Chatter dynamic analysis for Van der Pol Equation with impulsive effect via the theory of flow switchability

In this paper, the phenomenon of free vibrations in LC circuit was introduced as well as some restrictions in the application of triode. Then we optimize the problems and present a certain kind of Van der Pol Equations which can be considered as a class of second-order impulsive switched systems. To investigate the chatter dynamics on such system, we turn to look for conditions that keep the complex pulse phenomena absent. We introduce several conceptions of theory of flow switchability and analyze the flow’s dynamical behaviors such as transversal property at a boundary in the normal direction of separation surface by constructing generic mappings. Some sufficient conditions for the absence of pulse phenomena and numerical illustrations of periodic motions are obtained.     

Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination

The dynamical behavior of an SIR epidemic model with birth pulse and pulse vaccination is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the infection-free periodic solution and the epidemic periodic solution. By using the impulsive effects, a Poincaré map is obtained. The Poincaré map, center manifold theorem, and bifurcation theorem are used to discuss flip bifurcation and bifurcation of the epidemic periodic solution. Moreover, the numerical results show that the epidemic periodic solution (period-one) bifurcates from the infection-free periodic solution through a supercritical bifurcation, the period-two solution bifurcates from the epidemic periodic solution through flip bifurcation, and the chaotic solution generated via a cascade of period-doubling bifurcations, which are in good agreement with the theoretical analysis.     

Optimal threshold density in a stochastic resource management model with pulse intervention

Human activities and agricultural practices are having huge impacts on the development of fishery and land resources through different ways. To model such systems that involve harvesting, an impulsive model of natural resources with a stochastic noise perturbation element is formulated to study the relationship between (a) the maximal expectation of biomass after harvesting or fishing events and (b) the minimal expectation of pest biomass and the number of times pesticide is applied. Using a detailed analytical treatment, time estimation, and numerical demonstrations, we establish that the proposed mechanism is capable of maximizing fish populations at the end of a fishing season and minimizing pest numbers after a crop harvesting season once the intensity of the noise is relatively small. Investigations of the effects of different parameters reveal that theoretical predictions from the new stochastic model accord with those from the deterministic case. Recommendations for Resource Managers

• Various measures can be implemented to manage natural resources, such as adjusting fishing quantity and intensity to maximize fish population.
• In the natural environment, population growth is inevitably affected by the environment noise. So it is important to understand the noise effect to maintain sustainability of resources.
• Investigated methods are useful to converse resources and can be widely applied to control pests.

     

On the multiplicity of solutions of a simplified tumor growth model with a free boundary

We consider a mathematical model of tumor growth taking into account the spatial structure and include a different phases because of the lack of the nutriments. This model is expressed as a PDE on a spherical domain describing the tumor region. The nonlinearity of this PDE is discontinuous, and our problem can be regarded as a free boundary problem. In fact, the boundary of the tumor and another part in the interior are unknown. We obtain a multiplicity result of solutions together with some properties of the associated free boundaries.     

Classical model of interaction of 1-D helium with an intense ultrashort laser pulse

The dynamical process of interaction of one-dimensional helium with an intense ultrashort laser pulse has hen studied with a classical theory. Using the ensemble average method in the classical theory, the probability evolutions of the single and double ionization for the helium are simulated and the simulation results are analyzed.     

一类具有免疫时滞的病毒动力学模型的稳定性分析

建立和分析了一类具有CTL免疫反应且带有免疫时滞的病毒动力学模型.讨论了系统解的有界性,并获得了无病平衡点全局渐近稳定以及正平衡点稳定的条件.最后借助Matlab对模型进行了数值模拟.     

On a model for optical pulse propagation with saturation nonlinearity

We have suggested a new nonlinear equation for optical pulse propagation in the nonlinear medium with saturation type nonlinearity. The equation can be exactly analysed by Hirota's approach and we have studied the form of explicit one and two solition states.     

Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuron

This article is devoted to the study of a mathematical model arising in the mathematical modeling of pulse propagation in nerve fibers. A widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations. When considered as part of a particular initial boundary value problem the equation models the electrical activity in a neuron. A small perturbation parameter ε is introduced to the highest order derivative term. The parameter if decreased, speeds up the fast variables of the model equations whereas it does not affect the slow variables. In order to formally reduce the problem to a discussion of the moment of fronts and backs we take the limit ε → 0. This limit is singular and is therefore the solution tends to a slowly moving solution of the limiting equation. This leads to the boundary layers located in the neighborhoods of the boundary of the domain where the solution has very steep gradient. Most of the classical methods are incapable of providing helpful information about this limiting solution. To this effort a parameter robust numerical method is constructed on a piecewise uniform fitted mesh. The method consists of standard upwind finite difference operator. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives. A parameter uniform error estimate for the numerical scheme so constructed is established in the maximum norm. It is then proven that the numerical method is unconditionally stable and provides a solution that converges to the solution of the differential equation. A set of numerical experiment is carried out in support of the predicted theory, which validates computationally the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Perturbation analysis of self-sustained oscillations in a pulse combustion model

A lumped parameter model of pulse combustion is constructed which incorporates a valve submodel exhibiting continuous mass reactant flow. Unlike earlier models based on the orifice flow equations, the rate of change of the mass reactant flow, while discontinuous as the valve transitions from open to closed, exhibits a finite jump discontinuity. The model equations are analyzed by combining Fourier series expansions with regular perturbation expansions; this yields asymptotic expansions for both the combustion chamber pressure and tailpipe velocity oscillations. A numerical study of the perturbation expansions yields important information concerning the behavior of the pulse combustor as various physical and geometrical parameters are varied.     

Asymptotic analysis of a new dynamic semiparametric regression model with cross-effects of survivals

A new flexible and simple semiparametric model including the cases where hazard rates cross, go away, are proportional, approach, or converge is proposed. Semiparametric estimation procedures for censored data are given. A test for absence of hazard rates crossing is proposed. Bibliography: 6 titles.     

Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant

In this paper, we consider a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution. Further, it is globally asymptotically stable. The permanent condition of the investigated system is also analyzed by the theory on impulsive differential equation. Our results reveal that the chemostat environmental changes play an important role on the outcome of the chemostat.     

Theoretical analysis and numerical solution of laser pulse transformation for satellite laser ranging

The processes of the pulse transformation in satellite laser ranging (SLR) are analyzed, the analytical expressions of the transformation are deduced, and the effects of the transformation on Center-of-Mass corrections of satellite and ranging precision are discussed. The numerical solution of the transformation and its effects are also given. The results reveal the rules of pulse transformation affected by different kinds of factors. These are significant for designing the SLR system with millimeter accuracy.     

A semi-analytical model for dynamic analysis of non-uniform plates

Dynamic properties of the plate structures can be enhanced by introducing discontinuities of different kinds such as using surface-bonded discrete patches or spatially varying the stiffness and mass properties of the plate. Fast and reliable design of such complex structures requires efficient and accurate modeling tools. In this study, a novel semi-analytical model is developed for the dynamic analysis of plates having discrete and/or continuous non-uniformities. Two-dimensional Heaviside unit step functions are utilized to represent the discontinuities. Different from existing numerical methods based on Heaviside functions, a numerical technique is proposed for modeling the discontinuities that are not necessarily aligned with the plate axes. The governing equations are derived using Hamilton's principle and Rayleigh–Ritz method is used for determining the modal variables. The surface-bonded patches are used to demonstrate discrete non-uniformities where variable-stiffness laminates are selected to represent continuous non-uniform structures. Natural frequencies and mode shapes obtained using the proposed method are validated with finite element analyses and the existing results from the literature. The results show that the developed model performs accurately and efficiently.     

Conservative finite-difference scheme for the problem of a femtosecond laser pulse with an axially symmetric profile propagating in a medium with cubic nonlinearity

Conservative finite-difference schemes are constructed for the problem of a femtosecond laser pulse propagating in a cubically nonlinear medium in the axially symmetric case with allowance for temporal dispersion of the nonlinear response of the medium. The process is governed by the nonlinear Schrödinger equation involving the time derivative of the nonlinear term. The invariants of the differential problem are presented. It is shown that the difference analogues of these invariants hold for the solution to the finite-difference schemes proposed for the problem. As an example, the numerical results obtained for the self-focusing of a femtosecond light beam are presented.