Browsing by Browse by FOR 2008 "010204 Dynamical Systems in Applications"

We propose a theory that qualitatively predicts the stability and equilibrium structure of long-lived, quasi-steady flow states in decaying two-dimensional turbulence. This theory combines a maximum entropy principal with a nonlinear parameterization of the vorticity-stream-function dependency of such long-lived states. In particular, this theory predicts unidirectional-flow states that are bistable, exhibit hysteresis, and undergo large abrupt changes in flow topology; and a vortex-pair state that undergoes continuous changes in flow topology. These qualitative predictions are confirmed in numerical simulations of the two-dimensional Navier-Stokes equation. We discuss limitations of the theory, and why a reduced quantitative theory of long-lived flow states is difficult to obtain. We also provide a partial theoretical justification for why certain sets of initial conditions go to certain long-lived flow states.

Hopfield's Lyapunov function is used to view the stability and topology of equilibria in neuronal networks for visual rivalry and pattern formation. For two neural populations with reciprocal inhibition and slow adaptation, the dynamics of neural activity is found to include a pair of limit cycles: one for oscillations between states where one population has high activity and the other has low activity, as in rivalry, and one for oscillations between states where both populations have the same activity. Hopfield's Lyapunov function is used to find the dynamical mechanism for oscillations and the basin of attraction of each limit cycle. For a spatially continuous population with lateral inhibition, stable equilibria are found for local regions of high activity (solitons) and for bound states of two or more solitons. Bound states become stable when moving two solitons together minimizes the Lyapunov function, a result of decreasing activity in regions between peaks of high activity when the firing rate is described by a sigmoid function. Lowering the barrier to soliton formation leads to a pattern-forming instability, and a nonlinear solution to the dynamical equations is found to be given by a soliton lattice, which is completely characterized by the soliton width and the spacing between neighboring solitons. Fluctuations due to noise create lattice vacancies analogous to point defects in crystals, leading to activity which is spatially inhomogeneous.

As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and warming have been increasingly recognized as major ones. In this paper, we make use of three diffusive logistic models with free boundary in one space dimension to explore the impact of climate warming on the movement of mosquito range. First, a general model incorporating temperature change with location and time is introduced. In order to gain insights of the model, a simplified version of the model with the change of temperature depending only on location is analyzed theoretically, for which the dynamical behavior is completely determined and presented. The general model can be modified into a more realistic one of seasonal succession type, to take into account of the seasonal changes of mosquito movements during each year, where the general model applies only for the time period of the warm seasons of the year, and during the cold season, the mosquito range is fixed and the population is assumed to be in a hibernating status. For both the general model and the seasonal succession model, our numerical simulations indicate that the long-time dynamical behavior is qualitatively similar to the simplified model, and the effect of climate warming on the movement of mosquitoes can be easily captured. Moreover, our analysis reveals that hibernating enhances the chances of survival and successful spreading of the mosquitoes, but it slows down the spreading speed.

We present an improvement to the diffraction-enhanced imaging (DEI) method that permits imaging of moving samples or other dynamic systems in real time. The method relies on the use of a thin Bragg analyzer crystal and simultaneous acquisition of the multiple images necessary for the DEI reconstruction of the apparent absorption and refraction images. Theses images are conventionally acquired at multiple points on the reflectivity curve of an analyzer crystal which presents technical challenges and precludes imaging of moving subjects. We have demonstrated the potential of the technique by taking DEI "movies" of an artificially moving mouse leg joint, acquired at the Biomedical Imaging Centre at SPring-8, Japan.

Jupiter's Great Red Spot is a large swirling cloud mass of reddish-brown appearance (see figure in color plate section). Situated in Jupiter's southern hemisphere, it straddles the south tropical zone and, to the north of this, the south equatorial belt. The Great Red Spot (GRS) is roughly elliptical in shape, with the semi-major axis zonally aligned (east-west) and with dimensions approximately 22, 000 km (twice the diameter of the Earth) by 11, 000 km. The atmospheric motions associated with the GRS are visible in the cloud layer near the tropopause. It is generally agreed to be a vortex (Mitchell et aI., 1981); and Smith et aI. (1979a) give an estimate of the vorticity. This vortex is anticyclonic (rotating in the opposite sense to that induced by the planetary rotation), that is, anticlockwise, but with a weakly counter-rotating, or possibly quiescent inner region. The GRS is at high pressure and low temperature relative to its surroundings. A striking feature associated with the GRS is the turbulent oscillating cloud system to the northwest.

A new approach is proposed for X-ray dynamical diffraction theory in distorted crystals. The theory allows one to perform dynamical diffraction simulations between Bragg peaks for non-ideal crystals, using a simple approach of two distorted waves. It can be directly applied for reciprocal-space simulation. The formalism is used to analyse high-resolution X-ray diffraction data, obtained for an InSb/InGaSb/InSb/InAs superlattice grown on top of a GaSb buffer layer on a (001) GaSb substrate.

