Nonlinear Stability Theory Research Articles

Chaotification control of buck converter via time-delayed feedback

Chaotification is an inverse process of chaos control. This paper addresses the mechanism of chaotification of buck converter. A new chaotification model of buck converter is established. Then the chaotification rule of buck converter is discussed. The fact that using time-delay feedback at the input can realize chaotification in buck converter is proved. The ranges of control parameters are determined using nonlinear dynamics stability theory. Computer simulation is made to verify the proposed method and analyze the influence of different control parameters. The theoretical results are verified by experimental measurements.

Nonlinear stability analysis of thin micropolar film flows travelling down a vertical moving plate

This paper employs linear and nonlinear stability theories to conduct a numerical characterization of thin micropolar film flows travelling down a vertical moving plate. Having applied the long-wave perturbation method to derive the generalized kinematic equations for a free film surface condition, the multiple scales method is used to solve the micropolar film flow in the cases of a stationary vertical plate, a vertical plate moving in the downward direction and a vertical plate moving in the upward direction. The numerical results indicate that both subcritical instability and supercritical stability conditions may occur in the micropolar film flow system. No obvious change in the supercritical stability condition is observed when the plate moves vertically in the upward or downward direction. The numerical results indicate that a downward motion of the vertical plate tends to enhance the stability of the film flow.

New equilibrium function of traffic flow

Using nonlinear stability theory in macroscopic traffic flow model, we proposed a new equilibrium function starting from the existing equilibrium functions. The kinematic wave velocity at jam density yielded by this equilibrium function is consistent with the empirical value, and this equilibrium function is related to the critical stability in higher density interval.

Study on the bifurcation character of steering wheel self-excited shimmy of motor vehicle

Bifurcation theories of nonlinear dynamics and stability theories of ordinary differential equation have been considered to elaborate the bifurcation character of steering wheel self-excited shimmy of motor vehicle. It has been theoretically proved that the self-excited shimmy is a vibration phenomenon of stable limit cycle occurring after Hopf bifurcation. Some control objectives to attenuate the shimmy have been proposed on the basis of its bifurcation property, and preliminary exploration has been made on research methods of shimmy control strategies. A simple shimmy model of three degrees of freedom, parameters of which come from a truck with independent suspension, has been constructed as an example to conduct numerical analysis of bifurcation property, road test and numerical simulation. The results confirm the conclusions.

Nonlinear stability analysis of thin Newtonian film flowing down on the inner surface of a rotating vertical cylinder

The paper presents both of the linear and nonlinear stability theories for characterization of Newtonian film flows down on the inner surface of a rotating in.nite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, X, and the radius of cylinder, R, the film flow will make the flow system more stable.

Asymptotic Theory of Stability of a Viscous Fluid Jet Ejected along a Wall

A linear and weakly nonlinear theory of stability of a laminar viscous fluid wall jet is considered. At large Reynolds numbers (calculated from the characteristic jet length) the undisturbed stationary flow is concentrated in a narrow region adjacent to the rigid surface. Nonstationary perturbations of this flow can be described by equations similar to the equations used in the theory of free interaction of a boundary layer. Such a description is made possible by introducing asymptotic order relations between the Reynolds number and the perturbation amplitudes and wavelengths. The set of solutions of the dispersion relation for the linear stability problem includes the eigenvalues of the phase velocities and wave numbers corresponding to the neutral and unstable perturbation modes. For finite fluctuation amplitudes, the evolution of the wave fields obeys the Korteweg-de Vries equations.

Non-Linear Stability Analysis of a Thin Newtonian Film Flowing Down an Infinite Vertical Rotating Cylinder

Non-linear stability theories for the characterization of Newtonian film flow down an infinite vertical rotating cylinder is given. A generalized non-linear kinematic model is derived to represent the physical system and is solved by the long-wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviours. In the second step, an elaborated non-linear film flow model is solved by using the method of multiple scales to characterize flow behaviours at various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The modelling results indicate that by increasing the rotation speed, ω, and decreasing the radius of cylinder, R, the film flow becomes less stable, generally.

