Global stability and Hopf-bifurcation in a zooplankton–phytoplankton model

We deal with a predator–prey model, representing a resource (phytoplankton) and two predators (zooplankton) system with toxin-producing delay. The response function is assumed here to be concave in nature. Firstly, the stability criterion of the model is analyzed both from a local and a global point of view. Our results imply that the toxin’s intrinsic characteristics, such as toxic liberation rate and toxin-producing delay, will not change the stability of the system irreversibly. Secondly, Hopf bifurcation of both systems with delay and without delay can occur via system parameters pertaining to the toxin. Our results indicate that the toxin produced by phytoplankton may be used as a bio-control agent for the Harmful Algal Bloom problems. Furthermore, the explicit algorithm for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results.     

Non-isothermal film casting: Determination of draw resonance

The important industrial process of casting polymeric films suffers from the “draw resonance” instability that appears as sudden oscillations in the product dimensions. This instability influences the quality of the end-product and negatively limits productivity and efficiency of the process. The draw resonance originates when a material is being processed beyond the limits of its intrinsic properties. Research is conducted with the intention to find those process and material properties that allow to optimize the production process while keeping it stable.This paper concentrates on a non-isothermal analysis of the stability of the film casting. The mathematical model of the process is given by a quasi-linear system of first order PDEs with two point boundary conditions. The constitutive polymer behavior is approximated by the modified Giesekus model. Linear stability analysis combined with the Laplace transformation of the resulting linear system is applied to find parameters that determine mathematical and thus process instability. It all comes down to determining the spectrum of a compact operator; corresponding eigenfunctions can be regarded as the characteristic modes of the system. For implementation, the modification of Galerkin approach is used. The major advantage of the mathematical and numerical method is that the full spectrum is calculated in a matter of seconds. Our results agree perfectly with the ones from literature for isothermal case, and with the experimental data for the non-isothermal case. The results also indicate that non-isothermality is highly important and cannot be excluded from modeling.     

Stochastically perturbed hopf bifurcation

In this paper the influence of small stochastic parametric perturbations on systems exhibiting Hopf bifurcation is discussed in detail. The Markov diffusion approximation is used to obtain analytical results relating to the statistical properties of the stochastic response. For the linear system both moment stability and sample stability conditions are obtained. It is found that in the non-linear system, a shift of the bifurcation point takes place due to the presence of parametric noise.     

The research on Cournot–Bertrand duopoly model with heterogeneous goods and its complex characteristics

This paper details the research of the Cournot–Bertrand duopoly model with the application of nonlinear dynamics theory. We analyze the stability of the fixed points by numerical simulation; from the result we found that there exists only one Nash equilibrium point. To recognize the chaotic behavior of the system, we give the bifurcation diagram and Lyapunov exponent spectrum along with the corresponding chaotic attractor. Our study finds that either the change of output modification speed or the change of price modification speed will cause the market to the chaotic state which is disadvantageous for both of the firms. The introduction of chaos control strategies can bring the market back to orderly competition. We exert control on the system with the application of the state feedback method and the parameter variation control method. The conclusion has great significance in theory innovation and practice.     

随机ARNOLD系统的稳定性与分叉

本文详细讨论了当ｎ＝２时Ａｒｎｏｌｄ系统在小强度的随机参数激励扰动下，系统的运动稳定性及分叉。为了研究系统响应的统计特性，本文使用了Ｍａｒｋｏｖ近似技巧。在线性系统的情形，给出了系统矩稳定及样本稳定的充分必要条件。在非线性情形，本文的结果表明随机扰动可使系统的分叉点发生漂移     

A spectral approach for the stability analysis of turbulent open-channel flows over granular beds

A novel Orr–Sommerfeld-like equation for gravity-driven turbulent open-channel flows over a granular erodible bed is here derived, and the linear stability analysis is developed. The whole spectrum of eigenvalues and eigenvectors of the complete generalized eigenvalue problem is computed and analyzed. The fourth-order eigenvalue problem presents singular non-polynomial coefficients with non-homogenous Robin-type boundary conditions that involve first and second derivatives. Furthermore, the Exner condition is imposed at an internal point. We propose a numerical discretization of spectral type based on a single-domain Galerkin scheme. In order to manage the presence of singular coefficients, some properties of Jacobi polynomials have been carefully blended with numerical integration of Gauss–Legendre type. The results show a positive agreement with the classical experimental data and allow one to relate the different types of instability to such parameters as the Froude number, wavenumber, and the roughness scale. The eigenfunctions allow two types of boundary layers to be distinguished, scaling, respectively, with the roughness height and the saltation layer for the bedload sediment transport.     

