涂布流动的非线性阈值问题

本文研究了沿斜面流动薄层液体的非线性稳定性,即涂布流动的非线性稳定性问题。我们将周恒对平面Poiseuille流提出的弱非线性理论应用于涂布流动。文中对自由表面的世界条件提出了一个合理的简化方法,对亚临界时不同Reynolds数及扰动频率,求出了有限扰动的阈值。     

Effect of high-frequency stimulation on nerve pulse propagation in the FitzHugh–Nagumo model

We consider the FitzHugh–Nagumo model axon under action of a homogeneous high-frequency stimulation (HFS) current. Using a multiple scale method and a geometrical singular perturbation theory, we derive analytically the main characteristics of the traveling pulse. We show that the effect of HFS on the axon is determined by a parameter proportional to the ratio of the amplitude to the frequency of the stimulation current. When this parameter is increased, the pulse slows down and shrinks. At some threshold value, the pulse stops and its width becomes zero. The HFS prevents the pulse propagation when the parameter exceeds the threshold value. The analytical results are confirmed by numerical experiments performed with the original system of partial differential equations.     

接近亏损系统的矩阵摄动法

随着系统参数的变化，带有重频的系统可转变成带有密集频率的系统，反之亦然．本文讨论了亏损系统与接近亏损系统之间的关系．并提出了接近亏损系统的平均移位的摄动方法．算例表明了此方法的有效性．     

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic conservative perturbation method for linear time-varying Hamiltonian system.Firstly,the dynamic equation of Hamiltonian system is gradually changed into a high order perturbation equation,which is solved approximately by resolving the Hamiltonian coefficient matrix into a "major component" and a "high order small quantity" and using perturbation transformation technique,then the solution to the original equation of Hamiltonian system is determined through a series of inverse transform.Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes,the transfer matrix is a symplectic matrix;furthermore,the exponential matrices can be calculated accurately by the precise time integration method,so the method presented in this paper has fine accuracy,efficiency and stability.The examples show that the proposed method can also give good results even though a large time step is selected,and with the increase of the perturbation order,the perturbation solutions tend to exact solutions rapidly.     

A physical mechanism of large-scale structure generation in turbulent convection

The generation of large-scale structures during turbulent convection in a rotating layer of incompressible fluid heated by internal heat sources is considered. The results of a theoretical and experimental investigation of a physical mechanism of large-scale structure formation which operates under conditions of high-intensity small-scale turbulent convection and low boundary heat transfer are discussed. The theoretical investigation is based on a system of evolutionary equations obtained for the transverse space moments of the physical fields, which describes the motion in thin layers of rotating fluid. The stability of the solution of the mathematical model is studied using the small perturbation method. As a result, a condition of existence of longwave instability of the system and a criterion determining the threshold of its onset are obtained. The theoretical conclusions are confirmed by a series of experiments carried out on a laboratory model. The design of the laboratory apparatus and the experimental technique are described.Moscow, Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–29, September–October, 1996.     

Dynamic complexity of microbial pesticide model

In this paper, a mathematical model for the entomopathogenic nematode attacking the pest is investigated. This novel theoretical framework could result in an objective criterion on how to release the entomopathogenic nematode in order to control the pest population under the economic threshold (ET) which indicates the maximally admissible pest densities. Firstly, continuous release of the entomopathogenic nematode is taken. By using qualitative analysis method, the sufficient condition of the global stability of the positive equilibria and the existence and uniqueness of limit cycle of the system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. Using the Floquet’s theorem and small-amplitude perturbation, we obtain that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.     

Dynamics of a quantized circle homeomorphism with quasiperiodic perturbation

We introduce the notion of a quantized circle homeomorphism that is a discontinuous mapping of an interval shift, which is widely used in modern digital radio electronics. For a two-dimensional dynamical system given by a triangular mapping, which is a quantized circle homeomorphism with quasiperiodic perturbation, we prove, under some assumptions, that there exist an invariant absorbing belt and a repulsive contour, study properties of these structures, and get estimates for their sizes. To make the exposition complete, we, first, study the corresponding problems for three less complicated systems, namely, a proper circle homeomorphism, a proper circle homeomorphism with quasiperiodic perturbation, and a quantized circle homeomorphism without perturbation.     

Criteria of hydrodynamic instability of a phase interface in hydrothermal systems

The loss of stability of a vertical phase flow in a geothermal system in which a liquid layer overlies a vapor layer is considered. The loss of stability criteria are obtained in explicit form. It is found that when the physical parameters of the system are varied the transition to phase interface instability can be realized by means of one of the following mechanisms: the transition occurs spontaneously for any perturbation wavenumber (degenerate case); an unstable wavenumber arises at infinity; the instability threshold is determined by a double zero wavenumber. In the latter case the transition to instability is accompanied by simple resonance bifurcation. As a result of this bifurcation, secondary regimes dependent on the horizontal coordinate branch off from the basic regime describing the horizontally-homogeneous vertical phase flows.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 100–109. Original Russian Text Copyright © 2004 by Ilichev and Tsypkin.     

