Complicated Nonlinear Phenomena Research Articles

Asymptotic behavior of Riemann solutions to a hyperbolic system arising from a traffic flow model

This paper is mainly concerned with the Riemann problem for a hyperbolic system arising from a traffic flow model, whose Riemann solutions are obtained constructively in terms of the fine analysis of elementary waves. Especially, the Dirac delta shock wave is involved in the Riemann solution at some specially designated initial data. Then, the limiting behavior of Riemann solutions is investigated carefully from the current system to the pressureless Euler system by sending the perturbed parameter tend to zero, in which more complicated nonlinear phenomena of cavitation and concentration than before can be observed and analyzed during this limiting process. Finally, several representative numerical examples are offered to confirm our theoretical results.

Complicated nonlinear oscillations caused by maneuvering of a flexible spacecraft equipped with hinged solar arrays

We present a general methodology for the nonlinear dynamical modeling of a three-axis stabilized spacecraft equipped with flexible solar arrays. The large-span multi-panel solar arrays are modeled as flexible thin plates that are connected to the rigid central body of the spacecraft by means of nonlinear flexible hinges. We construct a low-dimensional yet accurate dynamical model by using a Galerkin expansion in terms of global modes of the system that we compute first by using the Rayleigh–Ritz method. The hinged connections between the rigid and flexible parts of the system are imposed employing Lagrange multipliers. The model is used to study the spacecraft response triggered by various maneuvering scenarios. We in particular focus on the coupling between vibrations of the flexible components and the rigid motion of the spacecraft induced by hinge nonlinearities during these orbital and attitude maneuvering operations. In all cases considered, it is found that four global modes are sufficient to accurately compute the system’s response. We also observe other complicated nonlinear dynamical phenomena, such as hysteresis and superharmonic resonance that may be of concern in spacecraft design. Our modeling approach can be applied to other multibody systems directly.

Novel Approximate Analytical Solutions to the Nonplanar Modified Kawahara Equation and Modeling Nonlinear Structures in Electronegative Plasmas

In this investigation, the nonplanar (spherical and cylindrical) modified fifth-order Korteweg–de Vries (nmKdV5) equation, otherwise known as the nonplanar modified Kawahara equation (nmKE), is solved using the ansatz approach. Two general formulas for the semi-analytical symmetric approximations are derived using the recommended methodology. Using the obtained approximations, the nonplanar modified Kawahara (mK) symmetric solitary waves (SWs) and cnoidal waves (CWs) are obtained. The fluid equations for the electronegative plasmas are reduced to the nmKE as a practical application for the obtained solutions. Using the obtained solutions, the characteristic features of both the cylindrical and spherical mK-SWs and -CWs are studied. All obtained solutions are compared with each other, and the maximum residual errors for these approximations are estimated. Numerous researchers that are interested in studying the complicated nonlinear phenomena in plasma physics can use the obtained approximations to interpret their experimental and observational findings.

An alternate state-space algorithm for dynamic solution, sensitivity analysis and parameter identification of dry friction systems

Dry friction is one of the most complicated nonlinear phenomena in mechanical systems, due to its discontinuous stick–slip property. This property certainly brings difficulty to the solution and identification of dry friction systems. To overcome the difficulty, an alternate state-space algorithm is proposed in this paper to transform the dry friction system into a piecewise linear system. In doing so, the stick–slip process is firstly modeled by a zero–one representation, with which the stick–slip transition conditions are deduced. Then, the transition points are detected by the transition conditions to break the system into several pure stick or slip phases; each phase corresponds to a linear system. Dynamic solution of the piecewise linear system is quickly acquired by precise time step integration over an arbitrary pure stick or slip phase and the sensitivity analysis, pertaining also to a piecewise linear equation, can proceed exactly in a similar way. As is noteworthy, the sensitivity may admit jumps at the transition points due to the discontinuity of the dry friction system and therefore, the affine relation between the sensitivity before and after a transition point is elaborately derived. Eventually, based on the acquired sensitivity and the Tikhonov regularization, a sensitivity approach is developed to inversely identify the parameters of the dry friction system. Numerical examples and an experimental case are studied to demonstrate the accuracy and efficiency of the proposed alternate state-space algorithm.

