Three-dimensional structure of ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers

We have investigated the spatial and temporal structure of certain cases of chaotic and organized vortex shedding in self-excited cylinder wakes at low Reynolds number (40–200) by extending hot-wire measurements to much larger downstream locations than our earlier measurements (Van Atta and Gharib, 1987) and performing concurrent smoke-wire flow visualizations. We describe here results for two cases in which chaotic vortex shedding is observed. The first is the single-vibration frequency locked-in case, in which the most highly disturbed regions of the vortex street, which produce chaotic spectra, are confined to compact spanwise-periodic propagating disturbed regions. The second is the multiple-vibration frequency case in which chaotic velocity spectra are observed at all spanwise locations.     

Dynamics of a close-loop controlled MEMS resonator

The dynamics of a close-loop electrostatic MEMS resonator, proposed as a platform for ultra sensitive mass sensors, is investigated. The parameter space of the resonator actuation voltage is investigated to determine the optimal operating regions. Bifurcation diagrams of the resonator response are obtained at five different actuation voltage levels. The resonator exhibits bi-stability with two coexisting stable equilibrium points located inside a lower and an upper potential wells. Steady-state chaotic attractors develop inside each of the potential wells and around both wells. The optimal region in the parameter space for mass sensing purposes is determined. In that region, steady-state chaotic attractors develop and spend most of the time in the safe lower well while occasionally visiting the upper well. The robustness of the chaotic attractors in that region is demonstrated by studying their basins of attraction. Further, regions of large dynamic amplification are also identified in the parameter space. In these regions, the resonator can be used as an efficient long-stroke actuator.     

On the aeroelastic stability and bifurcation structure of subsonic nonlinear thin panels subjected to external excitation

Dynamic behavior of panels exposed to subsonic flow subjected to external excitation is investigated in this paper. The von Karman’s large deflection equations of motion for a flexible panel and Kelvin’s model of structural damping is considered to derive the governing equation. The panel under study is two-dimensional and simply supported. A Galerkin-type solution is introduced to derive the unsteady aerodynamic pressure from the linearized potential equation of uniform incompressible flow. The governing partial differential equation is transformed to a series of ordinary differential equations by using Galerkin method. The aeroelastic stability of the linear panel system is presented in a qualitative analysis and numerical study. The fourth-order Runge-Kutta numerical algorithm is used to conduct the numerical simulations to investigate the bifurcation structure of the nonlinear panel system and the distributions of chaotic regions are shown in the different parameter spaces. The results shows that the panel loses its stability by divergence not flutter in subsonic flow; the number of the fixed points and their stabilities change after the dynamic pressure exceeds the critical value; the chaotic regions and periodic regions appear alternately in parameter spaces; the single period motion trajectories change rhythmically in different periodic regions; the route from periodic motion to chaos is via doubling-period bifurcation.     

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.     

Fluid dynamically caused cycle-to-cycle variations in piston-driven pipe expansion flows

A piston-driven pipe flow with a sudden expansion in cross-sectional area was studied experimentally using flow visualization and laser-Doppler anemometry as basic experimental techniques. To drive the flow, a piston was moved in the larger of the two pipes. It drew in the fluid into the pipe assembly during the upward stroke and removed fluid during the downward stroke. The time-varying flow field was investigated, yielding detailed information on the flow structure, its development, and variation with time. In the present paper, particular attention is given to those flow conditions that resulted in cycle-to-cylce variations of the flow field. For some of the conditions, repetitive cycle-to-cycle variations of the flow field occurred, yielding identical flows for even-numbered flow cycles and also for odd numbered cycles. However, the latter differed from the former. The presented data show that non-repetitive cycle-to-cycle variations of the flow field also exist and that these can be driven into a state where the flow appears to be nearly chaotic. The results of the investigation clearly demonstrate that fluid dynamically caused cycle-to-cycle flow variations can exist in piston-driven internal flows with separation.     

Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation

In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with single- and double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under two-frequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.