Nonlinear observer design to synchronize fractional-order chaotic systems via a scalar transmitted signal

In this paper, based on the idea of nonlinear observer and stability theory of fractional-order systems, a new systematic scheme to synchronize a class of fractional-order chaotic systems via a scalar transmitted signal is developed. The approach is simple, global and theoretically rigorous. It enables synchronization of fractional-order chaotic systems to be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method.

Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener–Hermite Expansions

A number of engineering problems, including laminar‐turbulent transition in convectively unstable flows, require predicting the evolution of a nonlinear dynamical system under uncertain initial conditions. The method of Wiener–Hermite expansion is an attractive alternative to modeling methods, which solve for the joint probability density function of the stochastic amplitudes. These problems include the “curse of dimensionality” and closure problems. In this paper, we apply truncated Wiener–Hermite expansions with both fixed and time‐varying bases to a model stochastic system with three degrees of freedom. The model problem represents the combined effects of quadratic nonlinearity and stochastic initial conditions in a generic setting and occurs in related forms in both classical dynamics, turbulence theory, and the nonlinear theory of hydrodynamic stability. In this problem, the truncated Wiener–Hermite expansions give a good account of short‐time behavior, but not of the long‐time relaxation characteristic of this system. It is concluded that successful application of truncated Wiener–Hermite expansions may require special adaptations for each physical problem.

Nonlinear Group Interactions of the Taylor–Görtler Disturbances in Supersonic Axisymmetric Jets

The nonlinear group interaction of the Taylor-Gortler disturbances (streamwise vortices) at the initial section of a supersonic axisymmetric jet is numerically studied within the framework of the weakly nonlinear theory of stability. The experimentally observed spectrum of disturbances is considered. The regular and specific features of the streamwise dynamics of various wave components for a turbulent jet are studied. It is shown that such an interaction in the coupled mode in resonant group triplets allows one to describe the experimentally observed elevated growth of background components of the real spectrum.

The nonlinear evolution of a wavetrain emanating from a point source in a boundary layer

The paper reports on an investigation into the nonlinear evolution of a wavetrain emanating from a point source in a flat-plate boundary layer. The work was mainly experimental, but calculations using linear and nonlinear stability theory and parabolized stability equations were also performed to support the conclusions. The amplitudes of the disturbances used were very small and, as a consequence, turbulence was not reached within the experimental domain. Nevertheless, interesting nonlinear behaviour was observed. The nonlinear regime was characterized by the appearance of a three-dimensional mean flow distortion in the form of longitudinal streaks. It was possible to distinguish two different stages of the nonlinear regime. In the first stage, the streak pattern displayed relatively low spanwise wavenumbers. The pattern appeared to have grown algebraically from a pair of counter-rotating streamwise vortices. Weakly nonlinear calculations suggested that the vortices arise from the interaction of the spanwise and the wall-normal velocity components. The second stage exhibited more streaks and higher spanwise wavenumbers. This stage was found to be associated with a peak-and-valley structure of the wave amplitude. A remarkable feature was that the streamwise position of the onset of the second stage of the nonlinear regime was not affected by the amplitude of the disturbance. Parabolized stability calculations suggested that the peak-and-valley structure was the outcome of a secondary instability of the fundamental type. An interaction similar to that of the first nonlinear stage involving the waves composing the peak-and-valley structure yielded other streamwise vortices and hence a different streak pattern. The results also suggested that, because the wave amplitudes were very small, the linear stability influenced the nonlinear evolution. Indeed, the onset of the second stage of the nonlinear regime appeared to be associated with the proximity of the second branch of the linear stability diagram.

Review of the study of nonlinear atmospheric dynamics in China (1999–2002)

Researches on nonlinear atmospheric dynamics in China (1999–2002) are briefly surveyed. This review includes the major achievements in the following branches of nonlinear dynamics: nonlinear stability theory, nonlinear blocking dynamics, 3D spiral structure in the atmosphere, traveling wave solution of the nonlinear evolution equation, numerical predictability in a chaotic system, and global analysis of climate dynamics. Some applications of nonlinear methods such as hierarchy structure of climate and scaling invariance, the spatial-temporal series predictive method, the nonlinear inverse problem, and a new difference scheme with multi-time levels are also introduced.