This volume grew out from the "International Conference on Reaction-Diffusion Systems and Viscosity Solutions" held at Providence University, Taiwan, during January 3-6,2007. It consists mostly of selected articles representing the recent progress of some important areas of nonlinear partial differential equations. Some of the articles are research papers by participants of the conference, but most are invited survey papers by leading experts in the field, not necessarily participant of the conference. The topics included here reflect the themes of the above-mentioned conference and the research interests of the editors, and therefore are naturally biased and incomplete. Nevertheless, they cover a wide range of partial differential equations, from regularity of viscosity solutions, to symmetry properties of positive solutions of parabolic equations, to nonlinear Schrodinger equations, to mention but a few. A complete list can be found from the content pages.

Binocular rivalry is investigated in a continuum model of the primary visual cortex that includes neural excitation and inhibition, stimulus orientation preference, and spike-rate adaptation. Visual stimuli consisting of bars or edges result in localized states of neural activity described by solitons. Stability analysis shows binocular fusion gives way to binocular rivalry when the orientation difference between left-eye and right-eye stimuli destabilizes one or more solitons. The model yields conditions for binocular rivalry, and two types of competitive dynamics are found: either one soliton oscillates between two stimulus regions or two solitons fixed in position at the stimulus regions oscillate out of phase with each other.

The existence of visual hallucinations with prominent temporal oscillations is well documented in conditions such as Charles Bonnett Syndrome. To explore these phenomena, a continuum model of cortical activity that includes additional physiological features of axonal propagation and synapto-dendritic time constants, is used to study the generation of hallucinations featuring both temporal and spatial oscillations. A detailed comparison of the physiological features of this model with those of two others used previously in the modeling of hallucinations is made, and differences, particularly regarding temporal dynamics, relevant to pattern formation are analyzed. Linear analysis and numerical calculation are then employed to examine the pattern forming behavior of this new model for two different forms of spatiotemporal coupling between neurons. Numerical calculations reveal an oscillating mode whose frequency depends on synaptic, dendritic, and axonal time constants not previously simultaneously included in such analyses. Its properties are qualitatively consistent with descriptions of a number of physiological disorders and conditions with temporal dynamics, but the analysis implies that corticothalamic effects will need to be incorporated to treat the consequences quantitatively.

The thalamus is introduced to a recent model of the visual cortex to examine its effect on pattern formation in general and the generation of temporally oscillating patterns in particular. By successively adding more physiological details to a basic corticothalamic model, it is determined which features are responsible for which effects. In particular, with the addition of a thalamic population, several changes occur in the spatiotemporal power spectrum: power increases at resonances of the corticothalamic loop, while the loop acts as a spatiotemporal low-pass filter, and synaptic and dendritic dynamics temporally low-pass filter the activity more generally. Investigation of the effect of altering parameters and gains reveals new parameter regimes where activity that corresponds to hallucinations is induced by both spatially homogeneous and inhomogeneous temporally oscillating modes. This suggests that the thalamus and corticothalamic loops are essential components of a model of oscillating visual hallucinations.

Spike-rate adaptation is investigated within a mean-field model of brain activity. Two different mechanisms of negative feedback are considered; one involving modulation of the mean firing threshold, and the other, modulation of the mean synaptic strength. Adaptation to a constant stimulus is shown to take place for both mechanisms, and limit-cycle oscillations in the firing rate corresponding to bursts of neuronal activity are investigated. These oscillations are found to result from a Hopf bifurcation when the equilibrium lies between the local maximum and local minimum of a given nullcline. Oscillations with amplitudes significantly below the maximum firing rate are found over a narrow range of possible equilibriums.

This book demonstrates how a 'triangulation method' can be used to model and control dynamical systems defined only from experimental data. For example, they are used to reconstruct the Hénon map. This is a simple two dimensional model found by a French astronomer Michel Hénon, which has a chaotic behaviour. A trajectory is shown in Figure 1.1. For details, see Section 4.3.1 on page 61 of this book. For comparison, we will apply the same control method to the true Hénon map. The idea of the 'triangulation method' is to construct a "net" of geometrical objects spanning the experimental data points. More precisely, the plane is filled with triangles, three dimensional space is filled with tetrahedra, and, in general, m-dimensional space is filled with simplices, which are simple m-dimensional objects, determined by the minimum number of points: 3 for the plane and m + 1 for an m-dimensional space. We require that all the vertices are the data points and no data point lies within any simplex. Now, we use this natural grid to approximate an unknown map by linear interpolation for every simplex separately.

Salient features instantly attract visual attention to their location and are crucial for object recognition. Experiments in ultra-fast visual perception have shown that object recognition can be surprisingly accurate given only ~20 ms of observation. Such short times exclude neural dynamics of top-down feedback and require fast mechanisms of low-level feature detection. We derive a neural model of the primary visual cortex with physiologically parameterized horizontal connections that reinforce salient features, and apply it to detect salient contours on ultra-fast time scales. Model performance qualitatively matches experimental results for human perception of contours, suggesting rapid neural mechanisms involving feedforward horizontal connections can be used to distinguish low-level objects.

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions.