Martin E Homer

TWO-PARAMETER DISCONTINUITY-INDUCED BIFURCATIONS OF LIMIT CYCLES: CLASSIFICATION AND OPEN PROBLEMS

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. 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 Poincare map from a neighborhood 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 grazing cycle is itself degenerate (e.g. nonhyperbolic) 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 with 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.

C-bifurcations and period-adding in one-dimensional piecewise-smooth maps

Abstract This paper examines the behaviour of piecewise-smooth, continuous, one-dimensional maps that have been derived in the literature as normal forms for grazing and sliding bifurcations. These maps are linear for negative values of the parameter and non-linear for positive values of the parameter. Both C 1 and C 2 maps of this form are considered. These maps display both period-adding and period-doubling behaviour. For maps with a squared or 3/2 term the stability and existence conditions of fixed points and period-2 orbits in the vicinity of the border-collision are found analytically. These agree with the Feigin classification proposed by di Bernardo et al. [Chaos Solitons and Fractals 10 (1999) 1881]. The period-adding behaviour is examined in these maps, where analytical solutions for the boundaries of periodic solutions are found. Implicit equations for the boundaries of periodic windows for varying power term are also found and plotted. Thus, it is proved that period-adding scenarios are generic in maps of this form.

Nonlinear Dynamics and Chaos: Where do we go from here?

In keeping with the spirit of the Colston conference on Nonlinear Dynam-ics and Chaos, this chapter emphasizes ideas more than details, describingmy vision of how the bifurcation theory of multiple time scale systemswill unfold. Multiple time scale dynamical systems are rife with compli-cated phenomena. The subject has a complicated history that interweavesthree di erent viewpoints: nonstandard analysis, classical asymptotics andgeometric singular perturbation theory. My perspective is decidedly geo-metric but draws upon asymptotic analysis, recognizing the fundamentalcontributions rst expressed in the language of nonstandard analysis. Thesuccess of dynamical systems theory in elucidating patterns of bifurcationin generic systems with a single time scale motivates the goal here, namelyto extend this bifurcation theory to systems with two time scales. Thereare substantial obstacles to realizing this objective, both theoretical andcomputational. Consequently, the nal shape that the theory will take isstill fuzzy.It may seem strange to talk about computational barriers to a math-ematical theory, so I give some explanation for this. Much of the progress

Mathematical modelling of the active hearing process in mosquitoes

Insects have evolved diverse and delicate morphological structures in order to capture the inherently low energy of a propagating sound wave. In mosquitoes, the capture of acoustic energy and its transduction into neuronal signals are assisted by the active mechanical participation of the scolopidia. We propose a simple microscopic mechanistic model of the active amplification in the mosquito species Toxorhynchites brevipalpis. The model is based on the description of the antenna as a forced-damped oscillator coupled to a set of active threads (ensembles of scolopidia) that provide an impulsive force when they twitch. This twitching is in turn controlled by channels that are opened and closed if the antennal oscillation reaches a critical amplitude. The model matches both qualitatively and quantitatively with recent experiments: spontaneous oscillations, nonlinear amplification, hysteresis, 2 : 1 resonances, frequency response and gain loss owing to hypoxia. The numerical simulations presented here also generate new hypotheses. In particular, the model seems to indicate that scolopidia located towards the tip of Johnstons organ are responsible for the entrainment of the other scolopidia and that they give the largest contribution to the mechanical amplification.

High-speed atomic force microscopy in slow motion-understanding cantilever behaviour at high scan velocities

Using scanning laser Doppler vibrometer we have identified sources of noise in contact mode high-speed atomic force microscope images and the cantilever dynamics that cause them. By analysing reconstructed animations of the entire cantilever passing over various surfaces, we identified higher eigenmode oscillations along the cantilever as the cause of the image artefacts. We demonstrate that these can be removed by monitoring the displacement rather than deflection of the tip of the cantilever. We compare deflection and displacement detection methods whilst imaging a calibration grid at high speed and show the significant advantage of imaging using displacement.

Experimental observation of contact mode cantilever dynamics with nanosecond resolution.

We report the use of a laser Doppler vibrometer to measure the motion of an atomic force microscope contact mode cantilever during continuous line scans of a mica surface. With a sufficiently high density of measurement points the dynamics of the entire cantilever beam, from the apex to the base, can be reconstructed. We demonstrate nanosecond resolution of both rectangular and triangular cantilevers. This technique permits visualization and quantitative measurements of both the normal and lateral tip sample interactions for the first and higher order eigenmodes. The ability to derive quantitative lateral force measurements is of interest to the field of microtribology/nanotribology while the comprehensive understanding of the cantilevers dynamics also aids new cantilever designs and simulations.

Mathematical modeling of the radial profile of basilar membrane vibrations in the inner ear

Motivated by recent experimental results, an explanation is sought for the asymmetry in the radial profile of basilar membrane vibrations in the inner ear. A sequence of one-dimensional beam models is studied which take into account variations in the bending stiffness of the basilar membrane as well as the potential presence of structural hinges. The results suggest that the main cause of asymmetry is likely to be differences between the boundary conditions at the two extremes of the basilar membranes width. This has fundamental implications for more detailed numerical simulations of the entire cochlea.

Impact of plant shoot architecture on leaf cooling: A coupled heat and mass transfer model

Plants display a range of striking architectural adaptations when grown at elevated temperatures. In the model plant Arabidopsis thaliana, these include elongation of petioles, and increased petiole and leaf angles from the soil surface. The potential physiological significance of these architectural changes remains speculative. We address this issue computationally by formulating a mathematical model and performing numerical simulations, testing the hypothesis that elongated and elevated plant configurations may reflect a leaf-cooling strategy. This sets in place a new basic model of plant water use and interaction with the surrounding air, which couples heat and mass transfer within a plant to water vapour diffusion in the air, using a transpiration term that depends on saturation, temperature and vapour concentration. A two-dimensional, multi-petiole shoot geometry is considered, with added leaf-blade shape detail. Our simulations show that increased petiole length and angle generally result in enhanced transpiration rates and reduced leaf temperatures in well-watered conditions. Furthermore, our computations also reveal plant configurations for which elongation may result in decreased transpiration rate owing to decreased leaf liquid saturation. We offer further qualitative and quantitative insights into the role of architectural parameters as key determinants of leaf-cooling capacity.

Basins of attraction in nonsmooth models of gear rattle

This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the systems behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit).

Comparison of nonlinear mammalian cochlear-partition models

Various simple mathematical models of the dynamics of the organ of Corti in the mammalian cochlea are analyzed and their dynamics compared. The specific models considered are phenomenological Hopf and cusp normal forms, a recently proposed description combining active hair-bundle motility and somatic motility, a reduction thereof, and finally a model highlighting the importance of the coupling between the nonlinear transduction current and somatic motility. It is found that for certain models precise tuning to any bifurcation is not necessary and that a compressively nonlinear response over a range similar to experimental observations and that the normal form of the Hopf bifurcation is not the only description that reproduces compression and tuning similar to experiment.