Krylov-Bogoliubov Method Research Articles

Theoretical and experimental study of strain localization phenomenon based on phase transition theory

In the study of large-scale localized strain features, localized strain can lead to energy release and seismic effects. From the macroscopic process of localized strain, it is found that localized deformation is accompanied by structural weakening, which is due to phase changes in rocks. The phase change corresponds to the secondary phase transition process in physics. The present study illustrates this phenomenon from the perspective of secondary phase transition theory in statistical physics, combining the localized strain features on a large scale. Theoretical analysis and experimental studies were carried out using three brittle rocks, including marble, granite and red sandstone. First, two perturbation methods, i.e., Krylov–Bogoliubov method and Poincare method, are used to calculate the higher-order control equations in analytical model. The influences of control equation coefficients on the strain localization process are analyzed. Then the uniaxial compression tests are carried out on the three rocks to record the strain process. Finally, the theoretical and experimental results are compared to analyze the strain localization phenomenon. The comparison results show that both the Krylov–Bogoliubov solution and the Poincare solution in the analytical model can well describe the evolution characteristics of localized strain. It indicates that the theoretical model is valid and has high accuracy. The Poincare method better simulates the nonlinear phenomenon of strain localization. The present study provides a new theoretical method to better understand the strain localization phenomenon.

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin’s method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well.

Coupled nonlinear dynamics of acoustic modes in a quasi 1-D duct with inhomogeneous mean properties and mean flow

This study investigates the coupled nonlinear evolution of acoustic modes in a quasi 1-D duct with axially inhomogeneous mean velocity and mean thermodynamic properties. A novel modification of the Krylov–Bogoliubov method of averaging (KBMA) is developed to analytically solve the modal-amplitude equations that are linearly and nonlinearly coupled with both quadratic and cubic nonlinearities. Approximate analytical solutions based on the method of multiple scales (MMS) are also derived for the coupled modal-amplitude equations with cubic nonlinearities. The KBMA and MMS solutions are presented for the cases with and without internal resonance among the acoustic modes. Furthermore, novel initial conditions are derived for the KBMA and MMS modal amplitude and phase equations that are essential to validate the analytical solutions using numerical solutions. While both the analytical solutions are seen to be in good to excellent agreement with the numerical solutions, only KBMA captures the low-frequency oscillations in the peak amplitude of the limit cycle oscillations. The role of the order of nonlinear terms in the evolution to limiting oscillations is elucidated by successively considering coupled modal equations with both quadratic and cubic nonlinearities, only cubic nonlinearities and only quadratic nonlinearities. It is seen that while the cubic nonlinearities determine the asymptotic temporal behavior, the quadratic nonlinearities give rise to complex oscillation patterns characterized by a range of non-integral frequencies. For coupled oscillators with only quadratic nonlinearities, it is observed that for mean temperature gradients below a threshold, the pressure fluctuations are characterized by intermittent bursts of high-amplitude fluctuations, as well as by a frequency spectrum that is essentially continuous. The effects of linear coupling of the modal-amplitude equations on the limit-cycle behavior of the solutions are also illustrated.

Chaotic vibrations of size-dependent flexible rectangular plates.

A mathematical model describing nonlinear vibrations of size-dependent rectangular plates is proposed. The plates are treated as the Cosserat continuum with bounded rotations of their particles (pseudo-continuum). The governing partial differential equations (PDEs) and boundary/initial conditions are obtained using the von Kármán geometric relations, and they are yielded by the energetic Hamilton principle. The derived mixed-form PDEs are reduced to ordinary differential equations and algebraic equations (AEs) using (i) the Galerkin-Krylov-Bogoliubov method (GKBM) in higher approximations, and then they are solved with the help of a combination of the Runge-Kutta methods of the second and fourth order, (ii) the finite difference method (FDM), and (iii) the Newmark method. The convergence of FDM vs the interval of the space coordinate grids and of GKBM vs the number of employed terms of the approximating function is investigated. The latter approach allows for achieving reliable results by taking account of almost infinite-degree-of-freedom approximation to the regular and chaotic dynamics of the studied plates. The problem of stability loss of the size-dependent plates under harmonic load is also tackled.

