Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects

In this paper, the propagation of a nonlinear delay SIR epidemic using the double epidemic hypothesis is modeled. In the model, a system of impulsive functional differential equations is studied and the sufficient conditions for the global attractivity of the semi-trivial periodic solution are drawn. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination, and nonlinear incidence have significant effects on the dynamics behaviors of the model. The conditions for the control of the infection caused by viruses A and B are given.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。     

The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and?harvesting prey at different fixed moments

In this paper, a delayed pest control model with stage-structure for pests by introducing a constant periodic pesticide input and harvesting prey (Crops) at two different fixed moments is proposed and analyzed. We assume only the pests are affected by pesticide. We prove that the conditions for global asymptotically attractive ??predator-extinction?? periodic solution and permanence of the population of the model depend on time delay, pulse pesticide input, and pulse harvesting prey. By numerical analysis, we also show that constant maturation time delay, pulse pesticide input, and pulse harvesting prey can bring obvious effects on the dynamics of system, which also corroborates our theoretical results. We believe that the results will provide reliable tactic basis for the practical pest management. One of the features of present paper is to investigate the high-dimensional delayed system with impulsive effects at different fixed impulsive moments.     

Permanence and periodic solutions in models with stage structure,delay, and pulse action

We obtain sufficient conditions for the permanence and existence of an asymptotically stable periodic solution for a model of evolution of a biological species with stage structure, delay, and pulse action and for a system of equations with delay and pulse action that models the dynamics of two competitive species with stage structure.     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays

This paper deals with dynamic behaviors on Hopfield type of ring neural network of four neurons having a pair of short-cut connections with multiple time delays. By suitable transformation and under certain assumptions on multiple time delays, the model is reduced to four dimensional nonlinear delay differential equations with three delays. Regarding these time delays as parameters, delay independent sufficient conditions for no stability switches of the trivial equilibrium of the linearized system are derived. Conditions for stability switching with respect to one delay parameter which is not associated with short-cut connection are obtained. Hopf bifurcations with respect to two other delays which are associated with short-cut connection are also obtained. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Using numerical simulations of the nonlinear model, different rich dynamical behaviors such as quasiperiodicity, torus attractor and chaotic-bands are also observed for suitable range of three delay parameters. Lyapunov exponents are also calculated using the AnT 4.669 tool for verification of chaotic dynamics.     

Dynamics of ripple bed vortices

 This paper presents the results of an experimental investigation into the dynamics of vortices formed by and shed from ripples under oscillatory motion. Particle Image Velocimetry (PIV) was used to acquire detailed velocity measurements at several phases of oscillation for each of four flow conditions: three purely horizontal oscillations and a ‘real’ wave with a vertical velocity component. For each of the flows, the vortices were located in the flow maps and defined at each time-step by their circulations and sizes. These calculated characteristics were compared with the predictions from a discrete-vortex model (DVM) developed at the University of Wales, Bangor. Received: 22 May 1997/Accepted: 29 January 1998     

Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg. 49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg. 58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg. 13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering 32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech. 53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech. 53, 1986, 855–863].     

A gas pendulum

It is shown that a periodic two-dimensional isentropic motion of a gas exists and it is described by an exact solution of the equations of gas dynamics. A polytropic gas that fills a circular cylinder rotates and oscillates (in the radial direction) simultaneously under the action of periodically changing external pressure. The solution obtained belongs to the class of solutions with a velocity field that is linear in the coordinates (with homogeneous deformation). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 115–119, September–October, 2000.     

Jounce and Impact in Cam-Tappet Conjunction Induced by the Elastodynamics of Valve train System

The paper highlights the importance of a multi-physics integrated approach in the design evaluation of complex valve trains. The multi-disciplinarity of the problems requires simultaneous solution to inertial dynamics, structural compliance of system elements and tribological conditions in load bearing conjunctions, all of which are critical to thermodynamic functional assurance of the system as the primary objective, and system efficiency in terms of reduced mechanical and frictional losses. This paper is an extended version of: ‘PREDICTION OF CAM-TO-TAPPET CONTACT LOSS WITH VALVE TRAIN ELASTODYNAMIC ANALYSIS’, 2004 AIMETA International Tribology Conference, September 14–17, 2004, Rome, Italy.     

Second-order interaction of irregular waves with a truncated column

A complete semi-analytical solution is obtained for second-order diffraction of plane bichromatic waves by a fixed truncated circular column. The fluid domain is divided into interior and exterior regions. In the exterior region, the second-order velocity potential is expressed in terms of ‘locked-wave’ and ‘free-wave’ components, both are solved using Fourier and eigenfunction expansions. The resulting ‘locked wave’ potential is expressed by one-dimensional Green's integrals with oscillating integrands. In order to increase computational efficiency, the far-field part of the integrals are carried out analytically. Solutions in both regions are matched on the interface by the potential and its normal derivative continuity conditions. Based on the present approach, the sum-and difference-frequency potentials are efficiently evaluated and are used to generate the quadratic transfer functions which correlates sum-frequency QTFs for a TLP column are present, which are compared for some frequency pairs with those from a fully numerical procedure. Satisfactory agreement has been obtained. QTF spectra for a case study TLP column, generated using the semi-analytical solution are presented. Also given are the results for nonlinear wave field around the column.     

