Search

In this paper, we perform a careful analysis of the forced PP04 model for climate change, in particular the behaviour of the ice ages. This system models the transition from a glacial to an inter-glacial state through a sudden release of oceanic carbon dioxide into the atmosphere. This process can be cast in terms of a Filippov dynamical system, with a discontinuous change in its dynamics related to the carbon dioxide release. By using techniques from the theory of non-smooth dynamical systems, we give an analysis of this model in the cases of both no insolation forcing and also periodic insolation forcing. This reveals a rich, and novel, dynamical structure to the solutions of the PP04 model. In particular, we see synchronized periodic solutions with subtle regions of existence which depend on the amplitude and frequency of the forcing. The orbits can be created/destroyed in both smooth and discontinuity-induced bifurcations. We study both the orbits and the transitions between them and make comparisons with actual climate dynamics.

Modelling the view factor, F, of a ‘grain-like’ observer near a tilted pool fire assumed to be cylindrical in shape, requires the observer (on the same ground level as the pool fire) to be as close as possible to the flame surface so that the flame surface is viewed as a plane. The orientation of the observer is to receive a maximum view, with the view factor integrated over the flame area seen by the observer. The derived expression for F was expressed in terms of standard functions and sensitive parameters: α, defined to ensure the observer receives a maximum view of the flame surface and β, the angle of inclination of the differential observer plane. Results from a planar approximation to F are compared with those of PHAST 7.2 simulated results in MATLAB. The values of the planar approximation to F was found to increase with increasing β. This suggests that the larger the value of β, the more the radiation received by a near observer. For β = 48 . 961 ∘ , horizontal distance, X = 30 m, α = 0.01 , flame length, L = 12.14 m, tilt angle, θ = 55 . 17 ∘ , and pool diameter, D = 5 m: planar and PHAST 7.2 approximations to F were found to coincide up to 6 significant digits and differ by 3.2 × 10 − 8 . The close agreement between both approximations depend heavily on the choice of flame properties and sensitive parameters. Interestingly, this result may depend sensitively on the choice of parameters, leading to prediction of planar approximation to F much higher or lower than those of PHAST 7.2 approximations. The advantage of this approach is that a knowledge of the plane where the largest concentration of radiation is located will help minimise loss of lives to fire hazards and improve the efficiency of risk safety assessment/management.

Turbochargers are a vital component for aiding engine manufacturers in meeting the latest emissions standards. However, their range of operation is limited for low mass flows by compressor surge. Operation in surge results in pressure and mass flow oscillations that are often damaging to the compressor and its installation. Since surge is a highly complex flow regime, full unsteady 3D models are generally too computationally expensive to run. The majority of current low-dimensional surge models use a cubic compressor characteristic that needs to be fitted to experimental data. Therefore, each time a compressor is studied using these models, costly experimental testing is required. In this paper, a new technique for obtaining an axisymmetric centrifugal compressor characteristic is presented. This characteristic is built using the equations of mass, momentum and energy from first principles in order to provide a more complete model than those currently obtained via experimental data. This approach enables us to explain the resulting cubic-like shape of the characteristic and hence to identify impeller inlet stall as a route into surge. The characteristic is used within a quasi-steady, map-based surge model in order to demonstrate its ability to predict the onset of surge while only providing geometric data as input. Validation is provided for this model by discussion of the qualitative flow dynamics and a good fit to experimental data, especially for low impeller speeds and pressure ratios.

In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg–de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p = 5 , solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p > 5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p = 5 . Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as p → 5 + . By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampere (M-A) equation with power source, with the following basic 2D model $$u_{t}=-|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where in two-dimensions $|D^{2}u|=u_{xx}u_{yy}-(v_{xy})^{2}$ and p > 1 is a fixed exponent. For a class of ‘dominated concave’ and compactly supported radial initial data $u_{0}(x)\geqslant0$ , the Cauchy problem is shown to be locally well posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented. The parabolic analogy of the parabolic M-A equation in 3D for which $|D^{2}u|$ is a cubic operator is $$u_{t}=|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{3}\times \mathbb{R}_{+},$$ and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similarblow-up in a cubic radial model related to the fourth-order M-A equation $$u_{t}=-|D^{4}u|+u^{3}\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where the cubic operator $|D^{4}u|$ is the catalecticant 3 × 3 determinant is also briefly discussed.

We study the self-similar solutions of the quasilinear parabolic equationWe show that there is an exponentsuch that if σ> then the equation admits a countable set {uk(x, t)} of self-similar blow-up solutions. These solutions have the formwhere T> 0 is a finite blow-up time, θ(ξ) solves a nonlinear ODE and each function uk(x, t) is nonconstant in a neighbourhood of the origin and has exactly k maxima and minima for x ≧ 0. There is a further critical exponent σ = ф such that if σ > ф there is a second set of self-similar solutions which are constant (in x) in a neighbourhood of the origin. We conjecture (and provide formal arguments and numerical evidence for) the existence of an infinite sequence σk→σ∞ of critical values, such that σ1 = 0 and uk exists only in the range σ>σk (when σ> 0 the equation has no nontrivial self-similar solutions). The proof of existence when σ>σ∞(σ>ф) is obtained by a combination of comparison and dynamical systems arguments and relates the existence of the self-similar solutions to a homoclinic bifurcation in an appropriate phase-space.