KB method for obtaining an approximate solution of slowly varing amplitude and phase of nonlinear differential systems with varying coefficients

To determine an approximate solutions of damped nonlinear ordinary differential system with varying coefficients of oscillatory procedure is envisaged based on the Krylov-Bogoliubov (KB) method. Our aim is to this paper of Krylov and Bogoliubov (KB) for determining an approximate solution of a nonlinear differential system with varying coefficients. Finally, results are discussed, especially to enrich the physical prospects, and shown graphically by utilizing Mathematica and Mathlab software. However, in some cases it’s feasible to alternate nonlinear differential equations with an associated linear equation closely enough to give helpful results. Some expository examples are given to exhibit the suitability and proficiency considered method.

On invariant measures and the asymptotic behavior of a stochastic delayed SIRS epidemic model

In this paper, we consider a stochastic epidemic model with time delay and general incidence rate. We first prove the existence and uniqueness of the global positive solution. By using the Krylov–Bogoliubov method, we obtain the existence of invariant measures. Furthermore, we study a special case where the incidence rate is bilinear with distributed time delay. When the basic reproduction number R0<1, the analysis of the asymptotic behavior around the disease-free equilibrium E0 is provided while when R0>1, we prove that the invariant measure is unique and ergodic. The numerical simulations also validate our analytical results.

Analytical and computational studies of the nonlinear vibrations of SWCNTs embedded in viscous elastic matrix using KBM method.

The forced vibration analysis of single wall carbon nanotubes (SWCNTs) embedded in the viscous elastic matrix subjected to axial parametric excitation has been investigated. The Euler Bernoulli beam model of the non-local continuum theory is used. The resonant and non-resonant solutions are analytically studied using the Krylov Bogoliubov and Mitropolsky method. It has been seen that the amplitude remains constant up to the second order of approximation. The resonant solutions are also found to analyze the possibility of chaos in the neighborhood of resonance. The computational techniques are used, and plots of time series, phase plot, and Poincaré surface of section are also drawn to confirm the chaotic behavior for certain values of parameters of SWCNTs, which may lead the aging process in the SWCNTs after a long time.

When an oscillating center in an open system undergoes power law decay

We have probed the condition of periodic oscillation in a class of two variable nonlinear dynamical open systems modeled with Lienard-Levinson-Smith(LLS) equation which can be a limit cycle, center or a very slowly decaying center type oscillation. Using a variety of examples of open systems like Glycolytic oscillator, Lotka-Volterra(L-V) model, a generalised van der Pol oscillator and a time delayed nonlinear feedback oscillation as a non-autonomous system, each of which contains a family of periodic orbits, we have solved LLS systems in terms of a multi-scale perturbation theory using Krylov-Bogoliubov(K-B) method and it is utilised to characterise the size and shape of the limit cycle and center as well as the approach to their steady state dynamics. We have shown the condition when the average scaled radius of a center undergoes a power law decay with exponent 1/2.

Bubble shape oscillations of finite amplitude

Shape oscillations arising from the spherical instability of an oscillating bubble can be sustained in a stationary acoustic field. Describing such a steady state requires that nonlinear saturation effects are accounted for to counteract the natural exponential growth of the instability. In this paper, we analyse the establishment of finite-amplitude bubble shape oscillations as a consequence of nonlinear interactions between spherical and non-spherical modes. The set of coupled dynamical equations describing the volume pulsation and the shape oscillations is solved using a perturbation technique based on the Krylov–Bogoliubov method of averaging. A set of first-order differential equations governing the slowly varying amplitudes and phases of the different modes allows us to reproduce the exponential growth and subsequent nonlinear saturation of the most unstable, parametrically excited, shape mode. Solving these equations for steady-state conditions leads to analytical expressions of the modal amplitudes and derivations of the conditionally stable and absolutely stable thresholds for shape oscillations. The analysis of the solutions reveals the existence of a hysteretic behaviour, indicating that bubble shape oscillations could be sustained for acoustic pressures below the classical parametric threshold.

