An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response

In this study we deal with a microelectromechanical system (MEMS) and develop a dynamical integrity analysis to interpret and predict the experimental response. The device consists of a clamped-clamped polysilicon microbeam, which is electrostatically and electrodynamically actuated. It has non-negligible imperfections, which are a typical consequence of the microfabrication process. A single-mode reduced-order model is derived and extensive numerical simulations are performed in a neighborhood of the first symmetric natural frequency, via frequency response diagrams and behavior chart. The typical softening behavior is observed and the overall scenario is explored, when both the frequency and the electrodynamic voltage are varied. We show that simulations based on direct numerical integration of the equation of motion in time yield satisfactory agreement with the experimental data. Nevertheless, these theoretical predictions are not completely fulfilled in some aspects. In particular, the range of existence of each attractor is smaller in practice than in the simulations. This is because these theoretical curves represent the ideal limit case where disturbances are absent, which never occurs under realistic conditions. A reliable prediction of the actual (and not only theoretical) range of existence of each attractor is essential in applications. To overcome this discrepancy and extend the results to the practical case where disturbances exist, a dynamical integrity analysis is developed. After introducing dynamical integrity concepts, integrity profiles and integrity charts are drawn. They are able to describe if each attractor is robust enough to tolerate the disturbances. Moreover, they detect the parameter range where each branch can be reliably observed in practice and where, instead, becomes vulnerable, i.e. they provide valuable information to operate the device in safe conditions according to the desired outcome and depending on the expected disturbances.     

Internal resonance in the higher-order modes of a MEMS beam: experiments and global analysis

This work investigates the dynamics of a microbeam-based MEMS device in the neighborhood of a 2:1 internal resonance between the third and fifth vibration modes. The saturation of the third mode and the concurrent activation of the fifth are observed. The main features are analyzed extensively, both experimentally and theoretically. We experimentally observe that the complexity induced by the 2:1 internal resonance covers a wide driving frequency range. Constantly comparing with the experimental data, the response is examined from a global perspective, by analyzing the attractor-basins scenario. This analysis is conducted both in the third-mode and in fifth-mode planes. We show several metamorphoses occurring as proceeding from the principal resonance to the 2:1 internal resonance, up to the final disappearance of the resonant and non-resonant attractors. The shape and wideness of all the basins are examined. Although they are progressively eroded, an appreciable region is detected where the compact cores of the attractors involved in the 2:1 internal resonance remain substantial, which allows effectively operating them under realistic conditions. The dynamical integrity of each resonant branch is discussed, especially as approaching the bifurcation points where the system becomes more vulnerable to the dynamic pull-in instability.

     

Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy

The nonlinear dynamical behavior of a single-mode model of noncontact AFM is analyzed in terms of attractors robustness and basins integrity. The model considered for the analyses, proposed in (Hornstein and Gottlieb in Nonlinear Dyn. 54:93–122, 2008), consistently includes the nonlinear atomic interaction and is studied under scan excitation (which appears as parametric excitation) and vertical excitation (which is prevalently external). Local bifurcation analyses are carried out to identify the overall stability boundary in the excitation parameter space as the envelope of system local escapes, to be compared with the one obtained via numerical simulations. The dynamical integrity of periodic bounded solutions is studied, and basin erosion is evaluated by means of two different integrity measures. The obtained erosion profiles allow us to dwell on the possible lack of homogeneous safety of the stability boundary in terms of robustness of the attractors, and to identify practical escape thresholds ensuring an a priori design safety target.     

Chaos and Hopf bifurcation of a finance system

The complex dynamical behavior of a finance system is investigated in this paper. The Ruelle–Takens route to chaos and strange nonchaotic attractors (SNA) are found through numerical simulations. Then the system with time-delayed feedback is considered and the stability and Hopf bifurcation of the controlled system are investigated. This research has important theoretical and practical meanings.     

