Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear coupling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A?sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dynamics by the Ge?CYu?CChen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchronization strategy.     

TEMPORAL STABILITY OF FLOW THROUGH VISCOELASTIC TUBES

The stability of the Hagen–Poiseuille flow of a Newtonian fluid in an incompressible, viscoelastic tube contained within a rigid, hollow cylinder is determined using linear stability analysis. The stability of the system subjected to infinitesimal axisymmetric or non-axisymmetric disturbances is considered. The fluid and wall inertia terms are retained in their respective equations of motion. A novel numerical strategy is introduced to study the stability of the coupled fluid–structure system. The strategy alleviates the need for aninitial guess and thus ensures that all the unstable modes within a given closed region in the complex eigenvalue plane will be found. It is found that the system is unstable to both axisymmetric and non-axisymmetric disturbances. Moreover, depending on the values of the control parameters, the first unstable mode can be either an axisymmetric mode with the azimuthal wavenumber n=0 or a non-axisymmetric mode withn =1. For a given azimuthal wavenumber, it is found that there are no more than two unstable modes within the closed region considered here in the complex plane. For both the axisymmetric and non-axisymmetric instabilities, one mode is a solid-based, flow-induced surface instability, while the other one is a fluid-based instability that asymptotes to the least-damped rigid-wall mode as the thickness of the compliant wall tends to zero. All four modes are stabilized, to different degrees, by the solid viscosity.     

Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters via nonlinear control

A new adaptive synchronization scheme by pragmatical asymptotically stability theorem is proposed in this paper. Based on this theorem and nonlinear control theory, a new adaptive synchronization scheme to design controllers can be obtained and especially the constraints for minimum values of feedback gain K in controllers can be derived. This new strategy shows that the constraint values of feedback gain K are related to the error of unknown and estimated parameters if the goal system is given. Through this new strategy, an appropriate feedback gain K can be always decided easily to obtain controllers achieving adaptive synchronization. Two identical Lorenz systems with different initial conditions and two completely different nonlinear systems with different orders, augmented R?ssler??s system and Mathieu?Cvan der Pol system, are used for illustrations to demonstrate the efficiency and effectiveness of the new adaptive scheme in numerical simulation results.     

Effect of rotation on ferromagnetic porous convection with a thermal non-equilibrium model

The effect of rotation on the onset of thermal convection in a horizontal layer of ferrofluid saturated Brinkman porous medium is investigated in the presence of a uniform vertical magnetic field using a local thermal non-equilibrium (LTNE) model. A two-field model for temperature representing the solid and fluid phases separately is used for energy equation. The condition for the occurrence of stationary and oscillatory convection is obtained analytically. The stability of the system has been analyzed when the magnetic and buoyancy forces are acting together as well as in isolation and the similarities as well as differences between the two are highlighted. In contrast to the non-rotating case, it is shown that decrease in the Darcy number Da and an increase in the ratio of effective viscosity to fluid viscosity Λ is to hasten the onset of stationary convection at high rotation rates and a coupling between these two parameters is identified in destabilizing the system. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. Besides, the influence of magnetic parameters and also parameters representing LTNE on the stability of the system is discussed and the veracity of LTNE model over the LTE model is also analyzed.     

Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting

In this paper, a predator–prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order k (k≥2) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.     

Delay induced stability switches in a viral dynamical model

In this paper, a delay-differential mathematical model that described HIV infection of CD4⁺ T cells is analyzed. The effect of time delay on stability of the equilibria of the infection model has been studied. And the sufficient criteria for stability switch of the infected equilibrium and the local and global asymptotic stability of the uninfected equilibrium are given. By using the geometric stability switch criterion in the delay-differential system with delay-dependent parameters, we present that stable equilibria become unstable as the time delay increases. Numerical simulations are carried out to explain the mathematical conclusions.

     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances

This paper studies the characteristics of a series of complex systems, which are in combination of interest rate, investment, and price index parameters meeting the condition c?b?abc??0, including stable node, saddle points, bifurcation, Hopf bifurcation, and chaos, and the corresponding mathematical expressions of Lyapunov index are given. Based on this, we perform complexity analysis of the system and study the change circumstances of Lyapunov index when one or two parameters synchronously change. Numerical simulation results verify the theoretical analysis and conclusion. The results of this economic and financial system provide reference to practical problems, and have a positive effect on the actual application of the system of this type.     