Dynamic Stability of a Thin Film Bonded to a Compliant Substrate Subjected to a Step Load with Damping

Abstract The flexible electronic structure is based on the buckling of a thin film on a compliant substrate. This paper studies the dynamic stability of this structure subjected to a uniaxial step load with damping. The equation of motion is derived by Hamilton’s principle and Euler-Lagrange equation. Through the qualitative analysis of the phase portraits of the dynamic equation, as well as the quantitative analysis of the responses according to Budiansky-Roth criterion, the critical dynamic load is determined. It is the same as that in static buckling. Affected by damping, at the stage of pre-buckling, the amplitude of the film vibrates and attenuates, where the maximum response is the initial amplitude. The structural damping can be derived by the logarithm of the ratio between two adjacent peaks of the vibration. At the stage of post-buckling, the amplitude of the film vibrates and tends to a stable response, which is the amplitude in static buckling. The upper and lower bounds of the post-buckling response are asymmetric and solved by modified Krylov-Bogoliubov method.

Stability analysis of rotor whirl under nonlinear internal friction by a general averaging approach

The two main sources of internal friction in a rotor-shaft system are the shaft structural hysteresis and the possible shrink-fit release of the assembly. The internal friction tends to destabilize the over-critical rotor running, but a remedy against this effect may be provided by a proper combination of some external damping in the supports and an anisotropic arrangement of the support stiffness, or at most by the support damping alone, depending on the system geometry. The present analysis reported here considers a general asymmetric rotor-shaft system, where the rotor is perfectly rigid and is constrained by viscous–flexible supports having different stiffnesses on two orthogonal planes. The internal friction is modelled by nonlinear Coulombian forces, which counteract the translational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are dealt with using an averaging approach based on the Krylov-Bogoliubov method with some adaptation to address the multi-degree-of-freedom nature of the problem. Stable limit cycles may be attained by the overcritical whirling motions, whose amplitudes are inversely proportional to the external dissipation applied by the supports. A noteworthy result is that the stiffness anisotropy of the supports is recognized as beneficial in reducing the natural whirl amplitudes, albeit mainly in the symmetric configuration of the rotor at the mid span and, to a rather lesser extent, in the asymmetric configuration, which then requires a stronger damping action in the supports.

On the limits of applicability of an approximate formula for the surface impedance of a rectangular groove

The problem of determination of the equivalent surface impedance of a rectangular groove is considered. The solution is obtained by the integral equations method. The Krylov–Bogoliubov method is used to solve integral equations numerically. The obtained dependence of the impedance value on the groove depth is compared to the dependence calculated using an approximate formula for different groove widths. The limits of applicability of the approximate formula are determined.

Generalization of the Krylov-Bogoliubov Method for Nonlinear Oscillators

In this paper the Krylov-Bogoliubov method for solving nonlinear oscillators is considered. Based on the original method, developed for the oscillator with small nonlinearity, a generalization is made to oscillators with strong nonlinearity. After rewriting the equation into two first order differential equations, the averaging procedure is introduced. Truly nonlinear differential equations are investigated where the linear term does not exist nor the linearization of the equation is possible. Solution is assumed in the form of the Ateb-function. After averaging the approximate solution for the oscillator is obtained. A numerical example is tested. It is shown that the difference between the analytical approximate solution and the exact numerical one is negligible.

Nonlinear effects in dynamics of cylindrical resonator of wave solid-state gyro with electrostatic control system

A new mathematical model of wave solid-state gyro is constructed, which describes interrelated electrical and mechanical oscillations in the case when the voltage is available at the electrodes. Wave pattern of resonator oscillations was studied using asymptotic Krylov-Bogoliubov method. Nonlinear electric processes in resonator control loop lead to additional gyro errors. A numerical example is provided.