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.

Evolution of a 2-D disturbance in a supersonic boundary layer and the generation of shocklets

Through direct numerical simulation, the evolution of a 2-D disturbance in a supersonic boundary layer has been investigated. At a chosen location, a small amplitude TS wave was fed into the boundary layer to investigate its evolution. Characteristics of nonlinear evolution have been found. Two methods were applied for the detection of shocklets, and it was found that when the amplitude of the disturbance reached a certain value, shocklets would be generated, which should be taken into consideration when nonlinear theory of hydrodynamic stability for compressible flows is to be established.

Perturbation analysis to the nonlinear stability characterization of thin condensate falling film on the outer surface of a rotating vertical cylinder

The linear and nonlinear stability theories for characterization of condensate film flow down on the outer surface of a rotating infinite vertical cylinder is investigated analytically. A generalized nonlinear kinematic model is derived to represent the physical system and is solved by the long-wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability and supercritical explosion. The modeling results indicate that by increasing the rotation speed, Ω, and decreasing the radius of cylinder, R, the film flow becomes less stable, generally.

Weakly nonlinear stability analysis of thin viscoelastic film flowing down on the outer surface of a rotating vertical cylinder

The paper presents both of the linear and nonlinear stability theories for characterization of viscoelastic film flows down on the outer surface of a rotating infinite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, Ω, and decreasing the radius of cylinder, R, the film flow will generally make the flow system less stable. In this study, the interaction of the rotation and the radius of cylinder are taken into account. Generally, Reynolds number is divided into three regions, which are Re<3, 3< Re<8 and 8< Re, for discussion corresponding to the pre-selected Rossby number ( Ro=0.1) and viscoelastic parameter ( k=0.05).

Determination of critical Rayleigh number for binary solutions in a nonlinear theory of convective stability

The Rayleigh–Benard convective instability in a cell with horizontal copper electrodes is considered. The space between the electrodes is filled with a CuSO4 solution. The dependence of the variation in the limiting current of copper deposition, ΔI, which is caused by the formation of convective dissipative structures in the system, on the solution concentration \(c_{{\text{CuSO}}_{\text{4}} } \) is determined experimentally. The results agree satisfactorily with a nonlinear theory of convective stability that accounts for the electrical forces generated by the formation of a space charge near the electrode, along with the buoyancy forces. This enables one to determine the critical Rayleigh number for the system by extrapolating the linear dependence of ΔI on \(\sqrt c \). The critical Rayleigh number thus obtained agrees satisfactorily with the theoretical value calculated within the linear theory of stability.

Three-wave interactions of disturbances in a supersonic boundary layer

Within the framework of the weakly nonlinear stability theory, group interaction of disturbances in a supersonic boundary layer is considered. The disturbances are represented by two spatial packets of traveling instability waves (wave trains) with multiple frequencies. The possibility of energy redistribution in such wave systems in the case of three-wave resonant interactions of packet constituents is considered. The model is used to test the dynamics of unstable waves arising due to introduction of controlled high-intensity disturbances into a supersonic boundary layer. It is found that this mechanism is not the main one for the features of streamwise dynamics of such nonlinear waves being observed.

Modal interactions in a Bickley jet: comparison of theory with direct numerical simulation

This study is concerned with three-dimensional transition in plane wakes and jets. Nonlinear stability theory has predicted that interactions between different instability modes can play an important role in that transition. In this paper, we consider interactions between varicose and sinuous oblique disturbances in the Bickley jet, using both nonlinear stability theory (in its nonlinear critical layer form) and direct numerical simulation using a spectral method. We will first present an overview of the nonlinear stability theory which indicates that a (nonlinear) interaction between the modes should occur, and then present simulations that appear to support the theory.

Nonlinear theory of combustion stability in liquid rocket engine based on chemistry dynamics

Nonlinear theory of combustion stability in liquid rocket engine based on chemistry dynamics