Dynamical behavior of a class of prey-predator system with impulsive state feedback control and Beddington–DeAngelis functional response

This paper studies systematically a Bedd-ington?CDeAngelis prey?Cpredator system with harvesting and impulsive state feedback control. Conditions for existence and stability of predator-free periodic solution are obtained. When the predator-free periodic solution loses its stability, the existence and stability of nontrivial period solution are also established. Furthermore, computer simulations show that this impulsive system displays a series of complex phenomena, including period-doubling bifurcation and cascade, period window, and chaotic bands. Through numerical simulation, it is also observed that capture capability can influence the amount of predator released and the interval of the stability for nontrivial period-1 solution. Moreover, the superiority of impulsive state feedback control strategy is also exhibited over the impulsive fixed-time control.     

简谐载荷下嵌入式单壁碳纳米管动态响应的Magnus方法

基于两端固支的弹性梁模型,研究嵌入式单壁碳纳米管在横向简谐载荷作用下的非线性振动问题。利用Galerkin方法对运动微分方程进行近似处理,将原方程从非线性动力学系统转化到二阶动力学系统,对于二阶动力学方程采用Magnus级数方法进行求解。通过数值实验,分析了嵌入式单壁碳纳米管非线性振动幅频特性,根据非线性动力学理论分析了碳纳米管动态响应,结果表明倍周期分岔产生混沌。     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.     

两自由度塑性碰撞振动系统的动力学研究

用三维映射表示具有单侧刚性约束的两自由度振动系统在塑性碰撞时的动力学方程。借助理论分析与数值方法研究了系统周期n-1振动的存在性与稳定性,描述了系统周期n-1振动的特点,讨论了碰撞振子与约束擦边引起的Poincare映射奇异性对系统全局分岔的影响。     

Homoclinic-Doubling Cascades

Cascades of period-doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos. In this paper we consider routes to onset of chaos involving homoclinic-doubling bifurcations. We show the existence of cascades of homoclinic-doubling bifurcations which occur persistently in two-parameter families of vector fields on ?³. The cascades are found in an unfolding of a codimension-three homoclinic bifurcation which occur an orbit-flip at resonant eigenvalues. We develop a continuation theory for homoclinic orbits in order to follow homoclinic orbits through infinitely many homoclinic-doubling bifurcations.     

周期激励浅拱分岔研究

研究了一阶和二阶模态在１：２内共振条件下浅拱的复杂动力学行为，指出当周期激励浅拱具有初始静变形时，系统的一阶模态和二阶模态会产生内共振，系统两共振模态之间会产生相互作用，系统的能量会在其低阶和高阶模态之间相互传递，对称破缺后的Ｈｏｐｆ分岔解会通过一系列的倍化周期分岔导致混沌，在混沌域中还会发现稳定的周期解窗口．     

The study of a ratio-dependent predator–prey model with stage structure in the prey

In this paper, a ratio-dependent predator–prey model with stage structure in the prey is constructed and investigated. In the first part of this paper, some sufficient conditions for the existence and stability of three equilibriums are obtained. In the second part, we consider the effect of impulsive release of predator on the original system. A sufficient condition for the global asymptotical stability of the prey-eradication periodic solution is obtained. We also get the condition, under which the prey would never be eradicated, i.e., the impulsive system is permanent. At last, we give a brief discussion.

     

Bifurcation and chaos of airfoil with multiple strong nonlinearities

The bifurcation and chaos phenomena of two-dimensional airfoils with multiple strong nonlinearities are investigated.First,the strongly nonlinear square and cubic plunging and pitching stiffness terms are considered in the airfoil motion equations,and the fourth-order Runge-Kutta simulation method is used to obtain the numerical solutions to the equations.Then,a post-processing program is developed to calculate the physical parameters such as the amplitude and the frequency based on the discrete numerical solutions.With these parameters,the transition of the airfoil motion from balance,period,and period-doubling bifurcations to chaos is emphatically analyzed.Finally,the critical points of the period-doubling bifurcations and chaos are predicted using the Feigenbaum constant and the first two bifurcation critical values.It is shown that the numerical simulation method with post-processing and the prediction procedure are capable of simulating and predicting the bifurcation and chaos of airfoils with multiple strong nonlinearities.     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Dynamical behaviors of the periodic parameter-switching system

