Suppressing chaos via Lyapunov–Krasovskii’s method

An algorithm for suppressing the chaotic oscillations in non-linear dynamical systems with singular Jacobian matrices is developed using a linear feedback control law based upon the Lyapunov–Krasovskii (LK) method. It appears that the LK method can serve effectively as a generalised method for the suppression of chaotic oscillations for a wide range of systems. Based on this method, the resulting conditions for undisturbed motions to be locally or globally stable are sufficient and conservative. The generalized Lorenz system and disturbed gyrostat equations are exemplified for the validation of the proposed feedback control rule.     

Decomposition and suboptimal control in dynamical systems

A non-linear controllable dynamical system described by Lagrange equations is considered. The problem of constructing bounded controlling forces which steer the system to a given state in a finite time is investigated. Sufficient conditions are indicated for the problem to be solvable. Under these conditions, the initial system splits into subsystems, each with the degree of freedom. On the basis of this decomposition, using a game-theoretic approach, a feedback control law is proposed which solves the problem posed above and is nearly time-optimal. It is shown that the control must be constructed with proper allowance for the maximum values of the non-linear terms and perturbations in the equations of motion. The perturbations may be ignored only if the ratio of the maximum level of the perturbation to that of the control does not exceed the “golden section”.     

Evolution of near-soliton initial conditions in non-linear wave equations through their Bäcklund transforms

A novel analytic technique for determining the evolution of near-soliton initial conditions in non-linear wave equations is introduced. It is based on the Bäcklund transform connecting soliton solutions of successive order. This transformation lowers the order of the initial condition rendering the determination of the evolution easier. The result of the evolution in this order is transformed to the higher order using again the Bäcklund transform. As a demonstration, the proposed technique is applied to the non-linear Schrödinger (NLS) and Korteweg–de Vries (KdV) equations. The results are in very good agreement with those obtained by other approaches based on the inverse scattering method. Finally, numerical simulations verify the validity of the proposed technique.     

Vibrations and stability of axially traveling laminated beams

In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh-Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system.     

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.     

Nonlinear stability and chaos in electrohydrodynamics

Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.     

A class of conjugate gradient methods for convex constrained monotone equations

The recent designed non-linear conjugate gradient method of Dai and Kou [SIAM J Optim. 2013;23:296–320] is very efficient currently in solving large-scale unconstrained minimization problems due to its simpler iterative form, lower storage requirement and its closeness to the scaled memoryless BFGS method. Just because of these attractive properties, this method was extended successfully to solve higher dimensional symmetric non-linear equations in recent years. Nevertheless, its numerical performance in solving convex constrained monotone equations has never been explored. In this paper, combining with the projection method of Solodov and Svaiter, we develop a family of non-linear conjugate gradient methods for convex constrained monotone equations. The proposed methods do not require the Jacobian information of equations, and even they do not store any matrix in each iteration. They are potential to solve non-smooth problems with higher dimensions. We prove the global convergence of the class of the proposed methods and establish its R-linear convergence rate under some reasonable conditions. Finally, we also do some numerical experiments to show that the proposed methods are efficient and promising.     

Synthesis of a control in a non-linear dynamical system based on decomposition

A non-linear controllable dynamical system with many degrees of freedom, described by Lagrange equations of the second kind, is considered. Geometric constraints are imposed on the magnitudes of the controls. It is assumed that, in the equations of motion, the kinetic energy matrix is close to a certain constant diagonal matrix. It is possible, for example, to reduce the equations of motion of robots, the drives of which have large gear ratios, to a system of this kind. A problem is formulated on the transfer of a system in a finite time from a specified initial state to a final state with zero velocities. The method of decomposition [1] is used to construct the equations. Sufficient conditions are found subject to which the maximum values of the non-linear terms in the equations of motion do not exceed the permissible magnitudes of the controls. In this case, non-linearities are treated as limited perturbations and the system is decomposed into independent, linear, second-order subsystems. A feedback control is specified for these subsystems which guarantees that each of them is brought into the final state for any permissible perturbations. The control has a simple structure. Applications of the proposed approach to problems in the control of manipulating robots are considered.     

Influence of variable heat flux on natural convection along a corrugated wall in porous media

In this work the coupled non-linear partial differential equations, governing the free convection from a wavy vertical wall under a power law heat flux condition, are solved numerically. For both Darcy and Forchheimer extended non-Darcy models, a wavy to flat surface transformation is applied and the governing equations are reduced to boundary layer equations. A finite difference scheme based on the Keller Box approach has been used in conjunction with a block tri-diagonal solver for obtaining the solution. Detailed simulations are carried out to investigate the effect of varying parameters such as power law heat flux exponent m, wavelength–amplitude ratio a and the transformed Grashof number Gr′. Both surface undulations and inertial forces increase the temperature of the vertical surface while increasing m reduces it. The wavy pattern observed in surface temperature plots, become more prominent with increasing m or a but reduces as Gr′ increases.     

Non-linear deformation properties of materials with stochastically distributed anisotropic inclusions

In the present contribution, the problem of non-linear deformation of materials with stochastically distributed anisotropic inclusions is considered on the basis of the methods of mechanics of stochastically non-homogeneous media. The homogenization model of materials of stochastic structure with physically non-linear components is developed for the case of a matrix which is strengthened by unidirectional ellipsoidal inclusions. It is assumed that the matrix is isotropic, deforms non-linearly; inclusions are linear-elastic and have transversally-isotropic symmetry of physical and mechanical properties. Stochastic differential equations of physically non-linear elasticity theory form the underlying equations. Transformation of these equations into integral equations by using the Green's function and application of the method of conditional moments allow us to reduce the problem to a system of non-linear algebraic equations. This system of non-linear algebraic equations is solved by the Newton-Raphson method. On the analytical as well as the numerical basis, the algorithm for determination of the non-linear effective characteristics of such a material is introduced. The non-linear behavior of such a material is caused by the non-linear matrix deformations. On the basis of the numerical solution, the dependences of homogenized Poisson's coefficients on macro-strains and the non-linear stress-strain diagrams for a material with randomly distributed unidirectional ellipsoidal pores are predicted and discussed for different volume fractions of pores. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).     

Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems

A sequential method is proposed to estimate boundary condition of the two-dimensional hyperbolic heat conduction problems. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton–Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of non-linear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. The example problem is used to demonstrate the characteristics of the proposed method. In the example, a well-known problem is used to demonstrate the validity of the proposed direct method and then the inverse solutions are evaluated. In the second example, the larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the boundary conditions in the two-dimensional inverse hyperbolic heat conduction problems.     

A complete characterization of the feedback-linearized input-output response

The problem of characterizing the feedback-linearized input–outputresponse of a single-input single-output nonlinear system viastatic state feedback is completely solved. The proposed approachreduces the problem of determining the structure of the mostgeneral linearized input–output response, together withthe respective control law, to that of solving a nonhomogeneoussystem of first-order partial differential equations. Necessaryand sufficient conditions for this system of equations to havea solution are established which are of an algebraic natureand henceforth easily checkable. Based on these conditions,the linearized input–output response is completely characterized,prior to determining the desired control law. Furthermore, thegeneral solution of this system of equations is derived whichleads to the determination of the general analytical expressionfor the control law that satisfies the input–output linearizationproblem.     

Regularity and convergence results for a phase–field model with memory

A phase–field model based on the Coleman–Gurtin heat flux law is considered. The resulting system of non-linear parabolic equations, associated with a set of initial and Neumann boundary conditions, is studied. Existence, uniqueness, and regularity results are proved. An asymptotic analysis is also carried out, in the case where the coefficient of the interfacial energy term tends to 0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

小垂度粘弹性索非线性响应及振动主动控制

研究具有初始应力的小垂度粘弹性索的非线性动态响应及振动主动控制。在假定索材料的本构关系为一般微分本构类型的基础上,建立小垂度粘弹性索的运动微分方程;应用Galerkin方法将其转化为可用Runge-Kutta数值积分方法求解的一系列三阶非线性常微分方程。在仅考虑面内的横向振动及忽略非线性的情况下得到了连续状态空间中的状态方程,将状态方程离散为差分方程形式,并用矩阵指数来逐步近似状态转移矩阵;基于二次性能指标的最小化得到了最优的控制力与状态向量。最后通过数值仿真研究说明了粘性参数对索动态响应的影响。     

Rational solutions of fifth-order evolutionary equations for describing waves on water

A method is proposed for obtaining the exact solutions of evolutionary equations in the form of a rational function. Invariant manifolds of the equations are used which have the same form of dependence on the required function and its derivatives as the generalized Riccati equations. Using fifth-order Kawahara and Korteweg–de Vries equations as an example, it is shown that their known particular solutions can be obtained using this method. New solutions of a non-linear fifth-order equation, which is encountered when describing long waves on water, are obtained.     

Nonaxisymmetric thermal stress state of laminated rotational bodies of orthotropic materials under nonisothermic loading

A procedure for numerical investigation of nonaxisymmetric temperature fields and the elastic stress-strain state of laminated rotational bodies of cylindrically and rectilinearly orthotropic materials under nonisothermal loading is proposed. The deformation of orthotropic materials is described by the equations of anisotropic elasticity theory. The equations of state are written in the form of Hookes law for homogeneous materials, with additional terms which take into account the thermal deformation, changes in the mechanical properties of materials in the circumferential direction, and their dependence on temperature. A semianalytic finite-element method in combination with the method of successive approximations is used. An algorithm for numerical solution of the corresponding nonlinear boundary problem is elaborated, which is realized as a package of applied FORTRAN programs. Some numerical results are presented.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 731–752, November–December, 2004.     

The possibility of a theoretical confirmation of the experimental values of the external critical pressure of thin-walled cylindrical shells

The problem of the buckling of elastic, isotropic, thin-walled cylindrical shells with small initial shape defects that are under the action of an external pressure is solved in a geometrically non-linear formulation. Equations that are identical to Marguerre's equations for a shallow cylindrical shell are used in formulating the problem. The solution is constructed by the Rayleigh–Ritz method with the points of the middle surface of the shell approximated by double functional sums over trigonometric and beam functions. The system of non-linear equations obtained is solved by arc-length methods. Cases of the clamped and supported shells when loading with a lateral and uniform hydrostatic pressure are considered. Its deflections from the limit points of the postbuckling branches of its loading trajectory are used as the initial imperfections. An inspection of the different forms of the initial imperfections when they have maximum values of up to 30% of the shell thickness made it possible to obtain practically the whole range of experimentally found critical pressures.     

A method of investigating weakly non-linear interaction between one-dimensional waves

A method of constructing asymptotic approximations of wide classes of solutions of weakly non-linear systems is proposed based on the averaging scheme developed in /1–3/.^****See also: Krylov A.V. and Shtaras A.L. Internal averaging of multidimensional weakly non-linear systems along characteristics, Dep. in LitNIINTI, 10.11.86, No.1750, 1986). The method enables one to obtain the conditions for the asymptotic decay of systems described by the Burgers, Korteweg-de Vries and similar scalar equations, and also enables one to investigate problems in which this decay does not occur. As an example we investigate the propagation of perturbations in an elastic non-uniform tube. The interaction between two waves is considered and the conditions for resonance are obtained.