter motions and the regions of sticking which exist. Finally we consider the types of chaotic motion that occur within the parameter range. We discuss the possibility in relating successive low velocity impacts, especially with respect to possible low dimensional mappings for such a system

Muscle Activation in Individuals Who are Status-Post a Stroke during Over Ground, Treadmill, and Body Weight Supported Gait: A Comparative Study

Purpose: The purpose of this study was to evaluate individuals who are status post stroke during ambulation over ground, on a treadmill without support, and on a treadmill with partial body weight support (PBWS) to determine if there are any differences in muscle activation of major muscle groups in the lower extremity. Subjects: Two subjects were recruited for this study. Subjects were included if they were over the age of 50 years, could fulfill a two-hour time commitment, and walk independently with or without the use of an assistive device. Subjects were excluded if they have had surgery or an existing orthopedic involvement of the lower extremities. Instrumentation: Sensor surface electrodes were used to pick up electromyography (EMG) activity. EMG activity was collected and final data is presented as percent of normalized EMG activity as an average of four to five gait cycles for each of the support trials. Procedure: Consent forms were reviewed and signed by each participant. Electrodes and a heel switch were placed on the involved lower extremity. Subjects performed two trial walks over level ground. Subjects also performed three treadmill ambulation conditions in random order: ambulation on the treadmill without a harness (trm), ambulation on the treadmill with a harness and no body weight support (trmh), and ambulation on the treadmill with a harness and PBWS of 15% (trms). Subjects ambulated for two minutes and EMG activity was recorded for 30 seconds at the end of each minute during each trial condition. Data Analysis: The mean EMG activity of the second trial for all ambulation conditions was calculated for both stance and swing phases of each subject. Descriptive statistics were then used to compare muscle activation across conditions, as well as rank EMG mean muscle activity for each trial condition from highest to lowest. Results: EMG rankings were inconsistent across conditions, but both subjects had the least gastrocnemius activity during the trms condition. Conclusion and Clinical Implication: There were no major findings or trends to suggest differences or similarities in muscle activation between any of the conditions for either subject. Therefore, further research is needed

Simple model of bouncing ball dynamics. Displacement of the limiter assumed as a cubic function of time

Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter's motion making analysis of chattering possible.Comment: 8 pages, 1 figure, presented at the DSTA 2011 conference, Lodz, Polan