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

In this paper, a delay-differential mathematical model that described HIV infection of CD4⁺ T cells is analyzed. The effect of time delay on stability of the equilibria of the infection model has been studied. And the sufficient criteria for stability switch of the infected equilibrium and the local and global asymptotic stability of the uninfected equilibrium are given. By using the geometric stability switch criterion in the delay-differential system with delay-dependent parameters, we present that stable equilibria become unstable as the time delay increases. Numerical simulations are carried out to explain the mathematical conclusions.     

Prescribed-time adaptive neural tracking control for a class of uncertain robotic manipulators with dead zone input

This paper solves the prescribed-time control problem for a class of robotic manipulators with system uncertainty and dead zone input. To make the system stable within a given convergence time T, a novel prescribed-time adaptive neural tracking controller is proposed by using the temporal scale transformation method and Lyapunov stability theory. Unlike the finite-time and the fixed-time stability where the convergence time depends on the controller parameters, the convergence time constant T is introduced into the proposed controller so that the closed-loop system will be stable within T. To cope with the system uncertainty, radial basis function neural networks (RBFNNs) are used and only need to update one parameter online. In addition, by choosing the same structure and parameters of RBFNNs, the proposed method can shorten the convergence time of the neural networks. Finally, simulation results are presented to demonstrate the effectiveness of the prescribed-time controller.

     

Robust adaptive nonlinear control for uncertain control-affine systems and its applications

This paper addresses the robust tracking control problem for a class of uncertain nonlinear systems with time-varying parameters, perturbed by external disturbances. The unknown time-varying parameters and disturbances are neither required to be periodic nor to have known bounds. Depending on the characteristics of disturbance signals, two adaptive-based control algorithms are developed. First, an adaptive H _∞ control is designed that achieves: (i) an H _∞ tracking performance when the external disturbances are L ₂ signals, and (ii) the convergence of tracking error to zero if the disturbances are bounded and L ₂ signals. Then a novel adaptive control algorithm is proposed, only with the assumption of boundedness of disturbances, to drive the tracking error to zero. The designed tracking controllers are then used for controlling a cart-pendulum system, as an underactuated mechanical system, and chaos synchronization of uncertain Genesio–Tesi chaotic system. Numerical simulations are also given to demonstrate the effectiveness of the proposed control schemes.     

Stress-driven morphological instability and catastrophic failure of microdevices

In microdevices, the competition between surface energy and elastic energy could lead at the phenomenon known as stress-driven morphological instability (MI), causing an increase of surface roughness with time. Several different mass transport mechanisms can trigger such a morphological alteration and operate simultaneously: surface and bulk diffusion, evaporation and condensation, chemical reactions. Unstable solids could eventually evolve towards crack-like surfaces thus altering mechanical, electrical and optical properties of the devices or even leading to catastrophic failures by supercritical crack propagation. In this work, a more general kinetic law is employed to estimate the onset of MI, considering the effect of the stress field on the atomic mobility. A more intuitive and straightforward approach is used to determine the stability conditions, where the rate of atomic mass motion is introduced as a stability parameter. The critical loads and wavelengths for the onset of MI, determined as a function of material parameters α and β, are compared with the limiting conditions for the supercritical crack propagation (SC) of a crack-like surface in order to asses if and under which situations catastrophic failures by SC can be observed. Two practical cases are investigated: fixed wavelength (Case I) and arbitrary rough surface with a fixed remote load (Case II). In Case I, absolute and relative threshold loads are found below which MI could never occur and a transitional wavelength over which MI would always lead to SC is introduced. In Case II, it is shown that dominant perturbation for MI would always lead to SC given enough time for the surface to evolve towards a crack-like profile. The influence of the material properties α and β on the critical parameters is also addressed.     

Variable structure based robust backstepping controller design for nonlinear systems

This study presents robust control architecture in the sense of variable structure control via a backstepping design. By using systematic backstepping design techniques, closed-loop behavior of an n-order nonlinear system can be transformed into a stability and convergence problem of a fast switched 2nd order system. There are two main parts contained within the proposed control algorithm; one is a nominal control effort generated according to the Lyapunov stability criterion during recursive backstepping processes, and the other belongs to a smooth robust control law designed to eliminate the effects of unknown lumped perturbations. Finally, a Genesio system is used as an illustrated example to demonstrate the robustness of the control algorithm. The feasibility and properties of the proposed method are given by numerical simulations.     