Lie symmetries,symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.     

Perturbation threshold and hysteresis associated with the transition to turbulence in sudden expansion pipe flow

The complex flow resulting from the laminar-turbulent transition in a sudden expansion pipe flow, with expansion ratio of 1:2, subjected to an inlet parabolic velocity profile and a vortex perturbation, is investigated by means of direct numerical simulations. It is shown that the threshold amplitude for disordered motion is described by a power law scaling, with -3 exponent, as a function of the subcritical Reynolds number. The instability originates from a region of intense shear rate, which results on the flow symmetry breakdown. Above the threshold, several unsteady states are identified using space-time diagrams of the centreline axial velocity fluctuation and their energy. In addition, the simulations show a small hysteresis transition mode due to the reestablishment of the recirculation region in the subcritical range of Reynolds numbers, which depends on: (i) The initial and final quasi-steady states, (ii) the observation time and (iii) the number of intermediate steps taken when increasing and decreasing the Reynolds number.     

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).     

Longwave Approximation in Film Flow Theory

An asymptotic longwave model which takes dispersive terms into account is constructed for describing the motion of thin films with finite deviations from the middle surface. An exact periodic solution describing a nonlinear capillary wave is constructed within the framework of the model. Small deviations from the nonlinear capillary wave are described by a linear system with periodic coefficients. It is shown that for wave perturbation periods greater than a certain critical value the monodromy matrix of this system has eigenvalues whose absolute values are equal to unity. For perturbation periods less than the critical period the absolute value of one of the eigenvalues becomes greater than unity.     

短波近似的保辛算法

WKBJ短波近似是最常用的有效求解方法之一。保守体系的微分方程可用Hamilton体系的方法描述,其特点是保辛。保辛给出保守体系结构最重要的特性。但WKBJ短波近似却未曾考虑保辛的问题。WKBJ近似可用自变量坐标变换,然后再给出其保辛摄动。数值例题展示了本文变换保辛算法的有效性。     

Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric Systems with Cubic Nonlinearities

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Global bifurcations and chaos in a harmonically excited and undamped circular plate

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.     

Chemical chaos in enzyme kinetics

In this paper, the effect of impulsive perturbation on enzyme kinetics is investigated. The impulsive perturbation is affected by introducing periodic constant input. The dynamical behavior of system is simulated and bifurcation diagrams are obtained. The results show that impulsive perturbation can easily give rise to complex dynamics, which includes: quasi-periodic oscillation, periodic doubling cascade, periodic halving cascade, attractor crisis and chaotic bands with periodic windows.     

Asymptotic model of the development of separations inside a boundary layer under the action of a traveling pressure wave

A model of the nonlinear interaction between a pressure perturbation traveling at a constant velocity and an incompressible boundary layer is constructed when its near-wall part is described by the “inviscid boundary layer” equations. A steady-state solution is managed to obtain in the finite form under the assumption that it exists in a moving coordinate system. It is shown that the boundary layer can easily overcome pressure perturbations whose amplitude is not higher than the dynamic pressure calculated from the velocity of the pressure perturbation. At the higher pressure perturbation amplitudes a vortex sheet sheds from the body surface to the boundary layer. The vortex sheet represents an unstable surface of the tangential discontinuity which separates the regions of the direct and reverse separation flows. In the case of an arbitrary shape of the pressure perturbation the surface of the tangential discontinuity sheds from the body surface at a finite angle with the formation of a stagnation point. An example of the pressure perturbation in which the vortex sheet sheds from the body surface along the tangent is constructed.     

Stability Analysis for Flows in Rotating Spherical Layers (Linear Theory)

We present a technique and the results of linear stability analysis for the viscous incompressible flows in an annulus between two concentric coaxial spheres, of which the outer remains stationary while the inner is rotated. We show that the characteristics of the main flow instability strongly depend on the layer thickness, qualitatively as well quantitatively. The critical perturbation for the main flow in a thin layer is monotonic and has the form of a system of ring vortices covering the entire flow region from one pole to the other; the vortices are axisymmetric but nonsymmetric about the equatorial plane. The critical perturbation in a thick layer is nonmonotonic, three-dimensional, and nonsymmetric about the equatorial plane. The shape of the critical perturbation depends on the layer thickness. A comparison with the experimental data is carried out.     

Using delay to quench undesirable vibrations

A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed ε=0 system. The resulting amplitude–delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration.     

有界噪声激励下单摆-谐振子系统的混沌运动

研究了具有同宿轨道和周期轨道的可积单摆-谐振子系统在弱Hamilton摄动(即弱耦合摄动)和弱非Hamilton摄动(即阻尼和有界噪声微扰)下的混沌运动．用Melnikov方程预测Hamilton系统中可能存在混沌运动的参数域，并用Poincare截面验证解析结果．用数值方法计算了有阻尼与有界噪声激励下系统的最大Lyapun0V指数和Poincare截面，结果表明有界噪声在频率上的扩散减小了引发系统产生混沌运动的效应。