Dynamic stability of electrostatically actuated initially curved shallow micro beams

Micro and nano devices incorporating bistable structural elements have functional advantages including the existence of several stable configurations at the same actuation force, extended working range, and tunable resonant frequencies. In this work, after a short review of operational principles of bistable micro devices, results of a theoretical and numerical investigation of the transient dynamics of an initially curved, shallow, double-clamped micro beam, actuated by distributed electrostatic and inertial forces are presented. Due to the unique combination of mechanical and electrostatic nonlinearities, typically not encountered in large scale structures, the device exhibits sequential snap-through and electrostatic (pull-in) instabilities. A phase plane analysis, performed using a consistently derived lumped model along with the numerical results, indicate that critical voltages corresponding to the dynamic snap-through and pull-in instabilities are lower than their static counterparts, while the minimal curvature required for the appearance of the dynamic snap-through is higher than in the static case. The boundaries of the bistability region of a quasi-statically loaded beam are found in terms of the geometrical and loading parameters and are shown to be bounded from above by the dynamic pull-in instability. Some of the post-buckling states cannot be reached under suddenly applied or quasi-statically increasing voltages: specially tailored loading schemes are suggested for realization of these configurations often beneficial in applications.     

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.     

粘弹性圆薄板的动力学行为

基于线性粘弹性力学的Boltzmann叠加原理,给出粘弹性圆薄板动力学分析的初边值问题。通过一定的简化后得到描述薄板力学行为的四维非线性非自治动力系统。综合使用非线性动力学中的数值分析方法,研究了参数对粘弹性圆薄板动力学行为的影响。同时计算了吸引子的Lyapunov维、相关维和点形维。     

Dynamical analysis of the generalized Sprott C system with only two stable equilibria

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.     

FPGA realizations of high-speed switching-type chaotic oscillators using compact VHDL codes

In this paper, a two-stage oligopoly game of semi-collusion in production is analyzed and expounded, where at first stage all firms compete in R&D and at second stage all firms coordinate the production activities in order to make their joint profit maximized. Not only the local stability of equilibriums, but also the existence, stability and direction of flip bifurcation of the discrete nonlinear model are investigated by using the normal form method and the center manifold theory. Then, the validity of the theoretical analysis is justified through numerical simulation. We find that the model we built can exhibit very complex dynamical behaviors, but it cannot undergo the Neimark–Sacker bifurcation. Also the coexistence of attractors is found through numerical simulation, and their basins of attraction are simulated. At last of this paper, the chaotic motion of the proposed model is controlled by delayed feedback control method.     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

Partially unstable attractors in networks of forced integrate-and-fire oscillators

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.     

Finding coexisting attractors using amplitude control

Nonlinear dynamical systems often have multiple stable states and thus can harbor coexisting and hidden attractors that may pose an inconvenience or even hazard in practical applications. Amplitude control provides one method to detect these coexisting attractors, and it explains the unpredictable and irreproducible behavior that sometimes occurs in carefully engineered systems. In this paper, two regimes of amplitude control are described to illustrate the method for detecting multistability and possible coexisting or hidden attractors.     

Expected escape times from attractor basins due to low intensity noise

In this paper, a numerical approach is described to estimate escape times from attractor basins when a dynamical system is subjected to noise or stochastic perturbations. Noise can affect nonlinear system response by driving solution trajectories to different attractors. The changes in physical behavior can be observed as amplitude and phase change of periodic oscillations, initiation or annihilation of chaotic motion, phase synchronization, and so on. Estimating probability of transitions from one attractor to another, and predicting escape times are essential for quantifying the effects of noise on the system response. In this paper, a numerical approach is outlined where probability transition maps are generated between grids. Then, these maps are iterated to find the probability distribution after long durations, wherein, a constant escape rate can be observed between basins. The constant escape rate is then used to estimate the average escape times. The approach is applicable to systems subjected to low-intensity stochastic disturbances and with long escape times, where Monte Carlo simulations are impractical. Escape times up to \(10^{13}\) periods are estimated without relying on computationally expensive computations.