This paper presents and analyzes a model for the kinetic behavior of certain ion transport mechanisms known to exist on bacterial cell membranes. Such models are crucial in understanding and predicting microbiological processes and have many important applications. In particular this work may be regarded as an initial step in the development of a more complex model relating to the effects of certain organic acids on homeostasis and osmoregulation in bacteria-these have far-reaching implications in the area of novel food preservation techniques. A model for the transport of hydrogen, H^+, and potassium, K^+, ions across membranes of Escherichia coli cells is first proposed. A special case of this general model, which takes the form of two coupled, nonlinear, ordinary differential equations, is chosen and studied analytically using simple phase-plane analysis. This not only demonstrates (by reference to existing experimental results) the viability of the model, but also provides information on appropriate magnitudes of many of the parameters appearing in the model. The work also suggests further experimental and theoretical study.

This paper presents a detailed analysis of a model for a positive corona discharge, and discusses the relationship between the model and the space charge equations which predict the macroscopic motion of charged ions in a gas. The study is restricted to the asymptotic behaviour of the discharge for large time, and it does not consider its initial or burst phases in detail.One prediction of this model is the existence of a steady state solution to the space charge equations that may lose stability to a travelling wave disturbance which then grows into a strongly pulsed oscillation.We compare some numerical calculations with an asymptotic analysis of the discharge and find good agreement.

In the present paper a single-degree-of-freedom, periodically forced impact oscillator is investigated by numerics and experiments. At first the different types of motion and bifurcation diagrams are determined from experiments and compared to numerical results on the basis of the identified impact model. Also the nonsmooth third order system shows a rich bifurcational behaviour. On the basis of the identified nonsmooth model further analytical and numerical analysis can be applied. The Lyapunov exponents and the winding number are evaluated for the non-smooth system.

In the last talk on impact oscillators we reviewed some of their general behaviour and looked at the simple smooth bifurcations that occur when studying the stability of the simple (1, n) orbits. We now proceed to investigate the bifurcations which can occur when we consider discontinuous changes in behaviour. These are associated with grazing impacts of the trajectory of the particle with the obstacle. A graze leads to quite remarkable behaviour including an immediate bifurcation from a periodic orbit to a chaotic one. Such behaviour is unique to impact oscillators although [Nordmark] has shown that some of the behaviour associated with the discontinuities in the impact oscillator can be viewed as a limit of smooth bifurcations occuring in systems with smooth models for the impacts.

In this paper we shall examine bifurcations which occur in the semilinear elliptic system $$\eqalign{ & \Delta {\text{u}}\, +,\lambda (u + {\text{u}}\left| {\text{u}} \right|^{{\text{p - 1}}} ) = 0, \cr & {\text{u|}}\partial {\text{B}}\,=0, \cr} $$ (1.1) where B is the unit ball in the space ℝ3. The system (1.1) has recently been the subject of much research stimulated by the paper by BREZIS & NIRENBERG [2] which studied the special case p = 5 : the critical Sobolev exponent for ℝ3. We shall study in this paper the two regimes p > 5 (supercritical) and p ≈ 5 (near critical) and will show that in this parameter region problem (1.1) has a very rich bifurcation structure. This includes an infinity of fold bifurcations, and, for special values of p > 5 an infinity of symmetry breaking bifurcations from non positive solution branches. These properties of the solution structure are intimately connected with a reformulation of problem (1.1) as a dynamical system which, for the particular case p = 5, is a perturbed Hamiltonian system. He thus will demonstrate that as p passes through the value 5 the uniqueness properties of the solutions of problem (1.1) change significantly.

In this paper we examine the symmetry breaking bifurcations which occur on the radially symmetric solution branches of the following semilinear elliptic equation: for ∆ the usual Laplacian operator and 0 < λ ∈ ℝ $$\left\{ {\begin{array}{*{20}{c}} {\Delta v + \lambda \left( {v + {v^5}} \right) = 0,{\text{ for }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{r} \in B,} \\ {{\text{ }}v = 0,{\text{ for }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{r} \in \partial B.} \end{array}} \right.$$ (1.1) .

In this paper large resistor-capacitor (RC) networks that consist of randomly distributed conductive and capacitive elements which are much larger than those previously explored are studied using an efficient algorithm. We investigate the emergent power-law scaling of the conductance and the percolation and saturation limits of the networks at the high and low frequency bounds in order to compare with a modification of the classical Effective Medium Approximation (EMA) that enables its extension to finite network sizes. It is shown that the new formula provides a simple analytical description of the network response that accurately predicts the effects of finite network size and composition and it agrees well with the new numerical calculations on large networks and is a significant improvement on earlier EMA formulae. Avenues for future improvement and explanation of the formula are highlighted. Finally, the statistical variation of network conductivity with network size is observed and explained. This work provides a deeper insight into the response of large resistor-capacitor networks to understand the AC electrical properties, size effects, composition effects and statistical variation of properties of a range of heterogeneous materials and composite systems.