Regular and chaotic dynamics of a matter-wave soliton near the atomic mirror

The dynamics of the soliton in a self-attractive Bose–Einstein condensate under the gravity are investigated. First, we apply the inverse scattering method, which gives rise to equation of motion for the center-of-mass coordinate of the soliton. We analyze the amplitude-frequency characteristic for nonlinear resonance. Applying the Krylov–Bogoliubov method for the small parameters the dynamics of soliton on the phase plane are considered. Hamiltonian chaos under the action of the gravity on the Poincaré map are studied.

Equivalent surface impedance of an infinite periodic array of slot impedance loads based on a semicylinder cavity

A problem of finding the equivalent surface impedance of an infinite periodic array of slot impedance loads based on a semicylinder cavity. The problem is solved by the integral-equation method. As the numerical method for solving the integral equation, the Krylov-Bogoliubov method is used. The results of the numerical experiment are presented in the form of dependences of the impedance on the cavity radius, slot width, conductor width, and angle of electromagnetic-wave incidence. The angular dependences of the impedance are compared to the earlier results obtained for an array of rectangular grooves and a single impedance load based on a semicylinder cavity.

Performance of galloping piezoelectric energy harvesters

A galloping piezoelectric energy harvester is composed of a cantilevered piezoelectric beam and a tip bluff body. Self-excited vibration is induced when the tip bluff body is subjected to airflow. Then, the piezoelectric materials can convert the mechanical energy into electrical energy. In this article, a coupled aero-electro-mechanical model is developed to analyze a galloping piezoelectric energy harvester. A dimensionless formulation is adopted so that it is very convenient to conduct scaling and performance comparison of different galloping piezoelectric devices. Analytical approximate solution approach is employed to solve these coupled nonlinear equations using the Krylov–Bogoliubov method. Tip bluff body is assumed to have either rectangular or square section. System parameters such as galloping velocity, limit cycle oscillation amplitude, and harvested energy are determined accordingly and presented in an explicit form. Performances of galloping piezoelectric energy harvesters with different tip bluff bodies and electrical loads are characterized. In addition, experimental data are used to validate our predictions with good agreement. The galloping piezoelectric energy harvester having square section shows better performance compared to the one with rectangular section. Furthermore, the performance can be substantially improved by exploring the inherent jump phenomenon as observed in the limit cycle oscillation hysteresis response of a galloping piezoelectric energy harvester with square bluff body.

A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

We introduce a gyro-gauge independent formulation of a simplified guiding-center reduction, which removes the fast time-scale from particle dynamics by Lie-transforming the velocity vector field. This is close to Krylov-Bogoliubov method of averaging the equations of motion, although more geometric. At leading order, the Lie-transform consists in the generator of Larmor gyration, which can be explicitly inverted, while working with gauge-independent coordinates and operators, by using the physical gyro-angle as a (constrained) coordinate. This brings both the change of coordinates and the reduced dynamics of the minimal guiding-center reduction order by order in a Larmor radius expansion. The procedure is algorithmic and the reduction is systematically derived up to full second order, in a more straightforward way than when Lie-transforming the phase-space Lagrangian or averaging the equations of motion. The results write up some structures in the guiding-center expansion. Extensions and limitations of the method are considered.

A generalized van der Pol type oscillator: Investigation of the properties of its limit cycle

A generalized van der Pol type oscillator is considered, with power-form nonlinearities in the restoring and damping force. The amplitude of the limit cycle is obtained by applying the generalized Krylov–Bogoliubov method for purely nonlinear oscillators. The influence of the values of the powers of the damping and the restoring force on this amplitude is investigated both analytically and numerically. The limiting values of this amplitude are found and the time for the amplitude to ‘nearly’ reach the limit cycle is estimated.

Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg–Landau equation with degenerate noise

In this paper, we consider the ergodicity of invariant measures for the stochastic Ginzburg–Landau equation with degenerate random forcing. First, we show the existence and pathwise uniqueness of strong solutions with H 1 -initial data, and then the existence of an invariant measure for the Feller semigroup by the Krylov–Bogoliubov method. Then in the case of degenerate additive noise, using the notion of asymptotically strong Feller property, we prove the uniqueness of invariant measures for the transition semigroup.