Complexity of triopoly price game in Chinese cold rolled steel market

This paper considers a Bertrand model based on nonlinear demand functions which are closer to reality and different from previous studies. We apply the model into Chinese cold rolled steel market and study game process of triopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Simulations show complex dynamical behaviors of the system. The results illustrate that altering the relevant parameters of system can affect the stability of Nash equilibrium point and cause chaos to occur, and the complex dynamical behaviors will disappear by parameters control method. The results have an important theoretical and practical significance to Chinese cold rolled steel market.     

非线性振动系统周期运动及其稳定性的数值研究

§1引言确定型非线性振动系统的运动可分类如下: 1.非定常运动;2.定常运动:(1)周期运动,(2)各态历经运动,(3)浑沌运动。其中非定常运动是一暂态过程,会随着时间的增长逐步衰减乃至实际上消失。定常运动中的各态历经运动,指系统至少有两个互不通约(即其比值为无理数)的振动频率,因此运动虽然局      

Modeling and simulation of aerial refueling by finite element method

The aerial refueling hose-and-drogue system is a special case of a generalized aerial cable towed system. The present work investigates the effect of pertinent parameters such as the cable tension, tow point disturbance and vortex wake on the dynamic behavior and stability of the generalized model by using the finite element method with an accurate and computationally efficient three-noded, curved beam element. The analysis results show that the conventional modal and spectrum analysis method is inappropriate for the dynamic stability analysis of the aerial cable towed system. This is because the mechanism of instability due to the tow point disturbance is not the resonance of the aerial cable towed system but the wave propagation downstream along the cable absorbing energy from the airflow when the wave propagation speed is less than the airflow speed. The study also demonstrates that the vortex wake has a significant impact on the dynamics of the aerial cable towed system. The short cable system will orbit with the vortex and the orbiting behavior will diminish as the cable length increases.     

谐波齿轮系统的快慢振荡机制研究

谐波齿轮减速器是一种新型的传动装置, 因其具有诸多的优点, 因而得到了广泛应用. 谐波齿轮减速器涉及不同振荡尺度之间的耦合作用, 这通常会诱发复杂的快慢振荡, 严重影响了谐波齿轮系统的正常工作. 本文考虑涉及扭转刚度非线性因素的谐波齿轮系统, 旨在研究系统的快慢动力学, 揭示新型的快慢振荡机制. 首先, 构建了非线性扭转刚度下的谐波齿轮系统的快慢动力学模型. 然后, 通过改变扭转刚度系数, 得到了系统从常规振荡向快慢振荡的转迁过程. 接着, 简要地论述了有关快慢系统的基础理论. 在此基础上, 采用快慢分析法研究了快子系统的动力学特性, 揭示了快慢振荡的产生机制. 研究表明, 当系统参数改变时, 快子系统的平衡点曲线并未发生失稳或分岔; 然而, 在某一点附近, 平衡点曲线能够产生急剧量变, 其特征是平衡点在局部小范围内可以在正坐标值与负坐标值之间快速转迁. 在此基础上, 揭示了一种诱发快慢振荡的新型动力学机制, 比较了这种诱发机制与其他相关机制之间的区别. 本文丰富了系统通向快慢振荡的路径, 为实际谐波齿轮传动系统中的快慢振荡机理与控制研究提供参考.      

判定广义Birkhoff系统稳定性的三重组合梯度方法

研究判定广义Birkhoff系统稳定性的三重组合梯度方法．首先,给出4类三重组合梯度系统的定义和微分方程;其次,得到广义Birkhoff系统成为三重组合梯度系统的条件,从而将广义Birkhoff系统化成三重组合梯度系统;最后,利用组合梯度系统的性质来研究系统的稳定性,举例说明结果的应用．     

Pointwise Green Function Bounds for Shock Profiles of Systems with Real Viscosity

Following the pointwise semigroup approach of [ZH,MZ.1], we establish sharp pointwise Green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic-parabolic systems of conservation laws of dissipative type, under the necessary assumptions ([Z.1,Z.3,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave; transversality of the connecting profile; and hyperbolic stability of the corresponding ideal shock of the associated inviscid system, with no additional assumptions on the structure or strength of the shock. These bounds are used in a companion paper [MZ.2] to establish nonlinear stability of small-amplitude Lax shocks of symmetrizable hyperbolic-parabolic systems.     

