Influence of yaw stiffness on the nonlinear dynamics of railway wheelset

A numerical simulation of the dynamic behavior of a railway wheelset is presented. The contact forces between the wheel and the rail are estimated using Johnson and Vermeulen theory of creepages. Nonlinear governing equations of motion of wheelset on a straight track are solved using fourth-order Runge–Kutta method. Both symmetric and asymmetric oscillations and chaotic motion are observed. The influence of yaw stiffness and axial velocity on the response of wheelset is studied. Broadband chaotic motion is developed at various velocity levels. The results are presented in the form of time evolution, phase plots, Poincare maps and bifurcation diagrams. The Lyapunov exponent is calculated and its variation with time is presented. Intermittency is observed. There is a shift in the bifurcation diagram by increasing the yaw stiffness. It indicates that chaotic behavior could be delayed with increasing yaw stiffness.     

Complex dynamic behaviors of a discrete-time predator-prey system

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto''s chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension

This study performs a systematic analysis of the dynamic behavior of a single degree-of-freedom spur gear system with and without nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a spur gear system and therefore serve as a useful source of reference for engineers in designing and controlling such systems.     

Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type III functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of $\mathbb{R}_+^2$. Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-$2$, $4$, $8$, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos

This study aims to analyze the dynamic behavior of bevel-geared rotor system supported on a thrust bearing and journal bearings under nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

Bifurcations and chaos control in a discrete-time predator-prey system of Leslie type

We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $\R^{2}_+$ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.     

Strong nonlinearity analysis for gear-bearing system under nonlinear suspension—bifurcation and chaos

This study performs a systematic analysis of the dynamic behavior of a gear-bearing system with nonlinear suspension, nonlinear oil-film force, and nonlinear gear mesh force. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

Dynamic analysis, controlling chaos and chaotification of a SMIB power system

The dynamic behaviors of a SMIB power system are studied in this paper. A single modal equation is used to analyze the qualitative behaviors of the system. The famous equation of motion is called “swing equation”. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. The bifurcation of the parameter dependent system is studied numerically. Besides, the phase portraits, the Poincaré maps, and the Lyapunov exponents are presented to observe periodic and chaotic motions. Further, the addition of periodic force and the feedback control are used to control chaos effectively. Finally, the chaotification problem of the SMIB power system is also issued.     

Chaotic responses on gear pair system equipped with journal bearings under turbulent flow

This study performs a systematic analysis of the dynamic behavior of a gear-bearing system with the turbulent flow effect, nonlinear suspension, nonlinear oil-film force and nonlinear gear mesh force. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient, unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic and chaotic behaviors. The results presented in this study provide a detailed understanding of the operating conditions under which undesirable dynamic motion takes place in gear-bearing system and therefore offer a useful source of reference for engineers in designing and controlling such systems.     

具有反馈控制离散Logistic系统的分支分析（英文）

The paper studies the dynamical behaviors of a discrete Logistic system with feedback control. The system undergoes Flip bifurcation and Hopf bifurcation by using the center manifold theorem and the bifurcation theory. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the system, such as the period-doubling bifurcation in periods 2, 4, 8 and 16, and quasi-periodic orbits and chaotic sets.     

Dynamical behaviors of a discrete two-dimensional competitive system exactly driven by the large centre

In this paper, a new discrete large-sub-center system is obtained by using the Euler and nonstandard discretization methods for the corresponding continuous system. It is surprised that all dynamic behaviors of the discrete system are exactly driven by the large-center equation, for example, the stabilities, the bifurcations, the period-doubling orbits, and the chaotic dynamics, etc. Additionally, the global asymptotical stability, the existence of exact 2-periodic solutions, the flip bifurcation theorem, and the invariant set of the sub-center equation is also given. These results reveal far richer dynamics of the discrete model compared with the continuous model. Through numerical simulation, we can observe some complex dynamic behaviors, such as period-doubling cascade, periodic windows, chaotic dynamics, etc. Especially, our theoretical results are also showed by those numerical simulations.     

Generating a new chaotic attractor by feedback controlling method

A new chaotic system is found by feedback controlling method in this paper. According to the definition of the generalized Lorenz system, the new chaotic system does not belong to generalized Lorenz systems. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map. The particular interest is that this novel system can generate two one‐scroll and one two‐scroll chaotic attractors with the variation of a single parameter. The obtained results show clearly that the system is a new chaotic system and deserves a further detailed investigation. Copyright © 2011 John Wiley & Sons, Ltd.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

一个新混沌系统及错位投影同步通讯方案

提出了一个新的混沌系统,该系统含有五个参数,每个状态方程均含有非线性乘积项.通过理论推导,数值仿真,Lyapunov指数、Lyapunov维数、分岔图研究其基本的动力学特性,并分析了改变参数时系统的动力学行为的变化.本文研究了该系统的错位投影同步,设计了非线性控制器,实现了两个初值不同的新系统的错位投影同步.另外,将该系统及错位投影同步方法应用到保密通信中,基于改进的混沌掩盖通讯原理,在发送端使用新系统信号对信息信号进行加密及传送,最后在同步后的接收端不失真地恢复出有用信号.数值仿真表明所设计的新的混沌系统具有复杂的动力学特性,适用于保密通讯.     

Complex dynamic behaviour of an economic cycle model

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark–Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

A new hyperchaotic system from the Lü system and its control

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D Lü system. Some complex dynamical behaviors such as Hopf bifurcation, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Furthermore, the design is illustrated with both simulations and experiments. Finally, the control problem of a new hyperchaotic system is investigated using negative feedback control. Ordinary feedback control, dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.     

Hopf bifurcation control in the XCP for the Internet congestion control system

In this paper, we investigated the Hopf bifurcation of the eXplicit Control Protocol (XCP) for the Internet congestion control system. These bifurcation behaviors may cause heavy oscillation of average queue length and induce network instability. A time-delayed feedback control method was proposed for controlling Hopf bifurcation in the XCP system. Numerical simulation results are presented to show that the time-delayed feedback controller is efficient in controlling Hopf bifurcation.     

The energy resources system with parametric perturbations and its hyperchaos control

This paper analyses the dynamic behavior of an energy resources system with parametric perturbations. By adding the small sinusoidal perturbations to the three-dimensional energy resource system established for two regions of China, the autonomous system becomes the non-autonomous system which has richer dynamical behaviors. Periodic, chaotic and hyperchaos behaviors are discovered in the system by means of Lyapunov exponents and bifurcation diagrams. A new energy resources hyperchaos attractor is obtained, which has not yet been reported in present literature. Furthermore, effective non-autonomous feedback controllers are designed for stabilizing hyperchaos to unstable periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results. This study will be instructive for the energy resource demand-supply in some regions of China.     

Bifurcations and chaos in a nonlinear discrete time Cournot duopoly game

A nonlinear discrete time Cournot duopoly game is investigated in this paper. The conditions of existence for saddle-node bifurcation, transcritical bifurcation and flip bifurcation are derived using the center manifold theorem and the bifurcation theory. We prove that there exists chaotic behavior in the sense of Marotto＇s definition of chaos. The numerical simulations not only show the consistence with our theoretical analysis, but also exhibit the complex but interesting dynamical behaviors of the model. The computation of maximum Lyapunov exponents confirms the theoretical analysis of the dynamical behaviors of the system.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.