Nonplanar free and forced vibrations of an imperfect nanobeam employing nonlocal strain gradient theory

In this paper, the free and forced vibrations of a three-dimensional nonplanar nanobeam with initial geometric imperfection are investigated. The Kelvin–Voigt model and nonlocal strain gradient theory (NSGT) are applied to the nanobeam with viscoelastic structural damping and size-dependent effect. A mechanical model of the nanobeam is established by applying Hamilton’s principle, and an initial displacement is used to describe the initial geometric imperfection. The differential quadrature method (DQM) is used to discretize the complex partial differential equation and simulate the mid-span vibration amplitude of the nonplanar nanobeam. Comparisons between the proposed model and published results show good agreement and further demonstrate the validity of the model. Numerical simulations are performed to illustrate how the geometric imperfection, the viscoelasticity, the nonlocal parameter, the strain gradient parameter and lateral motion influence the buckling phenomena, natural frequencies and vibrational responses. It is found that the nonlocal effect can reduce the natural frequencies and enhance the complicated nonlinear phenomena. However, the contrary phenomena are induced by the strain gradient effect. The geometric imperfection causes buckling phenomena which significantly affect the natural frequencies of the nanobeam. Moreover, the viscoelastic effect, imperfect effect and lateral motion can lead to more complicated and rich vibrational responses due to the greater contribution of nonlinearity.

The sintering and creep and cracking of thermal barrier coatings under thermal cycling

During service, thermal barrier coating will generally go through a series of complicated nonlinear phenomena. In this work, a constitutive framework is provided for the sintering and creep of ceramic top-coat under thermal cycling. The influence of the top-coat sintering and creep was investigated using a three-dimensional thermal barrier system model. It is found that the top-coat sintering gives rise to an obvious reduction in the low-temperature out-of-plane stress within the top-coat but hardly has an effect on the high-temperature out-of-plane stress. The bond-coat/TGO interface is subjected to significant normal stresses upon cooling to ambient temperature, which have greater magnitudes because of the top-coat sintering. Nonetheless, the top-coat creep has very limited effects on the out-of-plane stress within the top-coat and the normal stress across the bond-coat/TGO interface. The cohesive damage model was used to evaluate the delamination of bond-coat/TGO interface under thermal cycling. It is found that the interface delamination could promote the out-of-plane tension of the top-coat and then accelerate the cracking of the top-coat. A static crack is preset in the top-coat to analyze the influence of top-coat crack on the interface delamination. It is found that the top-coat crack hardly affects the normal stresses across the bond-coat/TGO interface and therefore is less likely to facilitate the interface delamination.

Analytical and numerical solutions to the (3 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa with time-dependent coefficients

In this paper, we implemented the Exp-function method and the extended tanh method to finds the analytical solutions of one of the important nonlinear partial differential equations with variable coefficients called the new (3 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa with time-dependent coefficients. Moreover, we find the numerical solutions to that equation using the finite difference method. The results demonstrate the effectiveness and convenience of the used methods. Comparison between our results through some tables and graphs to illustrate the accuracy of the applied methods. We obtained the different types of solutions-like bright, dark, periodic, rational and singular periodic soliton solutions using the different values of constant parameters. The obtained exact solutions may be useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in (3 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa with time-dependent coefficients model.

Two-Parameter Stability Analysis of Resistive Droop Control Applied to Parallel-Connected Voltage-Source Inverters

The present research work aims to study the large signal stability of two parallel-connected voltage-source inverters operating with resistive droop control through bifurcation analysis. This minimal setup can be considered a prototype of a small islanded ac microgrid studying where droop control acts as a synchronization scheme between inverters. Since switching power converters are strong nonlinear circuits, bifurcation analysis helps to understand nonlinear phenomena more deeply than the standard linear approach. Plenty of complicated nonlinear phenomena have been observed through bifurcation analysis, such as multiequilibria, period-doubling bifurcation, Hopf bifurcation, quasiperiodicity, border collision, bistability, and chaos. Thereby, a set of parameters under certain conditions can influence the equilibrium point and its stability, leading to some of the aforementioned nonlinear phenomena. In this sense, the main contribution of this article is to detect the nonlinear phenomena in considering islanded ac microgrid in order to determine a safe droop parameter region for more robust microgrid operation. This information can be summarized in bifurcation diagrams leading to practical rules for choosing the control and droop parameters. Experimental results are performed to validate the main instabilities predicted by the theoretical models.