     

N-scroll chaotic attractors from saturated function series employing CCII+s

The generation of n-scroll chaotic attractors by using saturated nonlinear function series (SNFS) realized with positive-type second generation current conveyors (CCII+s), is introduced. The nonlinear dynamical system is expressed by a third-order differential equation and to carry out numerical simulations, SNFS are ideally modeled by using staircase functions. Therefore, numerical simulations are introduced to approximate the swings, widths, breakpoints and equilibrium points of the n-scroll attractors by considering, as input variables: the dynamic range associated to active devices, gain of the nonlinear system and the number of scrolls. Therefore, its dynamical behavior is investigated in the state space. Besides, the CCII± is a versatile analog building block and it has been demonstrated to be very useful in several linear and nonlinear applications, since CCII-based implementations offer better performances that Opamps-based implementations in terms of accuracy and bandwidth. Therefore, the nonlinear system is synthesized with CCII+s to generate 3- and 4-scrolls. HSPICE simulations and experimental results are shown to verify the agreement on the behavior of the proposed circuit and the numerical simulations.     

Load carrying capacity of systems within a global safety perspective. Part II. Attractor/basin integrity under dynamic excitations

The effects of the dynamic excitation on the load carrying capacity of mechanical systems are investigated with reference to the archetypal model addressed in Part I, which permits to highlight the main ideas without spurious mechanical complexities. First, the effects of the excitation on periodic solutions are analyzed, focusing on bifurcations entailing their disappearance and playing the role of Koiter critical thresholds. Then, attractor robustness (i.e., large magnitude of the safe basin) is shown to be necessary but not sufficient to have global safety under dynamic excitation. In fact, the excitation strongly modifies the topology of the safe basins, and a dynamical integrity perspective accounting for the magnitude of the solely compact part of the safe basin must be considered. By means of extensive numerical simulations, robustness/erosion profiles of dynamic solutions/basins for varying axial load and dynamic amplitude are built, respectively. These curves permit to appreciate the practical reduction of system load carrying capacity and, upon choosing the value of residual integrity admissible for engineering design, the Thompson practical stability. Dwelling on the effects of the interaction between axial load and lateral dynamic excitation, this paper supports and, indeed, extends the conclusions of the companion one, highlighting the fundamental role played by global dynamics as regards a reliable estimation of the actual load carrying capacity of mechanical systems.     

Pull-in instability of a typical electrostatic MEMS resonator and its control by delayed feedback

Pull-in instability of the electrostatic microstructures is a common undesirable phenomenon which implies the loss of reliability of micro-electromechanical systems. Therefore, it is necessary to understand its mechanism and then reduce the phenomenon. In this work, pull-in instability of a typical electrostatic MEMS resonator is discussed in detail. Delayed position feedback and delayed velocity feedback are introduced to suppress pull-in instability, respectively. The thresholds of AC voltage for pull-in instability in the initial system and the controlled systems are obtained analytically by the Melnikov method. The theoretical predictions are in good agreement with the numerical results. It follows that pull-in instability of the MEMS resonator can be ascribed to the homoclinic bifurcation inducing by the AC and DC load. Furthermore, it is found that the controllers are both good strategies to reduce pull-in instability when their gains are positive. The delayed position feedback controller can work well only when the delay is very short and AC voltage is low, while the delayed velocity feedback will be effective under a much higher AC voltage and a wider delay range.     

The structure of symmetric attractors

We consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems. We show, for example, that the symmetry of an attractor cannot, in general, be an arbitrary subgroup of the group of symmetries. In addition, there are group-theoretic restrictions on the symmetry of connected components of a symmetric attractor. The symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time-average of a chaotic dynamical system.Our methods are topological in nature and exploit connectedness properties of the ambient space. In particular, we prove a general lemma about connected components of the complement of preimage sets and how they are permuted by the mapping.These methods do not themselves depend on equivariance. For example, we use them to prove that the presence of periodic points in the dynamics limits the number of connected components of an attractor, and, for one-dimensional mappings, to prove results on sensitive dependence and the density of periodic points.     

A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations.     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.