     

关于Mises桁架结构的非线性动力学研究初探

本文通过计算机仿真，观察和研究了Ｍｉｓｅｓ桁架的动力不稳定及其混沌特性．结果表明，对于调和外激励的某一频率和幅值，存在着周期或非周期的特性；而当动力外激励为阶跃函数时，不引起混沌响应．     

A new chaotic attractor and its robust function projective synchronization

We introduce a simple chaotic system that contains one multiplier and one quadratic term. The system is similar to the generalized Lorenz system but is not topologically equivalent. The properties of the proposed chaotic system are examined by theoretical and numerical analysis. An analog chaotic circuit is implemented that realizes the chaotic system for the verification of its attractor. Furthermore, we propose a robust function projective synchronization using time delay estimation. A numerical simulation of synchronization between the proposed system and the Lorenz system demonstrates that the proposed approach provides fast and robust synchronization even in the presence of unknown parameter variations and disturbances.     

Sensitivity vector fields for atomic force microscopes

The application of the sensitivity vector fields (SVFs) for tip–sample interactions in the context of multi-mode tapping dynamics of atomic-force microscopes (AFM) is presented. SVFs represent a novel approach to accurately determine simultaneous parameter variations by exploiting the geometric features of observed chaotic dynamics. The paper also demonstrates that, in certain operating conditions, higher modes are essential to correctly predict the AFM dynamics, and they cannot be neglected. The accuracy of the SVF approach is discussed as applied to a multi-mode AFM model where mode shapes vary due to multiple parameter variations. The identifiability of various parameters based on SVF reconstruction is investigated. Several calibration issues for parameter reconstruction are observed and resolved by a specialized sample filtering and a novel correction factor.     

Adaptive control of a chaotic permanent magnet synchronous motor

This paper proposes a simple adaptive controller design method for a chaotic permanent magnet synchronous motor (PMSM) based on the sliding mode control theory which has given an effective means to design robust controllers for nonlinear systems with bounded uncertainties. The proposed sliding mode adaptive controller does not require any information on the PMSM parameter and load torque values, thus it is insensitive to model parameter and load torque variations. Simulation results are given to verify that the proposed method can be successfully used to control a chaotic PMSM under model parameter and load torque variations.     

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models

In the present work hysteresis is simulated by means of internal variables. The analytical models of different types of hysteresis loops allow the reproduction of major and minor loops and provide a high degree of correspondence with experimental data. In models of this type adding an external periodic excitation or increasing the number of dimensions can lead to the occurrence of chaotic behaviour. Using an effective algorithm based on numerical analysis of the wandering trajectories [1–7], the evolution of the chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. The substantial influence of a hysteretic dissipation value on the form and location of these regions, as well as the restraining and generating effects of hysteretic dissipation on the occurrence of chaos, are ascertained. Conditions for pinched hysteresis are defined. Furthermore, autonomous coupled hysteretic oscillators under sliding friction are investigated. Conditions for the occurrence of chaotic behaviour in a two-degree-of-freedom (two-DOF) hysteretic system are found in the plane of maximal static friction forces of both oscillators versus belt velocity.     

Modeling Human Chaotic Behavior: Nonlinear Forecasting Analysis of Logistic Iteration

Does the unpredictability of human behavior arise from randomness, from deterministic but chaotic processes, or from humans' use of (possibly nonlinear deterministic) heuristics in coping with complicated situations? One way to find out might be to see whether humans can behave chaotically when asked to do so. Previous work showed that when humans are asked to generate a series of numbers according to a particular chaotic equation they can do so but not in exactly the way the equation would generate them. Nonetheless, their series of guesses do contain nonlinear deterministic structure, which is one indication that they may be generated by a chaotic process. Series of guesses generated by a computer simulation of a model that simulates the heuristic thought processes of the humans making the guesses also contain nonlinear deterministic structure of the same order as the logistic the humans are attempting to mimic. Thus, when faced with a chaotic process, humans seem to cope by using a heuristic process that approximates the chaotic process within the limitations of human memory and performance.     

万有引力场中陀螺体的混沌运动

研究万有引力场中沿圆轨道运行的非对称陀螺体的姿态运动,引入Ｄｅｐｒｉｔ正则变量建立系统的Ｈａｍｉｌｔｏｎ结构,利用Ｍｅｌｎｉｋｏｖ方法证明在万有引力短作用的昆体产生混沌运动的可能性。对Ｐｏｉｎｃａｒｅ截面的数值计算表明提高陀螺体的转子转速可对混沌起抑制作用。     

Adaptive lag synchronization in coupled chaotic systems with unidirectional delay feedback

Lag synchronization in unidirectionally coupled chaotic systems is investigated in this paper. Based on the invariance principle of differential equations, a new adaptive delay feedback scheme is proposed to realize the lag synchronization effectively in the coupled chaotic systems. As an example, numerical simulations for the coupled Hindmarsh-Rose (HR) neuron models are conducted, which is in good agreement with the theoretical analysis. More interestingly, it is found that there is a fine U-shaped structure in the lag synchronization curve for the HR neuron model. Furthermore, lag synchronization and the corresponding U-shaped structure are robust against the small mismatch of parameters and noisy disturbances.     

