The route to chaos for the Kuramoto-Sivashinsky equation

We present the results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. Our concern is with the asymptotic nonlinear dynamics as the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic (in time) attractor, to another steady fully modal attractor, to another time-periodic attractor, to a steady tetramodal attractor, to another time-periodic attractor having a full sequence of period-doublings (in the parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the lengths of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period-doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4.669201609.... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.This research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-18605 while the authors were in residence at the Institute of Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. Additional support for the second author was provided by ONR Grant N-00014-86-K-0691 while he was at UCLA.     

齿轮传动系统共存吸引子的不连续分岔

大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素.多吸引子共存时,运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换,对整个机器产生不良的影响.目前,一些隐藏的吸引子没有被发现,共存吸引子的分岔演化规律没有被完全揭示.考虑单自由度直齿圆柱齿轮传动系统,构建由局部映射复合的Poincaré映射,给出Jacobi矩阵特征值计算的半解析法.应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔,应用胞映射法计算共存吸引子的吸引域,讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响,揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为.倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞,使倍周期分岔呈现亚临界特性.鞍结分岔是共存周期吸引子出现或消失的主要原因.边界激变引起混沌吸引子及其吸引域突然消失,对应周期吸引子的分岔终止.     

Time delay Duffing’s systems: chaos and chatter control

The effect of a delay feedback control (DFC), realized by displacement in the Duffing oscillator, for parameters which generate strange chaotic Ueda attractor is investigated in this paper. First, the classical Duffing system without time delay is analysed to find stable and especially unstable periodic orbits which can be stabilized by means of displacement delay feedback. The periodic orbits are found with help of the continuation method using the AUTO97 software. Next, the DFC is introduced with a time delay and a feedback gain parameters. The proper time delay and feedback gain are found in order to destroy the chaotic attractor and to stabilize the periodic orbit. Finally, chatter generated by time delay component is suppressed with help of an external excitation.     

碰撞振动系统中周期轨擦边诱导的混沌激变

基于图胞映射理论, 提出了一种擦边流形的数值逼近方法, 研究了典型Du ng 碰撞振动系统中擦边诱导激变的全局动力学. 研究表明, 周期轨的擦边导致的奇异性使得系统同时产生1 个周期鞍和1 个混沌鞍. 当该周期鞍的稳定流形与不稳定流形发生相切时, 边界激变发生使得该混沌鞍演化为混沌吸引子. 噪声可以诱导周期吸引子发生擦边, 这种擦边导致了1 种内部激变的发生, 表现为该周期吸引子与其吸引盆内部的混沌鞍发生碰撞后演变为1 个混沌吸引子.     

Analytic evaluation of periodic responses of a forced nonlinear oscillator

The periodic responses of a strongly nonlinear, single-degree-of-freedom forced oscillator with weak excitation and damping are examined. The presented methodology is based on a regular perturbation expansion, whose first term is the solution of the unforced, and undamped nonlinear problem. Higher order approximations are computed by explicitly solving linear differential equations possessing a periodically varying coefficient. The general theory is used for studying the periodic steady state motions of the periodically forced system. Moreover, it is shown that the presented analysis can be used to analytically study the orbital stability of the identified steady state motions. The proposed method can also be used for studying periodic responses due to nonperiodic transient forces, provided that these responses are close to the O(1) periodic generating solution.     

Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam-mass system

The nonlinear dynamics of a base-excited slender beam carrying a lumped mass subjected to simultaneous combination parametric resonance of sum and difference type along with 1:3:5 internal resonances is investigated. Method of normal form is applied to the governing nonlinear temporal differential equation of motion to obtain a set of first-order differential equations which are used to obtain the steady-state, periodic, quasi-periodic and chaotic responses for different control parameters viz., amplitude and frequency of external excitation and damping. Frequency response, phase portraits, time spectra and bifurcation diagram are plotted to visualize the system behaviour with variation in the control parameters. Here, two distinct zones of trivial instability, blue sky catastrophe phenomena, jump down phenomena, simultaneous occurrence of periodic and chaotic orbits, period doubling of the mixed-mode periodic orbits leading to chaos, attractor merging crisis, boundary crisis, type II and on-off intermittencies are observed. Bifurcation diagram is plotted to facilitate the designer to choose a safe operating zone.     

Chemical chaos in enzyme kinetics

In this paper, the effect of impulsive perturbation on enzyme kinetics is investigated. The impulsive perturbation is affected by introducing periodic constant input. The dynamical behavior of system is simulated and bifurcation diagrams are obtained. The results show that impulsive perturbation can easily give rise to complex dynamics, which includes: quasi-periodic oscillation, periodic doubling cascade, periodic halving cascade, attractor crisis and chaotic bands with periodic windows.     

