Parametric oscillations of nonlinear flutter systems

We consider a special class of abstract nonlinear equations of hyperbolic type that contains the main boundary value problems of panel flutter theory. For these equations, we study the existence and stability of parametric oscillations bifurcating from the zero equilibrium in the case of small damping.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant tori in nonlinear oscillations

The boundedness of all the solutions for semilinear Duffing equationx″ + ω² x + φ(x) =p(t), ω ∈ ℝ⁺ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded.     

On forced nonlinear oscillations

The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$ and

$\text{lim sup}\text{|x|?+∞}\text{|f(t,x)|}\text{| x |}\text{<}\text{4π}{}^2\text{2π + 1}$

(ii) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$. Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases.     

Linearized oscillations in nonlinear delay difference equations

Consider the nonlinear delay difference equation We establish a linearized oscillation result of this equation, which is the extension of the result in the paper [1]. Supported by the National Natural Science Foundation of China     

Forced nonlinear oscillations of elastic membranes

An initial-boundary-value problem is considered for the spatially two-dimensional damped Boussinesq equation with a forcing term. The problem in question is argued to serve as a model for describing small nonlinear oscillations of a circular membrane under the influence of acoustic pressure. Eigenfunction expansion method is used for constructing the global-in-time solution of the problem in question. Existence and uniqueness follow from the construction. Long-time asymptotics is computed on the basis of the obtained series representation. Numerical simulations are conducted. The algorithm shows excellent convergence properties.     

On nonlinear oscillations in a suspension bridge system

In this paper, we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations. By applying the Leray-Schauder degree theory, it is proved that the suspension bridge system has at least two solutions, one is a near-equilibrium oscillation, and the other is a large amplitude oscillation.

     

Nonlinear convective oscillations in multilayer systems

The joint action of buoyancy and the thermocapillary effect in two- and three-layer systems, is studied. Two types of boundary conditions, periodic boundary conditions and rigid heat–insulated lateral walls, are considered. In the case of periodic boundary conditions, regimes of traveling waves have been found. In the case of rigid heat-insulated lateral walls, symmetric and asymmetric standing waves are observed. Transitions between different types of oscillatory motions are studied. The competition between buoyancy and the thermocapillary effect in a three-layer system may lead to the complex dynamics and different unexpected effects. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error bounds of the finite-difference method in the problem of approximate construction of forced oscillations in nonlinear systems

The study considers the finite-difference method in the problem of approximate construction of forced oscillation modes in nonlinear systems described by differential equations of the form with a constant matrix A and a nonlinearity f T-periodic in t. A guaranteed bound on the error of the method is derived for the case of a weak nonlinearity f.The study was financially supported by the Russian Foundation of Basic Research (93-012-869).Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 81–85, 1993.