In this paper, a periodic parameter-switching system about Lorenz oscillators is established. To investigate the bifurcation behavior of this system, Poincaré mapping of the whole system is defined by suitable local sections and local mappings. The location of the fixed point and the parameter values of local bifurcations are calculated by the shooting method and Runge–Kutta method. Then based on the Floquent theory, we conclude that the period-doubling and saddle-node bifurcations play an important role in the generation of various periodic solutions and chaos. Meanwhile, upon the analysis of the equilibrium points of the subsystems, we explore the mechanisms of different periodic switching oscillations.     

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.     

Impulsive control induced effects on dynamics of single and coupled ODE systems

The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.     

Chaos and instability in a power system: Subharmonic-resonant case

The response of a single-machine quasi-infinite busbar system to the simultaneous occurrence of principal parametric resonance and subharmonic resonance of order one-half is investigated. By numerical simulations we show the existence of oscillatory solutions (limit cycles), period-doubling bifurcations, chaos, and unbounded motions (loss of synchronism). The method of multiple scales is used to derive a second-order analytical solution that predicts (a) the onset of period-doubling bifurcations, which is a precursor to chaos and unbounded motions (loss of synchronism), and (b) saddle-node bifurcations, which may be precursors to loss of synchronism.     

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.     

Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system

This paper analyzes the hyperchaotic behaviors of the newly presented simplified Lorenz system by using a sinusoidal parameter variation and hyperchaos control of the forced system via feedback. Through dynamic simulations which include phase portraits, Lyapunov exponents, bifurcation diagrams, and Poincaré sections, we find the sinusoidal forcing not only suppresses chaotic behaviors, but also generates hyperchaos. The forced system also exhibits some typical bifurcations such as the pitchfork, period-doubling, and tangent bifurcations. Interestingly, three-attractor coexisting phenomenon happens at some specific parameter values. Furthermore, a feedback controller is designed for stabilizing the hyperchaos to periodic orbits, which is useful for engineering applications.     

Stability analysis for milling process

In this article, the stability of a milling process is studied by using a semi-discretization method. The model of the workpiece–tool system includes loss-of-contact effects between the workpiece and the tool and time-delay effects associated with the chip-thickness variation. In addition, feed-rate effects are also considered. The governing system of equations is a non-autonomous, delay-differential system with time-periodic coefficients. Stability of periodic orbits of this system is studied to predict the onset of chatter and numerical evidence is provided for period-doubling bifurcations and secondary Hopf bifurcations. Stability charts generated using the semi-discretization method are found to compare well with the corresponding results obtained through time-domain simulations.     

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.     

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral–partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force–response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.     

Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey

We investigate a stage-structured delayed predator-prey model with impulsive stocking on prey and continuous harvesting on predator. According to the fact of biological resource management, we improve the assumption of a predator-prey model with stage structure for predator population that each individual predator has the same ability to capture prey. It is assumed that the immature and mature individuals of the predator population are divided by a fixed age, and immature predator population does not have the ability to attach prey. Sufficient conditions are obtained, which guarantee the global attractivity of predator-extinction periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Numerical analysis is presented to illuminate the dynamics of the system.     

悬索在外激励作用下的1∶3内共振分析(II)∶数值结果

本研究的第一部分已经推导了悬索在第一阶面内对称模态主共振和第三阶面内对称模态主共振下的平均方程,其中考虑了这两阶模态之间1∶3内共振。本文对平均方程的稳态解、周期解以及混沌解进行了研究。利用Newton-Naphson方法和拟弧长的延拓算法确定了主共振情况下的幅频响应曲线,通过利用Jacobian矩阵的特征值判断幅频响应曲线中解的稳定性。在这些幅频响应曲线中,都存在超临界Hopf分叉,导致平均方程的周期解。以这些超临界Hopf分叉为起点,利用打靶法和拟弧长的延拓算法确定了两种主共振情况下的周期解分支,同时通过利用Floquet理论判断这些周期解的稳定性。然后利用数值结果研究了两种主共振情况下的周期解经过倍周期分叉通向混沌的过程。最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应。