Synchronization of four coupled van der Pol oscillators

It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill’s equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.     

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.     

Deterministic and stochastic analyses of chaotic and overturning responses of a slender rocking object

The relationship between chaos and overturning in the rocking response of a rigid object under periodic excitation is examined from both deterministic and stochastic points of view. A stochastie extension of the deterministic Melnikov function (employed to provide a lower bound for the possible chaotic domain in parameter space) is derived by taking into account the presence of random noise. The associated Fokker-Planck equation is derived to obtain the joint probability density functions in state space. It is shown that global behavior of the rocking motion can be effectively studied via the evolution of the joint probability density function. A mean Poincaré mapping technique is developed to average out noise effects on the chaotic response to reconstruct the embedded strange attractor on the Poincaré section. The close relationship between chaos and overturning is demonstrated by examining the structure of the invariant manifolds. It is found that the presence of noise enlarges the boundary of possible chaotic domains in parameter space and bridges the domains of attraction of coexisting responses. Numerical results consistent with the Foguel alternative theorem, which discerns asymptotic stabilities of responses, indicate that the overturning attracting domain is of the greatest strength. The presence of an embedded strange attractor (reconstructed using the mean Poincaré mapping technique) indicates the existence of transient chaotic rocking response.     

Primary resonance of traveling viscoelastic beam under internal resonance

Under the 3:1 internal resonance condition,the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied.The viscoelastic behaviors of the traveling beam are described by the standard linear solid model,and the material time derivative is adopted in the viscoelastic constitutive relation.The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes.For the first time,the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam.The undetermined coefficient method is used to approximately establish the real modal functions.The approximate analytical results are confirmed by the Galerkin truncation.Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses.To illustrate the effect of the internal resonance,the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.     

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.     

Solar sail chaotic pitch dynamics and its control in Earth orbits

The attitude dynamics and control for solar sail orbiting a celestial body (e.g., the Earth) are critical for the space missions. In the paper, the pitch dynamics is addressed by considering the torques by the center-of-mass and center-of-pressure offset, the gravity gradient, the internal damping and the control vane. The chaotic pitch motion is analytically detected for the sailcraft in the circular and elliptical orbits with small eccentricities using the Melnikov’s method. The validity of the Melnikov method is numerically verified by checking the Poincare surface of section and the power spectral density. The stability criterion method with some improvements is utilized to stabilize the chaotic pitch motion onto the reference unstable periodic motion embedded in the chaotic attractor. The reference unstable periodic motion is obtained based on the calculation of the close return pairs. The small control input torques and the stabilization effects are presented, and the advantages of the modified stabilization method are clarified based on the numerical simulations.     

Large-Amplitude Periodic Solutions for Differential Equations with Delayed Monotone Positive Feedback

The aim of this paper is to show that the structure of the global attractor for delayed monotone positive feedback can be more complicated than the union of spindle-like structures between consecutive stable equilibria with respect to the pointwise ordering. Large amplitude periodic orbits—in the sense that they are not between two consecutive stable equilibria—are constructed for nonlinearities close to a step function. For some nonlinearities there are exactly two large amplitude periodic orbits. By describing the unstable sets of these periodic orbits, a complete picture is obtained about the global attractor outside the spindle-like structures.     

Magnetorheological damping and semi-active control of an autoparametric vibration absorber

A numerical study of an application of magnetorheological (MR) damper for semi-active control is presented in this paper. The damper is mounted in the suspension of a Duffing oscillator with an attached pendulum. The MR damper with properties modelled by a hysteretic loop, is applied in order to control of the system response. Two methods for the dynamics control in the closed-loop algorithm based on the amplitude and velocity of the pendulum and the impulse on–off activation of MR damper are proposed. These concepts allow the system maintaining on a desirable attractor or, if necessary, to change a position from one attractor to another. Additionally, the detailed bifurcation analysis of the influence of MR damping on the number of periodic solutions and their stability is shown by continuation method. The influence of MR damping on the chaotic behavior is studied, as well.     