Hidden Multistability in a Memristor-Based Cellular Neural Network

In this paper, we report a novel memristor-based cellular neural network (CNN) without equilibrium points. Dynamical behaviors of the memristor-based CNN are investigated by simulation analysis. The results indicate that the system owns complicated nonlinear phenomena, such as hidden attractors, coexisting attractors, and initial boosting behaviors of position and amplitude. Furthermore, both heterogeneous multistability and homogenous multistability are found in the CNN. Finally, Multisim circuit simulations are performed to prove the chaotic characteristics and multistability of the system.

Generalized Anticipation Effect Models for Traffic Flow

Taking into account the startup behaviour of following vehicles in the waiting area of urban traffic signal, we propose an extended Nagel-Schreckenberg model for single-lane traffic flow, in which the dynamic behavior of each vehicle depends on not only its own headway but also the headway of the immediately preceding one. The numerical simulation of the present model reproduces some complicated nonlinear phenomena observed in real traffic such as free flow, ghostly blockage, synchronized flow and so on. For specific parameter combinations the flow-density relation of this model shows two meta-stable branches near the transition density from free flow to wide moving traffic jam. Finally, the analytical results of the model under some specific parameters are given by using the mean field theory.

Dynamic performance of a rotor system with an initial bow and coupling faults of imbalance-rub during whirling motion

The dynamic characteristics of a rotor system with an initial bow and coupling fault of imbalance-rub are investigated in this work. The geometrical nonlinearity of shafts becomes significant due to the large-amplitude whirling motion. Then, the influences of the initial bow and geometrical nonlinearity on the natural frequencies corresponding to the linear part of the rotor system are studied. Moreover, the coupling faults of imbalance-rub are introduced to the rotor system. Complicated nonlinear phenomena are revealed by bifurcation diagrams, time histories, Poincare sections, and spectra. The influences of several key design parameters, such as initial bow, shaft radius, and casing stiffness, are analyzed. One of the main findings of this investigation is that when initial bow and geometrical nonlinearity coexist in a system, the resonance characteristics are obviously affected by the initial bow’s degree. This coexistence can result in the jump phenomenon, which rapidly increases the amplitude of whirling motion. These findings are useful in the fault diagnosis and feature recognition of rotating machines.

Breather Turbulence: Exact Spectral and Stochastic Solutions of the Nonlinear Schrödinger Equation

I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also referred to as finite gap theory (FGT) in the Russian literature) and (2) quasiperiodic Fourier series, both of which enhance the physical and mathematical understanding of complicated nonlinear phenomena in water waves. The basic approach I refer to is nonlinear Fourier analysis (NLFA). The formulation describes wave motion with spectral components consisting of sine waves, Stokes waves and breather packets that nonlinearly interact pair-wise with one another. This contrasts to the simpler picture of standard Fourier analysis in which one linearly superposes sine waves. Breather trains are coherent wave packets that “breath” up and down during their lifetime “cycle” as they propagate, a phenomenon related to Fermi-Pasta-Ulam (FPU) recurrence. The central wave of a breather, when the packet is at its maximum height of the FPU cycle, is often treated as a kind of rogue wave. Breather turbulence occurs when the number of breathers in a measured time series is large, typically several hundred per hour. Because of the prevalence of rogue waves in breather turbulence, I call this exceptional type of sea state a breather sea or rogue sea. Here I provide theoretical tools for a physical and dynamical understanding of the recent results of Osborne et al. (Ocean Dynamics, 2019, 69, pp. 187–219) in which dense breather turbulence was found in experimental surface wave data in Currituck Sound, North Carolina. Quasiperiodic Fourier series are important in the study of ocean waves because they provide a simpler theoretical interpretation and faster numerical implementation of the NLFA, with respect to the IST, particularly with regard to determination of the breather spectrum and their associated phases that are here treated in the so-called nonlinear random phase approximation. The actual material developed here focuses on results necessary for the analysis and interpretation of shipboard/offshore platform radar scans and for airborne lidar and synthetic aperture radar (SAR) measurements.

