Chaos prediction and control in MEMS resonators

The chaotic dynamics of a micro mechanical resonator with electrostatic forces on both sides is investigated. Using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters. Detailed numerical studies including phase portrait, Poincare map and bifurcation diagram confirm the analytical prediction and reveal the effect of excitation amplitude on the system transition to chaos. Moreover a robust adaptive fuzzy control algorithm previously proposed by the authors is applied for controlling the chaotic motion. Additional numerical simulations show the effectiveness of the proposed control approach.     

Chaos of several typical asymmetric systems

The threshold for the onset of chaos in asymmetric nonlinear dynamic systems can be determined using an extended Padé method. In this paper, a double-well asymmetric potential system with damping under external periodic excitation is investigated, as well as an asymmetric triple-well potential system under external and parametric excitation. The integrals of Melnikov functions are established to demonstrate that the motion is chaotic. Threshold values are acquired when homoclinic and heteroclinic bifurcations occur. The results of analytical and numerical integration are compared to verify the effectiveness and feasibility of the analytical method.     

Chaos control for the plates subjected to subsonic flow

The suppression of chaotic motion in viscoelastic plates driven by external subsonic air flow is studied. Nonlinear oscillation of the plate is modeled by the von-Kármán plate theory. The fluid-solid interaction is taken into account. Galerkin’s approach is employed to transform the partial differential equations of the system into the time domain. The corresponding homoclinic orbits of the unperturbed Hamiltonian system are obtained. In order to study the chaotic behavior of the plate, Melnikov’s integral is analytically applied and the threshold of the excitation amplitude and frequency for the occurrence of chaos is presented. It is found that adding a parametric perturbation to the system in terms of an excitation with the same frequency of the external force can lead to eliminate chaos. Variations of the Lyapunov exponent and bifurcation diagrams are provided to analyze the chaotic and periodic responses. Two perturbation-based control strategies are proposed. In the first scenario, the amplitude of control forces reads a constant value that should be precisely determined. In the second strategy, this amplitude can be proportional to the deflection of the plate. The performance of each controller is investigated and it is found that the second scenario would be more efficient.     

Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback

The chaotic behavior of a double-well Duffing oscillator with both delayed displacement and velocity feedbacks under a harmonic excitation is investigated. By means of the Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. The analytical results reveal that for negative feedback the presence of time delay lowers the threshold and enlarges the possible chaotic domain in parameter space; while for positive feedback the presence of time delay enhances the threshold and reduces the possible chaotic domain in parameter space, which are further verified numerically through Poincare maps of the original system. Furthermore, the effect of the control gain parameters on the chaotic motion of the original system is studied in detail.     

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

The chaotic behaviors of a particle in a triple well ϕ⁶ potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.     

Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing

We examine the Melnikov criterion for a global homoclinic bifurcation and a possible transition to chaos in case of a single degree of freedom nonlinear oscillator with a symmetric double well nonlinear potential. The system was subjected simultaneously to parametric periodic forcing and self-excitation via negative damping term. Detailed numerical studies confirm the analytical predictions and show that transitions from regular to chaotic types of motion are often associated with increasing the energy of an oscillator and its escape from a single well.     

Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity

In this paper, we consider the dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. The amplitude equations are derived in the general case using the harmonic balance method. The model without discontinuity is first considered. The effects of the amplitude of the parametric modulation and some particular coefficients are found in the response curves. The transition to chaotic behavior is found using numerical simulations of the equations of motion. We find that chaos appears in the model between the quasi-periodic and periodic orbits when the amplitude of the external excitation E₀ vary. An adaptive Lyapunov control strategy enables us to drive the system from the chaotic states to a targeting periodic orbit. The effects of elasticity and damping on the dynamics of the self-sustained electromechanical system are also derived.     

Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation

A harmonic function with constant amplitude and random frequency and phase is called bounded noise. In this paper, the effect of bounded noise on the chaotic behavior of the Duffing oscillator under parametric excitation is studied in detail. The random Melnikov process is derived and a mean-square criterion is used to detect the chaotic dynamics in the system. It is found that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the noise in frequency increases. The threshold of bounded noise amplitude for the onset of chaos is also determined by the numerical calculation of the largest Lyapunov exponents. The effect of bounded noise on the Poincaré map and power spectra is also investigated. The numerical results qualitatively confirm the conclusion drawn by using the random Melnikov process with mean-square criterion for larger noise intensity.     

Chaotic vibration of a quarter-car model excited by the road surface profile

The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case of a quarter car model excited kinematically by the road surface profile. By analyzing the potential an analytic expression is found for the homoclinic orbit. By introducing an harmonic excitation term and damping as perturbations, the critical Melnikov amplitude of the road surface profile is found, above which the system can vibrate chaotically.     

Homoclinic bifurcations and chaotic dynamics of non-planar waves in axially moving beam subjected to thermal load

The homoclinic bifurcations and nonplanar chaotic waves in axially moving beam (AMB) under thermal excitation are investigated. By the multiple scale technique, the equivalent nonlinear system is derived to explore qualitatively the dynamical characteristics of AMB system for the case of primary resonance. Using Melnikov approach as well as geometric analysis, the criterion for homoclinic chaos and complex nonplanar motions for AMB system is discussed. The theoretical predictions are tested by the numerical approach. For the design and application of the AMB, some inspiration and guidance are provided by the results from theory and simulation.     