Non-feedback technique to directly control multistability in nonlinear oscillators by dual-frequency driving

A novel method to control multistability of nonlinear oscillators by applying dual-frequency driving is presented. The test model is the Keller–Miksis equation describing the oscillation of a bubble in a liquid. It is solved by an in-house initial-value problem solver capable to exploit the high computational resources of professional graphics cards. During the simulations, the control parameters are the two amplitudes of the acoustic driving at fixed, commensurate frequency pairs. The high-resolution bi-parametric scans in the control parameter plane show that a period-2 attractor can be continuously transformed into a period-3 one (and vice versa) by proper selection of the frequency combination and by proper tuning of the driving amplitudes. This phenomenon has opened a new way to drive the system to a desired, pre-selected attractor directly via a non-feedback control technique without the need of the annihilation of other attractors. Moreover, the residence in transient chaotic regimes can also be avoided. The results are supplemented with simulations obtained by the boundary-value problem solver AUTO, which is capable to compute periodic orbits directly regardless of their stability, and trace them as a function of a control parameter with the pseudo-arclength continuation technique.     

Numerical analysis of the oscillations of human vocal cords

Nonlinear differential equations of the fifth order, which govern the oscillations of the human vocal cords, are analyzed. A combined numerical technique is used to observe the regular and irregular solutions of these equations. A scenario from periodic, via quasiperiodic to chaotic attractor is demonstrated. A perturbation of the strange chaotic attractor to principal, combination and primary resonances is also discussed and illustrated.     

A novel strange attractor with a stretched loop

This short paper introduces a new 3D strange attractor topologically different from any other known chaotic attractors. The intentionally constructed model of three autonomous first-order differential equations derives from the coupling-induced complexity of the well-established 2D Lotka?CVolterra oscillator. Its chaotification process via an anti-equilibrium feedback allows the exploration of a new domain of dynamical behavior including chaotic patterns. To focus a rapid presentation, a fixed set of parameters is selected linked to the widest range of dynamics. Indeed, the new system leads to a chaotic attractor exhibiting a double scroll bridged by a loop. It mutates to a single scroll with a very stretched loop by the variation of one parameter. Indexes of stability of the equilibrium points corresponding to the two typical strange attractors are also investigated. To encompass the global behavior of the new low-dimensional dissipative dynamical model, diagrams of bifurcation displaying chaotic bubbles and windows of periodic oscillations are computed. Besides, the dominant exponent of the Lyapunov spectrum is positive reporting the chaotic nature of the system. Eventually, the novel chaotic model is suitable for digital signal encryption in the field of communication with a rich set of keys.     

Nonlinear Dynamics of Floating Cranes

The nonlinear dynamic responses of moored crane vessels to regular wavesare investigated experimentally and theoretically. The main subject ofinterest are nonlinear phenomena like bifurcations and the existence ofmultiple attractors. In the experimental part of the work, a mooredmodel of a crane vessel has been excited by regular waves in a wavetank. A special mechanism has been developed to model the nonlinearbehavior of real mooring systems. The theoretical part of the workconcerns the mathematical modeling of the floating cranes. Twomathematical models of different levels of complexity are presented. Twodifferent tools are used to systematically determine the responses ofthe systems to periodic forcing of waves. Firstly, the path-followingtechniques in combination with numerical integration of equations ofmotion applied to a full nonlinear model give insight into the dynamicsin time domain. Secondly, the multiple scales method allows for ananalytical investigation of simplified nonlinear models in frequencydomain. Many results of computations for two crane vessels, barge andship, are presented. Special attention is paid to oscillations near thefrequencies of primary resonances and to subharmonic motions. Anexcellent agreement is found between the results of time-domain andfrequency-domain analysis. The computational examples chosen correspondto the models used not only in the present experiments but in theexperiments of others as well. The results presented in the work allow usto draw several important conclusions concerning the dynamic behavior offloating cranes during offshore operations. Both the developed modelsand the analytical tools can be used to identify the limits of theoperating range of floating cranes.     

Non-linear dynamics of a flexible single link Cartesian manipulator

The present work deals with the non-linear vibration of a harmonically excited single link roller-supported flexible Cartesian manipulator with a payload. The governing equation of motion of this system is developed using extended Hamilton's principle, which is reduced to the second-order temporal differential equation of motion, by using generalized Galerkin's method. This equation of motion contains both cubic non-linearities of geometric and inertial type in addition to linear forced and non-linear parametric excitation terms. Method of multiple scales is used to solve this non-linear equation and study the stability and bifurcations of the system. Influence of amplitude of the base excitation and mass ratio on the steady state response of the system is investigated for both simple and subharmonic resonance conditions. Critical bifurcation points are determined from the fixed-point responses and periodic, quasi-periodic responses are also found for different system parameters. The results obtained using the perturbation analysis are compared with the previously published experimental work and are found to be in good agreement. This work will be useful for the designer of a flexible manipulator.