On simple stochastic diffusion models

This paper expands the diffusion of information models developed by Funkhouser and McCombs (1972) to include situations involving simple interaction processes and more complicated situations involving both mass mediated messages and interactively mediated messages. This paper develops discrete time models of information diffusion as opposed to the continuous time models developed by Coleman (1964) and Bartholomew (1967) and others.     

An efficient center manifold technique for Hopf bifurcation of n-dimensional multi-parameter systems

The center manifold theory with respect to the simple Hopf bifurcation of a n-dimensional nonlinear multi-parametric system is treated via a proper symbolic form. Analytical expressions of the involved quantities are obtained as functions of the parameters of the system via effective algorithms based on the followed procedure and carried out using a symbolic computation software. Moreover the normal form of a codimension 1 Hopf bifurcation, as well as the corresponding Lyapunov coefficient and bifurcation portrait, can be computed for any system under consideration. Here the computational procedure is applied to two nonlinear three-dimensional, three-parametric systems and graphical results are obtained as concerns the stability regions, the bifurcation portraits, as well as emerged limit cycles with respect to both the supercritical and the subcritical case of bifurcation.     

Hopf bifurcation in structural population models

We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely, counter‐migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration. Copyright © 2016 John Wiley & Sons, Ltd.     

Turing bifurcation in a system with cross diffusion

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction–diffusion equations subject to Neumann boundary data. Hence, we assume that there are two substances in a two-dimensional bounded spatial domain where they are diffusing according to Fick's law: the velocity of the flow of diffusing substance is directed opposite to the (spatial) gradient of the density and is proportional to its modulus, but the spatial flow of each substance is influenced not only by its own but also by the other one's density (cross diffusion). The domains in which the substances are diffusing are of three type: a regular hexagon, a rectangle and an isosceles rectangular triangle. It will be assumed that there is no migration across the boundary of these domains. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises.     

Hopf bifurcation in models for pertussis epidemiology

Pertussis (whooping cough) incidence in the United States has oscillated with a period of about four years since data was first collected in 1922. An infection with pertussis confers immunity for several years, but then the immunity wanes, so that reinfection is possible. A pertussis reinfection is mild after partial loss of immunity, but the reinfection can be severe after complete loss of immunity. Three pertussis transmission models with waning of immunity are examined for periodic solutions. Equilibria and their stability are determined. Hopf bifurcation of periodic solutions around the endemic equilibrium can occur for some parameter values in two of the models. Periods of about four years are found for epidemiologically reasonable parameter values in two of these models.     

Ill-posedness and multi-parameter regularization in an identification problem for jump diffusion processes

In many financial applications jump diffusion processes are used to describe price dynamics of assets. Therefore, methods to estimate the parameters of this stochastic process from some market data are required. Here we briefly discuss the unique and stable solvability of this specific inverse problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.     

Pressure and flow in two dimensional numerical bifurcation models

The finite element method has been used to solve the Navier-Strokes equations for steady flow conditions in bifurcations. The results are presented as pressure, velocity and streamline plots at different Reynolds number. The three bifurcations considered have rigid walls and bifurcation angles of 0°, 20° and 180°. For the bifurcation with branch angles 0° and 20° there is flow separation along the inner wall of the outlet branches and large spatial pressure variations, these phenomena being more pronounced at the higher Reynolds numbers. For the bifurcation with a branch angle of 180° the high pressure gradients occured at the outer corner and for the high Reynolds number a vortex formation developed downstream of this corner.     

Global bifurcation of positive equilibria in nonlinear population models

Existence of nontrivial nonnegative equilibrium solutions for age-structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.     

Reverse-time diffusion equation models

Reverse-time stochastic diffusion equation models are defined and it is shown how most processes defined via a forward-time or conventional diffusion equation model have an associated reverse-time model.     

On one bifurcation in relaxation systems

We establish conditions under which, in three-dimensional relaxation systems of the form {fx066-01}, where 0 < ε << 1, |μ| << 1, and ƒ, g ∈ C ^∞, the so-called “blue-sky catastrophe” is observed, i.e., there appears a stable relaxation cycle whose period and length tend to infinity as μ tends to a certain critical value μ_*(ε), μ_*(0) 0 = 0. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 63–72, January, 2008.     

A global bifurcation theorem for Darwinian matrix models

Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extinction equilibrium at a value of a model parameter at which the extinction equilibrium destabilizes. We give criteria for the global shape of the continuum, including local direction of bifurcation and its relationship to the local stability of the bifurcating positive equilibria. We discuss a relationship between backward bifurcations and Allee effects. Illustrative examples are given.     

Stability and bifurcation analysis of a delayed innovation diffusion model

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage(time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period(time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.     

Wavelet estimation of the diffusion coefficient in time dependent diffusion models

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.     

分数扩散模型的参数估计及其应用

研究分数扩散模型的参数估计及其应用问题.分数扩散模型是一类由分数Brownian运动驱动的随机微分方程.主要结果有:(1)利用二次变差方法给出模型中扩散系数的估计量,通过最小二乘法给出模型中漂移系数的估计量;(2)证明这些估计量的一致收敛性和渐近正态性;(3)利用MCMC方法对此估计量进行验证,并通过R软件将上述模型以及参数估计量应用到SHIBOR利率中进行实证研究.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

Finite-dimensional models of diffusion chaos

Some parabolic systems of the reaction-diffusion type exhibit the phenomenon of diffusion chaos. Specifically, when the diffusivities decrease proportionally, while the other parameters of a system remain fixed, the system exhibits a chaotic attractor whose dimension increases indefinitely. Various finite-dimensional models of diffusion chaos are considered that represent chains of coupled ordinary differential equations and similar chains of discrete mappings. A numerical analysis suggests that these chains with suitably chosen parameters exhibit chaotic attractors of arbitrarily high dimensions.