Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Robust synchronization of a chaotic mechanical system with nonlinearities in control inputs

Centrifugal flywheel governors are known as chaotic non-autonomous mechanical devices used for automatic control of the speed of engines. The main characteristic of them is avoiding the damage caused by sudden change of the load torques. In this paper, the problem of robust finite-time synchronization of centrifugal flywheel governor systems is studied. The effects of unknown parameters, model uncertainties, external noises, and input nonlinearities are fully taken into account. We propose some adaptive laws to overcome the side effects of the unknown parameters of the system on the synchronization performance. Then, a robust adaptive switching controller is introduced to synchronize centrifugal flywheel governors with nonlinear control inputs in a given finite time. The finite-time fast convergence property of the proposed scheme is analytically proved and numerically illustrated.     

Amplitude modulated motions in a two degree-of-freedom system with quadratic nonlinearities under parametric excitation: experimental investigation

We conducted an experimental investigation of amplitude modulated response of a two degree-of-freedom mechanical structure with quadratic nonlinearities to parametric excitation. The linear natural frequencies of the system were tuned so that they were approximately in the ratio of two-to-one, and the excitation frequency was in principal parametric resonance with the first mode. We observed periodically amplitude-modulated motions and chaotically amplitude-modulated motions.     

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.     

Phase-locking in a jet forced with two frequencies

This study investigates the effect of forcing a shear layer at more than one frequency. Multiple frequency forcing permits the phase and initial relative amplitudes among unstable waves to be manipulated. More control can be imposed on vortex merging and mixing. Various vortex merging modes were observed and explained by the relative strength of the instability waves and their phase alignment. The vortex phase and path jitterings present in single-frequency forcing cases are greatly reduced when forced at more than one frequency. The observed cycle-to-cycle variation was small. This enables phase-lock measurements to be performed more easily. The phase-lock data show excellent agreement with the flow visualization results even when averaged over only a few cycles.     

Stable amplitude chimera states and chimera death in repulsively coupled chaotic oscillators

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation.     

Onset of chaotic motion in a gyroscopic system

We study forced vibrations of a gimbal gyro occurring if the inner ring is subjected to a perturbing torque that is the sum of the viscous friction torque and a periodic small-amplitude torque. In the absence of the perturbing torque, there exist two steady-state motions of the gimbal gyro, in which the gimbal rings are either orthogonal or coincide. These motions are respectively stable and unstable. We obtain an equation for the unperturbed system, whose separatrix passes through hyperbolic points. The distance between these points (the Melnikov distance) is calculated to find a condition for the intersection of the separatrices of the perturbed system. We find a domain in the parameter space where the distance changes sign, which indicates the onset of chaotic motion.     

双面理想完整约束系统的机械能守恒律

研究双面理想完整约束系统在约束不是定常且主动力不是有势时的机械能守恒律. 建立系统的能量变化方程,给出存在机械能守恒律的充分必要条件. 分析有机械能守恒律的12种情况. 最后给出说明性算例.     

Nonlinear and chaotic analysis of a financial complex system

In this paper,determination of the characteristics of futures market in China is presented by the method of the phase-randomized surrogate data.There is a significant difference in the obtained critical values when this method is used for random timeseries and for nonlinear chaotic timeseries.The singular value decomposition is used to reduce noise in the chaotic timeseries.The phase space of chaotic timeseries is decomposed into range space and null noise space.The original chaotic timeseries in range space is restructured.The method of strong disturbance based on the improved general constrained randomized method is further adopted to re-deternination.With the calculated results,an analysis on the trend of futures market of commodity is made in this paper.The results indicate that China＇s futures market of commodity is a complicated nonlinear system with obvious nonlinear chaotic characteristic.     

Experimental study of regular and chaotic transients in a non-smooth system

This paper focuses on thoroughly exploring the finite-time transient behaviors occurring in a periodically driven non-smooth dynamical system. Prior to settling down into a long-term behavior, such as a periodic forced oscillation, or a chaotic attractor, responses may exhibit a variety of transient behaviors involving regular dynamics, co-existing attractors, and super-persistent chaotic transients. A simple and fundamental impacting mechanical system is used to demonstrate generic transient behavior in an experimental setting for a single degree of freedom non-smooth mechanical oscillator. Specifically, we consider a horizontally driven rigid-arm pendulum system that impacts an inclined rigid barrier. The forcing frequency of the horizontal oscillations is used as a bifurcation parameter. An important feature of this study is the systematic generation of generic experimental initial conditions, allowing a more thorough investigation of basins of attraction when multiple attractors are present. This approach also yields a perspective on some sensitive features associated with grazing bifurcations. In particular, super-persistent chaotic transients lasting much longer than the conventional settling time (associated with linear viscous damping) are characterized and distinguished from regular dynamics for the first time in an experimental mechanical system.     

Multiple attractors and strange nonchaotic dynamical behavior in a periodically forced system

Nonlinear Dynamics - This work investigates the dynamical behaviors of the third-order chaotic oscillator with threshold controller as its nonlinear element. The system reveals several complex...     

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

Nonlinear Dynamics - A general method to calculate multi-parameter bifurcation diagram in the parameter space is designed based on top Lyapunov exponent and Floquet multiplier to study the effect...     

Synchronization and chimera state in a mechanical system

Nonlinear Dynamics - Synchronization phenomenon appears in several natural systems being associated with physical, chemical and biological processes. In brief, synchronization may be understood as...     

Dynamical properties and simulation of a new Lorenz-like chaotic system

This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rössler system, Chen system, and includes Lü systems as its special case. By using the center manifold theorem, the stability character of its non-hyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.     

Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan?CYork dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.     

Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system

In this paper we revisit a 3D autonomous chaotic system, which contains both the modified Lorenz system and the conjugate Chen system, presented in [Huang and Yang, Chaos Solitons Fractals 39:567–578, 2009]. First by citing two examples to show the errors and limitations for the local stability of the equilibrium point S ₊ obtained in this literature, we formulate a complete determining criterion for the local stability of S ₊ of this system. Although the local bifurcation problem of this system, mainly for Hopf bifurcation, etc., has been studied, the invoking of incorrect proposition leads to an incorrect result for Hopf bifurcation. We then renew the study of the Hopf bifurcation of this system by utilizing the Project Method. The global bifurcation problem, relatively speaking, should be more difficult than the local bifurcation problem for a given system. However, the global bifurcation problem of this system, to the best of our knowledge, has not been investigated yet in the literatures. So next we consider the global bifurcation problem for this system, mainly for the existence of homoclinic and heteroclinic orbits. Our results, one of which shows the existence of two heteroclinic orbits, not only correct and further supplement the ones obtained in the literature, but also give something new to theoretically help fully understand the occurrence of chaos.