Anti-controlling quasi-periodic impact motion of an inertial impact shaker system

In this paper, anti-controlling quasi-periodic impact motion of an inertial impact shaker system is addressed. There exist two aspects of difficulty in the anti-control design: one is from the implicit Poincaré map of the system itself and the other from the limitation of the classical critical criterion of Hopf bifurcation described by the properties of eigenvalues. Through using linear feedback control method in the original differential system and applying an explicit criterion of Hopf bifurcation without using eigenvalues to the Poincaré map of the close-loop system, the two difficulties above can be overcome and the control design for creation of the quasi-periodic impact motion at a specified system parameter location is achieved. Numerical simulation shows that the stable quasi-periodic impact motion of the system is created at a desired parameter location by adjusting control parameter appropriately.     

Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators

We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincaré map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically.     

A four-wing and double-wing 3D chaotic system based on sign function

Many four-wing chaotic systems have been developed based on cross product or quadratic operations. Differently, we construct a three-dimensional chaotic system generating four-wing or double-wing attractors by virtue of sign function. Dynamical properties such as equilibrium points, Poincaré map, Lyapunov exponent spectra, Hopf bifurcations and bifurcation diagrams of the system are theoretically and numerically analyzed. Results of mathematical analyses and simulation tests indicate that the proposed chaotic system can keep chaotic to generate four-wing or double-wing attractors within a large scope of parameters. The system also shows hyperchaotic behaviors in some parameter range. Besides, circuit implementation of the chaotic system is studied. That proves the system is physically realizable.     

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincar'e map of the system is constructed. Using the Poincar'e map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.     

Analysis of the T-point-Hopf bifurcation

In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-2 heteroclinic cycle occurs. If the parameter space is three-dimensional, such a bifurcation is located generically on a curve. A more degenerate scenario appears when this curve reaches a surface of Hopf bifurcations of one of the equilibria involved in the heteroclinic cycle. We are interested in the analysis of this codimension-3 bifurcation, which we call T-point-Hopf. In this work we propose a model, based on the construction of a Poincaré map, that describes the global behavior close to a T-point-Hopf bifurcation. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol-Duffing electronic oscillator.     

Two routes to chaos in the fractional Lorenz system with dimension continuously varying

The object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives. We take the fractional Lorenz system as example and regard one or three of its orders as bifurcation parameters. In this framework, we compute the corresponding bifurcation diagrams via an optimal Poincaré section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3. One is the process of cascaded period-doubling bifurcations and the other is a crisis (boundary crisis) which occurs in the evolution of chaotic transient behavior. We would like to point out that our investigation is the first to find out that a fractional differential equations (FDEs) system can evolve into chaos by the crisis. Furthermore, we observe rich dynamical phenomena in these processes, such as two-stage cascaded period-doubling bifurcations, chaotic transients, and the transition from coexistence of three attractors to mono-existence of a chaotic attractor. These are new and interesting findings for FDEs systems which, to our knowledge, have not been described before.     

振动筛系统的两类余维三分岔与非常规混沌演化

建立了振动筛系统的动力学模型,推导出了其周期运动的Poincaré 映射,基于Poincaré 映射方法着重研究了系统Flip-Hopf-Hopf余维三分岔、三次强共振条件下的Hopf-Hopf余维三分岔以及三种非常规的混沌演化过程.研究结果表明,此两类余维三分岔点附近的动力学行为变得更加复杂和新颖,在分岔点附近出现了三角形吸引子、3T²环面分岔以及“五角星型”、“轮胎型”概周期吸引子,揭示了环面爆破、环面倍化以及T²环面分岔向混沌演化的过程,这些结果对于振动筛系统的动力学优化设计提供了理论参考. 关键词： 余维三分岔 非常规混沌演化 T²环面分岔')" href="#">T²环面分岔     

A Fractional-Order Sinusoidal Discrete Map

In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.     

Parametric excitation of two internally resonant oscillators

The response of two-degree-of-freedom systems with quadratic non-linearities to a principal parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order non-linear ordinary differential (averaged) equations governing the modulation of the amplitudes and the phases of the two modes. These equations are used to determine steady state responses and their stability. When the higher mode is excited by a principal parametric resonance, the non-trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude, whereas the amplitude of the lower mode, which is indirectly excited through the internal resonance, increases with the amplitude of the excitation. However, in addition to Poincaré-type bifurcations, this response exhibits a Hopf bifurcation leading to amplitude- and phase-modulated motions. When the lower mode is excited by a principal parametric resonance, the averaged equations exhibit both Poincaré and Hopf bifurcations. In some intervals of the parameters, the periodic solutions of the averaged equations, in the latter case, experience period-doubling bifurcations, leading to chaos.     

Experiments on routes to chaos in ball bearings

The theoretical motion of a ball bearing has been studied in a previous paper. Using a control parameter, different routes to chaos were described. The aim of this paper is to study the experimental routes to chaos in a ball bearing and to confirm whether theoretical predictions of the phenomena are realistic.An experimental test bench has been used and a numerical procedure has been proposed for observing Poincaré maps. As the control parameter varies the bearing clearly shows the appearance of instability in its motion. Two different routes to chaos are described as expected from the theory.The first route is related to the first resonant frequency of the bearing. It is a sub-harmonic route. The second route, associated with the second resonant frequency, is a quasi-periodic route.     