Investigation of physical issues of the initial conditions of free water surface due to underwater disturbances on the basis of SPH numerical simulation of the phenomenon

The current paper discusses the physical impacts of the various initial boundary conditions of the free surface of a waterbody on the initiation and propagation characteristics of water waves due to the underwater perturbations. Differences between traditional point of view and applied numerical method in this paper for exertion the initial conditions of the generated waves by surface deformation were surveyed in the Lagrangian domain vs. Eulerian. In this article, the smoothed-particle hydrodynamics (SPH) technique was applied for simulating of wave generation process using initial boundary condition of water surface deformation through utilizing DualSPHysics numerical code and comparing the modeling results with recorded data. As a distinct approach, we studied the effects of discrete water particles on properties of produced surface waves by using the Lagrangian analytical capability of SPH model. Illustrative compatibility on simulation results with experimental data proves that meshless techniques such as applied in DualSPHysics software can reproduce physical properties of the event very well, and this is a suitable alternative to existing classical approaches for prediction of shock occurrences with nonlinear behavior such as generated surface water waves by underwater disturbance. Besides, the waveforms and their characteristics behave more realistic by considering the thrown upward water mass which was not directly considered in old formula and theories. The results of numerical modeling indicated rational agreement between numerical and empirical data proving that a complicated nonlinear phenomenon could be predicted by an SPH model which modified initial boundary conditions were supposed into the model with actual assumptions.

Similarity and vortex-acoustic lock-on behavior in thermoacoustic oscillation involving vortex shedding

In engineering, the combustion chamber with a backward step is very popular, and it is a kind of flame stabilizer. In this type of combustion chamber, there will be shedding vortices at the step due to the instability of the flow field. The shedding vortices will carry reactants to move downstream and burn, resulting in unstable heat release and then pressure and velocity fluctuations of the sound field, thereby, finally, forming a combustion-vortex-acoustic interaction process. If a positive feedback loop is formed between the unstable heat release and the pressure fluctuation of sound field, combustion instability will occur, and it is also referred to as thermoacoustic oscillation due to vortex shedding. Combustion instability frequently occurs in many practical systems or equipment, and its induced significant pressure oscillations have a serious influence on the normal operation of the equipment. Recently, the combustion instability has been extensively studied experimentally, but the theoretical investigation on its nature is still rare. Since combustion instability is a complicated nonlinear phenomenon, it is necessary to study its nature from the viewpoint of nonlinear dynamics. Based on the one-dimensional simplified model of thermoacoustic instability involving vortex shedding proposed by Matveev and Culick, the typical nonlinear phenomenon in thermoacoustic oscillation induced by vortex shedding is studied. The study focuses on the initial value sensitivity of the system, the influence of key parameters on thermoacoustic oscillation, and the phenomenon of vortex-acoustic lock-on. Firstly, the Galerkin method is used to approximate the governing equation, and the partial differential equations are reduced to a set of ordinary differential equations. Then, the first ten modes are selected, and the pressure and velocity fluctuations of sound field under different system parameters are obtained by MATLAB program. Finally, the thermoacoustic instability of the system under different initial disturbances, the influences of different steady flow velocity on the thermoacoustic oscillation of the system, and the phenomenon of vortex-acoustic lock-on in thermoacoustic oscillation are studied in detail. The results show that the system of thermoacoustic oscillation involving vortex shedding is extremely sensitive to initial values, and there are a rich variety of nonlinear phenomena. With steady flow velocity increasing, the amplitude of pressure fluctuation augments generally. However, the similar structures are found in several intervals of steady flow velocity, and the amplitude first decreases and then increases. In particular, it is verified that the system oscillates periodically by integer (fp/fs) multiple of the vortex impinging frequency (fs), that is, the vortex-acoustic frequency locking with the number of revolutions fp/fs, which is found in experiment and can be regarded as an important characteristic of periodic thermoacoustic oscillation.

Review of deployment technology for tethered satellite systems

Tethered satellite systems (TSSs) have attracted significant attention due to their potential and valuable applications for scientific research. With the development of various launched on-orbit missions, the deployment of tethers is considered a crucial technology for operation of a TSS. Both past orbiting experiments and numerical results have shown that oscillations of the deployed tether due to the Coriolis force and environmental perturbations are inevitable and that the impact between the space tether and end-body at the end of the deployment process leads to complicated nonlinear phenomena. Hence, a set of suitable control methods plays a fundamental role in tether deployment. This review article summarizes previous work on aspects of the dynamics, control, and ground-based experiments of tether deployment. The relevant basic principles, analytical expressions, simulation cases, and experimental results are presented as well.

Numerical Investigation on Physical Foaming and Decay Process Using Multicomponent Thermal Lattice Boltzmann Model

Foam materials produced by physical process are common in the field of material engineering such as foamed asphalt. Considering heat and phase transfer, the complicated nonlinear phenomenon of foaming and decay process is difficult to be described numerically. In this paper, a multicomponent thermal lattice Boltzmann model for simulation of physical foaming and decay process is proposed. The model combines a thermal single-component phase change model with a multicomponent model, and the two models are verified separately. The physical foaming and decay process is solved numerically based on the proposed model. Simulation results show the influence of the distribution and content of the phase-change component on the expansion rate and decay time during physical foaming and decay process.