Synchronization between uncertain chaotic systems with a diverse structure based on a second-order sliding mode control

Synchronization between uncertain chaotic systems with a diverse structure is investigated using a second-order sliding mode control. Sliding surface, adaptive laws of the unknown parameters, and the sliding mode controller are designed based on stability theory. The Van der Pol system with chaotic behavior in physics is taken as a target system; the Duffing system with unknown parameters is taken as a response system. The artificial simulation results show that this method is still effective.     

Nonlinear stochastic analysis of subharmonic response of a shallow cable

The paper deals with the subharmonic response of a shallow cable due to time variations of the chord length of the equilibrium suspension, caused by time varying support point motions. Initially, the capability of a simple nonlinear two-degree-of-freedom model for the prediction of chaotic and stochastic subharmonic response is demonstrated upon comparison with a more involved model based on a spatial finite difference discretization of the full nonlinear partial differential equations of the cable. Since the stochastic response quantities are obtained by Monte Carlo simulation, which is extremely time-consuming for the finite difference model, most of the results are next based on the reduced model. Under harmonical varying support point motions the stable subharmonic motion consists of a harmonically varying component in the equilibrium plane and a large subharmonic out-of-plane component, producing a trajectory at the mid-point of shape as an infinity sign. However, when the harmonical variation of the chordwise elongation is replaced by a narrow-banded Gaussian excitation with the same standard deviation and a centre frequency equal to the circular frequency of the harmonic excitation, the slowly varying phase of the excitation implies that the phase difference between the in-plane and out-of-plane displacement components is not locked at a fixed value. In turn this implies that the trajectory of the displacement components is slowly rotating around the chord line. Hence, a large subharmonic response component is also present in the static equilibrium plane. Further, the time variation of the envelope process of the narrow-banded chordwise elongation process tends to enhance chaotic behaviour of the subharmonic response, which is detectable via extreme sensitivity on the initial conditions, or via the sign of a numerical calculated Lyapunov exponent. These effects have been further investigated based on periodic varying chord elongations with the same frequency and standard deviation as the harmonic excitation, for which the amplitude varies in a well-defined way between two levels within each period. Depending on the relative magnitude of the high and low amplitude phase and their relative duration the onset of chaotic vibrations has been verified.     

Chaos in the three-well potential system

A new approach based on analysis of the wandering trajectories is applied to investigate an appearance of chaotic vibrations in many-well potential systems. The chaotic behavior regions were found in the both amplitude–frequency of excitation and amplitude–damping coefficient plane. The phase plane of initial conditions has been investigated taking into account different values of an external periodic excitation. It demonstrated remarkable agreement with investigations based on homoclinic and heteroclinic bifurcation criteria for chaos, computations of Lyapunov exponents and fractal basin boundaries. The presented technique is very effective, convenient to use, and can be applied to the investigation of a wide class of problems.     

Designing chaotic S-boxes based on time-delay chaotic system

S-box structures used in the encryption architecture are of great importance for constructing powerful block encryption systems, which hold an important place in modern cryptology. The design of S-boxes with sound cryptographic characteristics is of utmost importance for constructing powerful encryption systems. In this study, an S-box design algorithm based on time-delay chaotic systems is proposed. The proposed algorithm is considered relative to other algorithms in the literature as more useful according to such criteria as simplicity and efficient implementation. Theoretical analysis and computer simulations demonstrated that the proposed algorithm meets all the performance requirements for the S-box design criteria, and also verified the efficient and practical structure of the algorithm.     

Curvature and torsion of material lines in chaotic flows

The deformation of a material line as it evolves in a chaotic flow is considered. Of particular interest are the curvature and torsion as a function of arclength or material coordinate along the line and their near singular behavior that develops as the deformation proceeds. Recent work [Thiffeault 2004. Physica D 198, 169–181] has established an interesting connection between local line stretch and curvature for regions of high curvature. The present effort confirms this connection and develops a similar relation between torsion and curvature for regions of high absolute torsion. In addition it is shown that certain properties of the strain tensor play a key role in the location and structure of the near singular behavior in curvature.