Étude de la convergence numérique d'une méthode vortex pour un écoulement a grand nombre de Reynolds dans un mélangeurNumerical convergence of vortex method for a high Reynolds number 2D bluff-body flow

In this paper the convergence of the RVM for a complex flow is studied, in function of three discretization parameters. Two of these parameters are related to the spatial discretization of the vorticity Γ (sheet or blob strength) and h (sheet length or core radius) and the third one to the time discretization Δt. Two main events are observed: first, the computation works but the convergence is not attained, secondly the computation fails. The first behaviour is attributed to a lack of accuracy and the second to a lack of numerical stability. Once the stability conditions are satisfied, decreasing the value of parameters always leads to convergence. To cite this article: I. Mortazavi et al., C. R. Mecanique 330 (2002) 409–416.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

ON THE STABILITY OF A FUNCTIONALLY GRADED RECTANGULAR MICRO-PLATE SUBJECTED TO HYDROSTATIC AND NONLINEAR ELECTROSTATIC PRESSURES

This article studies the stability of a functionally graded clamped-clamped micro-plate subjected to hydrostatic and electrostatic pressures. Equilibrium positions of the micro-plate are determined and shown in the state control space. To study the stability of the equilibrium positions, the motion trajectories are given for different initial conditions in the phase plane. Effects of the electrostatic and hydrostatic pressure changes on the deflection and stability of the micro-plate for some sample value of k are studied and values of the applied voltage and hydrostatic pressure leading system to unstable conditions by undergoing a saddle node and homoclinic bifurcations are determined.     

Birkhoff 系统稳定性的动力学控制

本文研究 Birkhoff 系统和广义 Birkhoff 系统平衡稳定性的动力学控制. 首先建立系统的运动方程和平衡方程. 其次,研究 Birkhoff 系统中控制参数出现在 Birkhoff 函数中平衡稳 定性的动力学控制. 方法是通过选取控制参数使得 Birkhoff 函数 $B$ 成为定号函数,而其时间导数 $\dot {B}$ 为与 $B$ 反号的常号函数. 再次,研究广义 Birkhoff 系统平衡稳定性的动力学控制,通过选取 Birkhoff 函数或附加项中包含控制参数的方法,使得 Birkhoff 函数是定号函数,而其时间导数为反号的常号函数,从而控制系统的平衡稳定性. 最后举例说明结果的应用.      

Modified iterative method for augmented system

The successive overrelaxation-like(SOR-like) method with the real parameters ω is considered for solving the augmented system. The new method is called the modified SOR-like(MSOR-like) method. The functional equation between the parameters and the eigenvalues of the iteration matrix of the MSOR-like method is given. Therefore, the necessary and sufficient condition for the convergence of the MSOR-like method is derived. The optimal iteration parameter ω of the MSOR-like method is derived. Finally, the proof of theorem and numerical computation based on a particular linear system are given, which clearly show that the MSOR-like method outperforms the SOR-like(Li, C. J., Li, B. J., and Evans, D. J. Optimum accelerated parameter for the GSOR method. Neural, Parallel Scientific Computations, 7(4), 453–462(1999)) and the modified symmetric SOR-like(MSSOR-like) methods(Wu, S. L., Huang, T. Z., and Zhao, X. L. A modified SSOR iterative method for augmented systems.     

Linear parameter varying feedforward control synthesis using parameter-dependent Lyapunov function

This paper presents the dynamic feedforward control synthesis for linear parameter varying (LPV) systems. It is assumed that all system matrices are dependent on varying parameters, which are measurable with sensor or observable. The parameters have bounded variation rates. Parameter-dependent Lyapunov function is used for the feedforward control synthesis such that the robust stability is assured for all varying parameters at the time of the operation. The method is formulated in terms of linear matrix inequalities for LPV feedforward controller that guarantees the stability of the transfer matrix having \(L_{2}\) -gain. This compensator is designed by adding on the feedback controller in two degrees of freedom control configuration. This controller can be used for the disturbance attenuation or decreasing the tracking error. The numerical examples and simulations are given to provide the applicability of the proposed solution.     

Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan?CYork dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.