Chaotic response of a quarter car model forced by a road profile with a stochastic component

The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case of a quarter car model excited kinematically by a road surface profile consisting of harmonic and noisy components. By analyzing the potential an analytic expression is found for the homoclinic orbit. The road profile excitation including harmonic and random characteristics as well as the damping are treated as perturbations of a Hamiltonian system. The critical Melnikov amplitude of the road surface profile is found, above which the system can vibrate chaotically. This transition is analyzed for different levels of noise and illustrated by numerical simulations.     

Bifurcation and Chaos in a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation

In this paper the dynamics of a weakly nonlinear system subjected to combined parametric and external excitation are discussed. The existence of transversal homoclinic orbits resulting in chaotic dynamics and bifurcation are established by using the averaging method and Melnikov method. Numerical simulations are also provided to demonstrate the theoretical analysis.     

Chaos in a new fractional-order system without equilibrium points

Chaotic systems without equilibrium points represent an almost unexplored field of research, since they can have neither homoclinic nor heteroclinic orbits and the Shilnikov method cannot be used to demonstrate the presence of chaos. In this paper a new fractional-order chaotic system with no equilibrium points is presented. The proposed system can be considered “elegant” in the sense given by Sprott, since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. When the system order is as low as 2.94, the dynamic behavior is analyzed using the predictor–corrector algorithm and the presence of chaos in the absence of equilibria is validated by applying three different methods. Finally, an example of observer-based synchronization applied to the proposed chaotic fractional-order system is illustrated.     

The transversal homoclinic solutions and chaos for stochastic ordinary differential equations

We consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a Brownian force. By Lyapunov–Schmidt reduction, the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces. It is shown that, for almost all sample paths of the Brownian motion, the perturbed system exhibits chaos.     

Parametric Perturbations and Dynamic System Control

The article analyzes dynamical systems with externally applied periodic perturbations in a general setting. We provide a rigorous justification of an approach that reduces such systems to autonomous systems and thus simplifies the analysis. The behavior of families of quadratic one-dimensional maps and circle maps in the presence of parametric perturbations is studied in detail. We prove the existence of periodic perturbations acting strictly on a chaotic subset that stabilize the dynamics and induce the emergence of stable cycles in initially chaotic maps. The analytical results are supplemented with numerical data. It is shown that chaos may be suppressed by a sufficiently complex periodic perturbation.     

非线性振动系统的异宿轨道分叉,次谐分叉和混沌

在参数激励与强迫激励联合作用下具有van der Pol阻尼的非线性振动系统,其动态行为是非常复杂的.本文利用Melnikov方法研究了这类系统的异宿轨道分叉、次谐分叉和混沌.对于各种不同的共振情况,系统将经过无限次奇阶次谐分叉产生Smale马蹄而进入混沌状态.最后我们利用数值计算方法研究了这类系统的混沌运动.所得结果揭示了一些新的现象.     

Modelling of hysteresis using Masing–Bouc-Wen’s framework and search of conditions for the chaotic responses

In the present work hysteresis is simulated by means of internal variables. It was shown that Masing’s imitating mechanism of the energy dissipation presented in the differential equations of Bouc-Wen’s structure allows to simulate hysteresis from very different fields. The constructed analytical models of different types of hysteresis loops are simple, enable major and minor loops reproducing and provide a high degree of correspondence with experimental data. The models of such structure are convenient for the further investigation. Hysteretic systems under harmonic excitation described by models of such structure may reveal chaotic behaviour. Using an effective algorithm based on analysis of the wandering trajectories [1], [2], [3], [4], [22], [23], an evolution of chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. Substantial influence of a hysteretic dissipation value on the form and location of these regions, and also restraining and generating effects of the hysteretic dissipation on a chaos occurrence are ascertained. Conditions for pinched hysteresis are defined.     

Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays

Time delays are often sources of complex behavior in dynamic systems. Yet its complexity needs to be further explored, particularly when multiple time delays are present. As a purpose to gain insight into such complexity under multiple time delays, we investigate the mechanism for the action of multiple time delays on a particular non-autonomous system in this paper. The original mathematical model under consideration is a Duffing oscillator with harmonic excitation. A delayed system is obtained by adding delayed feedbacks to the original system. Two time delays are involved in such system, one of which in the displacement feedback and the other in the velocity feedback. The time delays are taken as adjustable parameters to study their effects on the dynamics of the system. Firstly, the stability of the trivial equilibrium of the linearized system is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or double Hopf bifurcation may occur. Then, the chaotic behavior of such system is investigated in detail. Particular emphasis is laid on the effect of delay difference between two time delays on the chaotic properties. A Melnikov’s analysis is employed to obtain the necessary condition for onset of chaos resulting from homoclinic bifurcation. And numerical analyses via the bifurcation diagram and the top Lyapunov exponent are carried out to show the actual time delay effect. Both the results obtained by the two analyses show that the delay difference between two time delays plays a very important role in inducing or suppressing chaos, so that it can be taken as a simple but efficient “switch” to control the motion of a system: either from order to chaos or from chaos to order.     

Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a ϕ6 potential

This paper investigates the chaotic behavior of an extended Duffing Van der pol oscillator in a ϕ⁶ potential under additive harmonic and bounded noise excitations for a specific parameter choice. From Melnikov theorem, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation in the case of the ϕ⁶ potential is bounded, which are complemented by the numerical simulations from which we illustrate the bifurcation surfaces and the fractality of the basins of attraction. The results show that the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further validated by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps is also investigated.     

1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下,利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式,利用广义Melnikov方法研究该系统的多脉冲混沌动力学,并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下,提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下,矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．