多尺度法在时滞弹性关节机械臂非线性振动问题中的应用

针对非线性弹性关节机械臂,研究传动过程中的时滞效应对机械臂系统周期振动的影响．本文改进了具有弹性关节的非线性机械臂动力学模型,引入时滞参数,应用多尺度法,得到系统的近似解析解,考察了时滞对机械臂系统周期运动的影响规律．数值软件计算结果表明解析解与数值解具有较好的吻合度．从而验证了本文多尺度方法的有效性和正确性．     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

From Rate-Dependent to Rate-Independent Brittle Crack Propagation

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.     

Stability and Hopf bifurcation of a two-dimensional supersonic airfoil with a time-delayed feedback control surface

The time-delayed feedback control for a supersonic airfoil results in interesting aeroelastic behaviors. The effect of time delay on the aeroelastic dynamics of a two-dimensional supersonic airfoil with a feedback control surface is investigated. Specifically, the case of a 3-dof system is considered in detail, where the structural nonlinearity is introduced in the mathematical model. The stability analysis is conducted for the linearized system. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo the stability switches with the variation of the time delay. The nonlinear aeroelastic system undergoes a sequence of Hopf bifurcations if the time delay passes the critical values. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation and stability of Hopf-bifurcating periodic solutions are determined. Numerical simulations are performed to illustrate the obtained results.     

Non-Linear Dynamics of a Rotor-Bearing-Seal System

The non-linear dynamic behaviors of a rotor-bearing-seal coupled system are investigated by using Muszynska’s non-linear seal fluid dynamic force model and non-linear oil film force, and the result from the numerical analysis is in agreement with the one from the experiment. The bifurcation of the coupled system is analyzed under different operating conditions. It is indicated that the dynamic behavior of the rotor-bearing-seal system depends on the rotation speed, seal clearance and seal pressure of the rotor-bearing-seal system. The system state trajectory, Poincaré maps, frequency spectra and bifurcation diagrams are constructed to analyze the dynamic behavior of the rotor center. Various non-linear phenomena in the coupled system, such as periodic motion and quasi-periodic motion are investigated. The results show that the system has the potential for chaotic motion. The study may contribute to a further understanding of the non-linear dynamics of such a rotor-bearing-seal coupled system.     

Unsteady Flux-Vector-Based Green Element Method

This article extends the mathematical formulation and solution procedure of the modified ‘q-based’ GEM to unsteady situations, namely to the modified unsteady ‘q-based’ GEM. Solutions that provide information on the evolution of the pressure and the flux over long time intervals are available by incorporating the additional dimension of time into steady problems. This approach is first tested by solving an example for which an analytical solution is available. The numerical results for this example is found to be in excellent agreement with the analytical solution. Several problems involving geological features, such as wells and faults, are then investigated, with different properties applying to the faults. A strong influence of the low permeability faults is in evidence in these problems.     

The nonlinear interaction of vortex rings with a free surface

Nonlinear interactions of vortex rings with a free surface are considered in an incompressible, ideal fluid using the vortex contour dynamics technique and the boundary integral equation method. The flow is axisymmetric and the vorticity is linearly distributed in the vortex. Effects of the gravity and the surface tension as well as the initial geometric parameter of the vortex on the interaction process are investigated in considerable detail. The interaction process may be divided into three major stages: the vortex free-traveling stage, the collision stage, and the vortex stretching and rebounding stage. Time evolutions of both the vortex and free surface under various conditions are provided and analyzed. Two kinds of waves exist on the free surface during interaction. In a special case where the gravity and surface tension are very weak or the vortex is very strong, an electric-bulb-like ‘cavity’ is formed on the free surface and the vortex is trapped in the ‘cavity’ for quite a long time, resulting in a large amount of fluid above the mean fluid surface. The project supported by the National Education Commission of China and NASA under cooperative grant agreement # NCC5-34     

Creep—Slip Capture as a Possible Source of Squeal During Decelerated Sliding

Friction-induced vibration of a two-degree-of-freedom mass-damper-spring system interacting with a decelerating rigid strip is investigated. The friction law is approximated by an analytical function to facilitate the analyses and numerical integrations. It is shown that, after a quasi-harmonic transient period, accompanied by viscous energy dissipation, a short period of intensive ‘creep-slip’ vibration occurs, which generates a series of ‘micro-impacts’ on the strip. Because of the impulsive character of such kind of loading, its Fourier spectrum is rich and quite broadband. Using an averaging technique, the ‘normal form’ equations of motion show that the out-of-phase vibration mode absorbs more energy from the decelerating strip when its natural frequency satisfies certain resonance conditions. The study is then applied to an automotive disc brake model to gain useful insight into the generation of squeal. It is shown that the out-of-phase creep-slip vibration (in the longitudinal direction) of the brake pads generates an impulsive bending moment on the decelerating strip (disc rotor). This impulsive load may be considered as a possible source for brake squeal. The technique developed in this paper may be extended to other ‘squealing systems’ including models for geophysical faults (earthquakes).     

An analysis of subgrid-resolved scale interactions with use of results from direct numerical simulations

Subgrid nonlinear interaction and energy transfer are analyzed using direct numerical simulations of isotropic turbulence. Influences of cutoff wave number at different ranges of scale on the energetics and dynamics have been investigated. It is observed that subgrid-subgrid interaction dominates the turbulent dynamics when cut-off wave number locates in the energy-containing range while resolved-subgrid interaction dominates if it is in the dissipation range. By decomposing the subgrid energy transfer and nonlinear interaction into ‘forward’ and ‘backward’ groups according to the sign of triadic interaction, we find that individually each group has very large contribution, but the net of them is much smaller, implying that tremendous cancellation happens between these two groups.