Exact Traveling Wave Solutions of the Space Fractional (2+1)-Dimensional Breaking Soliton Equations

The fractional order differential equation is used to describe the phenomenons of physical, engineering and biological fields. In this paper, the exp (-Φ(ξ))-expansion method along with the Jumarie's modified Riemann-Liouville derivatives is proposed to solve the space fractional (2+1)-dimensional breaking soliton equations. The traveling wave solutions of the equations are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. In this work, it has been shown that the proposed method is very effectual and easily to find the exact traveling wave solutions to the fractional nonlinear evolution equations.

Meminductive Wein-bridge chaotic oscillator

A meminductor is a new type of memory device. It is of importance to study meminductor model and its application in nonlinear circuit prospectively. For this purpose, we present a novel mathematical model of meminductor, which considers the effects of internal state variable and therefore will be more consistent with future actual meminductor device. By using several operational amplifiers, multipliers, capacitors and resistors, the equivalent circuit of the model is designed for exploring its characteristics. This equivalent circuit can be employed to design meminductor-based application circuits as a meminductor emulator. By employing simulation experiment, we investigate the characteristics of this meminductor driven by sinusoidal excitation. The characteristic curves of current-flux (i-φ), voltage-flux (v-φ), v-ρ (internal variable of meminductor) and φ-ρ for the meminductor model are given by theoretical analyses and simulations. The curve of current-flux (i-φ) is a pinched hysteretic loop passing through the origin. The area bounding each sub-loop deforms as the frequency varies, and with the increase of frequency, the shape of the pinched hysteretic loop tends to be a straight line, indicating a dependence on frequency for the meminductor. Based on the meminductor model, a meminductive Wien-bridge chaotic oscillator is designed and analyzed. Some dynamical properties, including equilibrium points and the stability, bifurcation and Lyapunov exponent of the oscillator, are investigated in detail by theoretical analyses and simulations. By utilizing Lyapunov spectrum, bifurcation diagram and dynamical map, it is found that the system has periodic, quasi-periodic and chaotic states. Furthermore, there exist some complicated nonlinear phenomena for the system, such as constant Lyapunov exponent spectrum and nonlinear amplitude modulation of chaotic signals. Moreover, we also find the nonlinear phenomena of coexisting bifurcation and coexisting attractors, including coexistence of two different chaotic attractors and coexistence of two different periodic attractors. The phenomenon shows that the state of this oscilator is highly sensitive to its initial valuse, not only for chaotic state but also for periodic state, which is called coexistent oscillation in this paper. The basic mechanism and potential applications of the existing attractors are illustrated, which can be used to generate robust pseudo random sequence, or multiplexed pseudo random sequence. Finally, by using the equivalent circuit of the proposed meminducive model, we realize an analog electronic circuit of the meminductive Wien-bridge chaotic system. The results of circuit experiment are displayed by the oscilloscope, which can verify the chaotic characteristics of the oscillator. The oscillator, as a pseudo random signal source, can be used to generate chaotic signals for the applications in chaotic cryptography and secret communications.

Steady-state response characteristics of a dual-rotor system induced by rub-impact

In this paper, the steady-state responses and their stability of a dual-rotor system with rub-impact are investigated. The nonlinear equations of motion in eight d.o.f.s are obtained with the consideration of the gyroscopic effect. The multi-harmonic balance combined with the alternating frequency/time domain technique (MHB–AFT) is utilized to calculate the accurate amplitude of each harmonic component. Arc-length continuation is embedded in the MHB–AFT procedure to trace the branch of the periodic solutions, and the Floquet theory is used to discuss the stability of the obtained solutions. Through the numerical calculation, complicated nonlinear phenomena, such as combined harmonic vibrations, hysteresis and resonant peak shifting are obtained when the rub-impact occurs. The result also shows that the control parameters such as mass eccentricity, inter-shaft stiffness and rotational speed ratio make significant but different influences on the dynamic characteristics of the two rotors. Therefore, the contribution of this study is to provide a further understanding of the steady-state response characteristics of the dual-rotor system with rub-impact.

Solutions to Class of Linear and Nonlinear Fractional Differential Equations

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.