Hopf Algebras and Integrable Flows Related to the Heisenberg–Weil Coalgebra

On the basis of the structure of Casimir elements associated with general Hopf algebras, we construct Liouville–Arnold integrable flows related to naturally induced Poisson structures on an arbitrary coalgebra and their deformations. Some interesting special cases, including coalgebra structures related to the oscillatory Heisenberg–Weil algebra and integrable Hamiltonian systems adjoint to them, are considered.     

On the creation of stable periodic behavior of parametrically excited dynamical systems

The problem of the realization of stable periodic behavior of dynamical systems is considered. It is shown analytically that in certain cases it is possible to achieve by parametric perturbation stable periodic behavior of systems that in the autonomous case possess only unstable oscillatory or stationary regimes or are in a stable equilibrium position.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 507–512, September, 1995.     

Oscillation properties for the equation of vibrating beam with irregular boundary conditions

In this paper we study the oscillatory properties for the eigenfunctions of some fourth-order eigenvalue problems, where the boundary conditions are irregular in the sense of the classification of [S. Janczewski, Oscillation theorems for the differential boundary value problems of the fourth order, Ann. of Math. 29 (1928) 521–542]. In this case, we show that these oscillatory properties are different from those of the Sturm–Liouville problem.     

基于修正的Magnus方法的高振荡动力系统的数值积分方法

基于建立于一般线性动力系统上的Magnus数值积分方法,针对随时间而高频率振荡的二阶动力系统,给出了有效的修正Magnus数值积分算法．首先,将二阶动力系统重新表示为一阶系统的形式,通过引进新变量进行参考坐标变换,使动力系统的高振荡性质保留在新形式内;进而基于局部线性化技术用修正的Magnus方法求解新形式下的系统方程;最后,通过一系列数值实验说明了文中方法的有效性．     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

Adiabatic invariants of nonlinear dynamic systems with small parameter

We develop a symplectic method of finding the adiabatic invariants of nonlinear dynamic systems with small parameter. We show that a necessary and sufficient condition for the existence of quasi-Hamiltonian adiabatic invariants of nonlinear dynamic systems with regular dependence on a small parameter is that the Cauchy problem be well-posed for an equation of Lax type in the class of nongradient local functionals on the cotangent manifold of the phase space. It is established that scalar nonlinear dynamic systems always have a priori complete evolution invariants, not only adiabatic invariants. We also consider typical applications in hydrodynamics and oscillatory systems of mathematical physics.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 179–185.     

Dynamical behavior of Lotka–Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

This article is concerned with the study of trajectory behavior of Lotka–Volterra competition bistable systems and systems with telegraph noises. We proved that for bistable systems, there exists a unique solution, bounded above and below by positive constants. The oscillatory situation of systems with telegraph noises is pointed out.     

Homogenization for semilinear hyperbolic systems with oscillatory data

The behavior of multi-dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple-scale initial data is also established.     

Long-Time Oscillatory Energy Conservation of Total Energy-Preserving Methods for Highly Oscillatory Hamiltonian Systems

For an integrator when applied to a highly oscillatory system,the near conservation of the oscillatory energy over long times is an important aspect.In this paper,we study the long-time near conservation of oscillatory energy for the adapted average vector field(AAVF)method when applied to highly oscillatory Hamiltonian systems.This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly.This paper is devoted to analysing another important property of AAVF method,i.e.,the near conservation of its oscillatory energy in a long term.The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion.A similar result of the method in the multi-frequency case is also presented in this paper.     

Quadrature methods for highly oscillatory linear and non-linear systems of ordinary differential equations: part II

In this paper we present efficient numerical approximation for systems of highly oscillatory ordinary differential equations with matrices of variable coefficients. We assume that the spectrum of the matrix is purely imaginary and the frequency of oscillation grows large. We develop the asymptotic and the Filon-type methods for linear systems with time dependent matrices and we integrate oscillatory quadrature rules with waveform relaxation methods employing the WRF method for non-linear systems. We solve matrix exponential in Lie groups employing Magnus expansion. The methods are illustrated in several numerical examples of interest.     

Preconditioned defect corrected averaging for parabolic PDEs

Modelling physical systems with fast moving components leads to PDEs with highly oscillatory sources. Often, the time scale of the oscillation is much below the scale of the interesting variables. Time integrators must follow the scale of the fast motion, leading to long simulation times. For mechanical systems, methods like heterogeneous multiscaling and stroboscopic averaging are quite satisfactory. In case of semidiscretized PDEs, their advantage is limited. Here, we derive a smooth source term which generates a solution that coincides with the solution of the oscillatory system in stroboscopic points. The derivation involves the solution of a linear system with the solution operator of the PDE being the linear operator. Several preconditioners are developed and compared for those systems. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Analysis of the Synchronization of Coupled Chemical Oscillators

A reaction–diffusion model describing a system of coupled oscillators is constructed and investigated. The oscillators in this study are chemical oscillators that represent an oscillatory heterogeneous catalytic reaction in a granular catalyst layer. The oscillators are arranged serially in the reagent stream and are coupled through the gaseous phase. The dynamic behavior of the system is investigated as a function of the main external parameter — the partial pressure of one of the reagents in the gaseous phase. Existence regions of regular and chaotic oscillations are identified. Synchronization conditions are established for the oscillations in such a chain of coupled chemical oscillators.     

Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10     

Controllability and observability of one-dimensional oscillations of mechanical systems with nested masses

Controllability and observability conditions are derived for a three-mass oscillatory system with nested masses. Modal controllers and observers are constructed that take the system to a prescribed spectral region. Cases are considered when the state vector function or a linear combination of its components is observed.Kiev University. Tadzhik University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 103–107, 1991.     

Calogero–FranÇoise Flows and Periodic Peakons

The completely integrable Hamiltonian systems discovered by Calogero and FranÇoise contain the finite-dimensional reductions of the Camassa–Holm and Hunter–Saxton equations. We show that the associated spectral problem has the same form as that of the periodic discrete Camassa–Holm equation. The flow is linearized by the Abel map on a hyperelliptic curve. For two-particle systems, which correspond to genus-1 curves, explicit solutions are obtained in terms of the Weierstrass elliptic functions.     

Multi-revolution composition methods for highly oscillatory differential equations

We introduce a new class of multi-revolution composition methods for the approximation of the \(N\) th-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods.     

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.     

Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: part I

This work presents methods of efficient numerical approximation for linear and nonlinear systems of highly oscillatory ordinary differential equations. We show how an appropriate choice of quadrature rule improves the accuracy of approximation as the frequency of oscillation grows. We present asymptotic and Filon-type methods to solve highly oscillatory linear systems of ODEs, and WRF method, representing a special combination of Filon-type methods and waveform relaxation methods, for nonlinear systems. Numerical examples support this paper. Dedicated to the memory of Rudolf Khanamiryan. AMS subject classification (2000)  65L05, 34E05, 34C15