Thermocapillary Instability in a Two-Layer System with a Deformable Interface

The stability of equilibrium of a system consisting of two plane immiscible viscous liquid layers under conditions of weightlessness is considered. The surface tension force, which depends linearly on the temperature, acts on the deformable interface between the media. Three model systems are considered. In each system the development of instability is due to a particular asymmetry of the properties of the liquids or the layer thicknesses. The conditions of excitation of longitudinal thermocapillary and transverse capillary waves supported by the thermocapillary effect on the interface are discussed.     

判定非自治Birkhoff系统稳定性的广义组合梯度方法

研究判定非自治Birkhoff系统稳定性的广义组合梯度方法．首先,给出非自治Birkhoff系统和非自治广义Birkhoff系统的运动微分方程;其次,给出一类将广义梯度系统和广义斜梯度系统组合而成的广义组合梯度系统,并讨论广义组合梯度系统的一些性质;最后,将非自治Birkhoff系统和非自治广义Birkhoff系统在一定条件下表示成广义组合梯度系统,并用广义组合梯度系统的性质研究了这两类Birkhoff系统的稳定性．举例说明结果的应用．     

Technical Stability Conditions for Systems of Variable Structure with Derivatives of the Control Function

We derive sufficient conditions for the technical stability of forced states of automatic control systems with a variable structure depending on the derivatives of the control function. For arbitrary admissible initial perturbations from a measurable set of initial states of a control system with filtering, the technical stability conditions do not depend on the sliding conditions in the given domain of the system parameters. We find how the properties of the eigenvalues of the quadratic forms of the Lyapunov functions are related to the technical stability conditions for automatic control systems of variable structure with filtering. The technical stability conditions are analyzed for automatic control systems with third-order filtering where the forming signal creates eight structures     

Stability and stabilization of the family of pseudolinear differential systems

New methods for the analysis of the robust stability of equilibrium states are developed for some classes of nonlinear differential systems. We formulate sufficient conditions for the stability of the trivial solutions of the families of pseudolinear controlled systems with undetermined coefficient matrices and feedback from the measured output. A method is developed for the analysis of stability according to the first approximation to the family of nonlinear systems. The application of the obtained results is reduced to the solution of systems of linear differential matrix inequalities. We present an example of a system of stabilization of a double inverted pendulum.     

Control and synchronization of chaos systems using time-delay estimation and supervising switching control

In this paper, we present a new technique, developed using time-delay estimation (TDE) and supervising switching control (SSC), for the control and synchronization of chaos systems. The proposed technique consists of three units: a time-delay estimation unit that cancels system dynamics, a pole placement control unit that shapes error dynamics, and an SSC unit that is activated when the system dynamics are rapidly changing. We prove the stability of the closed-loop system using the Lyapunov analysis method. To verify the control and synchronization performance of the proposed technique (TDE-SSC), we compare it with TDC using numerical simulation. Our results indicate that the proposed scheme is an easily understood, numerically efficient, robust, and accurate solution for the control and synchronization of chaos systems.     

A New Methodology for the Analysis of Periodic Systems

A new numerical-analytical method for the combined global-localanalysis of nonlinear periodic systems referred to as an ExpandedPoint Mapping (EPM) is presented. This methodology combines thecell to cell mapping and point mapping methods toinvestigate the basins of attraction and stability characteristics ofequilibrium points and periodic solutions of nonlinear periodicsystems. The proposed method is applicable to multi-degrees-of-freedomsystems, multi-parameter systems, and allows analytical studies oflocal stability characteristics of steady state solutions. Inaddition, the EPM approach allows the study of stabilitycharacteristics as function of system parameters to obtain analyticalconditions for bifurcation. In the paper, the theoretical basis forthe EPM method is formulated and a procedure for the analysis ofnonlinear dynamical systems is presented. Analysis of a pendulum witha periodically excited support in the plane is used to illustrate themethod. The results demonstrate the efficiency and accuracy of theproposed approach in analyzing nonlinear periodic systems.     

Nonlinear vibrations and dynamic stability of a viscoelastic circular cylindrical shell with shear strain and inertia of rotation taken into account

We consider nonlinear vibration and dynamic stability problems for a viscoelastic circular cylindrical shell according to the refined Timoshenko theory, which takes into account the shear strain and the inertia of rotation, in a geometrically nonlinear setting. The problem data are reduced to systems of nonlinear integro-differential equations with singular relaxation kernels, which can be solved by the Bubnov-Galerkin method combined with a numerical method based on quadrature formulas. We study the numerical convergence of the Bubnov-Galerkin method. We analyze the shell dynamic behavior in a wide range of physical-mechanical and geometric parameters. We demonstrate the influence of the viscoelastic properties of the material on the nonlinear vibrations and dynamic stability of a circular cylindrical shell. We also compare the results obtained according to different theories.