Dynamics analysis and synchronization of a new chaotic attractor

In this paper, a new simple chaotic system is discussed. Basic dynamical properties of the new attractor are demonstrated in terms of phase portraits, equilibria and stability, Lyapunov exponents, a dissipative system, Poincaré mapping, bifurcation diagram, especially Hopf bifurcation. Next, based on well-known Lyapunov stability theorem, backstepping design is proposed for synchronization of the new chaotic system. At last, numerical studies are provided to illustrate the effectiveness of the presented scheme.     

Hopf bifurcation and uncontrolled stochastic traffic- induced chaos in an RED-AQM congestion control system

We study the Hopf bifurcation and the chaos phenomena in a random early detection-based active queue management (RED-AQM) congestion control system with a communication delay. We prove that there is a critical value of the communication delay for the stability of the RED-AQM control system. Furthermore, we show that the system will lose its stability and Hopf bifurcations will occur when the delay exceeds the critical value. When the delay is close to its critical value, we demonstrate that typical chaos patterns may be induced by the uncontrolled stochastic traffic in the RED-AQM control system even if the system is still stable, which reveals a new route to the chaos besides the bifurcation in the network congestion control system. Numerical simulations are given to illustrate the theoretical results.     

Nonlinear dynamics of two harmonically excited elastic structures with impact interaction

In this paper, non-smooth dynamics of two elastic beams excited by harmonic force with impact interaction is studied through analyses, simulations, and experiments. A two degree-of-freedom vibro-impact model is improved by applying the Galerkin approach and Newton's impact law for the two cantilever beams with impact interaction. Numerical analysis is taken to investigate the vibro-impact motions of cantilever beams excited by harmonic force. The l-adding periodic motions and k=1/1, k=2/2, k=3/4, and k=4/4 type of stable periodic motions of the impacted cantilever beam are presented. Poincaré map is established and the Floquet multipliers of the periodic motions are obtained through semi-analytical method to determine the stability of the motions near the bifurcation point. Through associated experiments, typical bifurcations and chaos of the non-smooth system are examined, which are in good agreement with numerical results.     

分段线性电路切换系统的复杂行为及非光滑分岔机理

分析了分段线性电路系统在周期切换下的复杂动力学行为及其产生的机理. 基于平衡点分析, 给出了两子系统Fold分岔和Hopf分岔条件. 考虑了在不同稳定态时两子系统周期切换的分岔特性, 产生了不同的周期振荡, 并揭示了其产生的机理. 在不同的周期振荡中, 切换点的数量随参数变化产生倍化, 导致切换系统由倍周期分岔进入混沌. 关键词： 分段线性电路 切换系统 非光滑分岔     

Delay-induced dynamics of an axially moving string with direct time-delayed velocity feedback

The local dynamics of an axially moving string under aerodynamic forces is investigated with a time-delayed velocity feedback controller. The retarded differential difference governing equation is obtained in modal coordinates of a two-degree-of-freedom system through Galerkin’s discretization procedure. The stability of trivial equilibrium is examined with the change of counting multiplicity of eigenvalue with positive real part. The Hopf bifurcation curves are determined in the controlling parameter spaces. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation of one complex variable on the center manifold. The approximate analytical solutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. The curves of excitation-response and frequency-response curves are shown with the effect of time delay. The stability analysis for steady-state periodic solutions of the reduced system indicates the onset of local control parameter for vibration control and response suppression. Moreover, the Poincaré-Bendixson theorem and energy considerations are used to investigate the existences and characteristics of quasi-periodic solutions when stability of the periodic solution is lost. Numerical results demonstrate the validity of the analytical prediction. Two different kinds of quasi-periodic solutions are found.     

Bifurcation analysis and control of periodic solutions changing into invariant tori in Langford system

Bifurcation characteristics of the Langford system in a general form are systematically analysed, and nonlinear controls of periodic solutions changing into invariant tori in this system are achieved. Analytical relationship between control gain and bifurcation parameter is obtained. Bifurcation diagrams are drawn, showing the results of control for secondary Hopf bifurcation and sequences of bifurcations route to chaos. Numerical simulations of quasi-periodic tori validate analytic predictions.     

Bifurcation analysis of the logistic map via two periodic impulsive forces

The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincare′ map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.     

Dynamic stability of weakly damped oscillators with elastic impacts and wear

The dynamics of non-linear oscillators comprising of a single-degree-of-freedom system and beams with elastic two-sided amplitude constraints subject to harmonic loads is analyzed. The beams are clamped at one end, and constrained against unilateral contact sites near the other end. The structures are modelled by a Bernoulli-type beam supported by springs using the finite element method. Rayleigh damping is assumed. Symmetric and elastic double-impact motions, both harmonic and sub-harmonic, are studied by way of a Poincaré mapping that relates the states at subsequent impacts. Stability and bifurcation analyses are performed for these motions, and domains of instability are delineated. Impact work rate, which is the rate of energy dissipation to the impacting surfaces, is evaluated and discussed. In addition, an experiment conducted by Moon and Shaw on the vibration of a cantilevered beam with one-sided amplitude constraining stop is modelled. Bifurcation observed in the experiment could be captured.     

Dynamical behaviors of a chaotic system with no equilibria

Based on Sprott D system, a simple three-dimensional autonomous system with no equilibria is reported. The remarkable particularity of the system is that there exists a constant controller, which can adjust the type of chaotic attractors. It is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping and period-doubling route to chaos are analyzed with careful numerical simulations.