The weak damped oscillation in a random medium

The paper gives a solution of the random differential equation with random coefficient, that is , where K (t) is a random process and is a damp coefficient, it is a little parameter.     

A scaling law near the primary resonance of the quasiperiodic Mathieu equation

We analyze the quasiperiodic damped Mathieu equation

The study on the chaotic motion of a nonlinear dynamic system

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On Asymptotic Properties of Continuously Differentiable Solutions of Quasilinear Functional Differential Equations

We establish new properties of C ¹ [τ(1), + ∞)-solutions of the quasilinear functional differential equation

A spectral resolving method for analyzing linear random vibrations with variable parameters

This paper is a development of ref. [1]. Consider the following random equation: Z(t)+2βZ(t)+ω₀²Z(t)=(a⁰+a¹Z(t))I(t)+c in which excitation I(t) and response Z(y) are both random processes, and it is proposed that they are mutually independent. Suppose that a(t) is a known function of time and I(t) is a stationary random process. In this paper, the spectral resolving form of the random equation stated above, the numerical solving method and the solutions in some special cases are considered.     

Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable,Hyperbolic Case

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}(t)&=f(\theta ,x(t),\epsilon x(t-\tau (x(t))))\\ \dot{\theta }(t)&=\omega . \end{aligned} \right. \end{aligned}$$

Under the assumption of exponential dichotomies for the \(\epsilon =0\) case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

     

The Damped Nonlinear Quasiperiodic Mathieu Equation Near 2:2:1 Resonance

We investigate the damped cubic nonlinear quasiperiodic Mathieu equation $$ \frac{d^2x}{dt^2}+(\delta+\varepsilon \cos t+\varepsilon \mu \cos\omega t)x+\varepsilon \mu c\frac{dx}{dt}+\varepsilon \mu \gamma x^3=0$$ in the vicinity of the principal 2:2:1 resonance. By using a double perturbation method which assumes that both ε and μ are small, we approximate analytical conditions for the existence and bifurcation of nonlinear quasiperiodic motions in the neighborhood of the middle of the principal instability region associated with 2:2:1 resonance. The effect of damping and nonlinearity on the resonant quasiperiodic motions of the quasiperiodic Mathieu equation is also provided. We show that the existence of quasiperiodic solutions does not depend upon the nonlinearity coefficient γ, whereas the amplitude of the associated quasiperiodic motion does depend on γ.     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

2:2:1 Resonance in the Quasiperiodic Mathieu Equation

In this work, we investigate regions of stability in the vicinity of 2:2:1 resonance in the quasiperiodic Mathieu equation $$\frac{{d^2 x}}{{dt^2 }} + \left( {\delta + \varepsilon \cos t + \varepsilon \mu \cos \left( {1 + \varepsilon \Delta } \right)t} \right)x = 0,$$ using two successive perturbation methods. The parameters ∈ andμ are assumed to be small. The parameter ∈ serves forderiving the corresponding slow flow differential system and μserves to implement a second perturbation analysis on the slow flowsystem near its proper resonance. This strategy allows us to obtainanalytical expressions for the transition curves in the resonantquasiperiodic Mathieu equation. We compare the analytical results withthose of direct numerical integration. This work has application toparametrically excited systems in which there are two periodicdrivers, each with frequency close to twice the frequency of theunforced system.     

Experimentelle und numerische Untersuchung laminarer,axisymmetrischer Freistrahlen mit und ohne Auftrieb

Zusammenfassung Das Verhalten axisymmetrischer, laminarer und inkompressibler Freistrahlen mit und ohne Auftrieb in einer homogenen Umgebung wird experimentell und numerisch untersucht. Die dazu erstellte Versuchsanlage wird kurz beschrieben. Charakteristische Grö\en von Fluidstrahlen ohne Auftrieb lassen sich unter Beachtung der beschreibenden Kenngrö\en parameterfrei darstellen. Auftriebsbehaftete Fluidstrahlen werden durch drei Parameter, die Reynoldszahl, die Grashofzahl und die Prandtlzahl vollständig beschrieben. Die Einflüsse der einzelnen Grö\en werden anhand der numerischen Lösung diskutiert, welche ihrerseits mit asymptotischen Verfahren kontrolliert wird. Die experimentellen Ergebnisse stimmen mit den berechneten Werten sehr gut überein. Die Versuche zur Stabilität laminarer Strahlen lassen sich gut mit einem Impulsstromparameter korrelieren.

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Experimental and numerical study of laminar, axisymmetrical jets with and without buoyancy

Axisymmetrical, laminar and incompressible jets with and without buoyancy in homogeneous surroundings are investigated experimentally and numerically. The experimental set up is described. Characteristics of jets without buoyancy are presented in a parameterless form. Buoyancy — induced jets are completely determined by three parameters, the Reynolds-Number, the Grashof-Number and the Prandtl-Number. The influence of the characteristic numbers to the numerical solution is discussed. On the other hand this result is controlled by analytical solutions. The experimental results are in good agreement with the predicted values. The experiments for stability of laminar jets are correlated with a parameter of momentum.

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Formelzeichen a Temperaturleitfähigkeit - a₁...a₄ Polynomkoeffizienten - b Breite - c_p spez. Wärmekapazität bei konstantem Druck - D Durchmesser - E kinetische Energie - g Erdbeschleunigung - l Länge - L laminare Lauflänge - m Massenstrom - p Parameter - Q Energiestrom - R Radius - T Temperatur - Geschwindigkeitsvektor - dimensionsloser Geschwindigkeitsvektor - U Parameter - Längenvektor - x_o Korrekturlänge - X Parameter - \ thermischer Ausdehnungsbeiwert - Grenzschichtdicke - dynamische Viskosität - T=T_o-T übertemperatur - =(T-T)/T dimensionslose Temperatur - kinematische Viskosität - dimensionsloser Längenvektor - Dichte - Stromfunktion - Funktion - dimensionslose Stromfunktion Indizes A au\en - I innen - m Werte auf Symmetrieachse - Th Thermoelement - – Mittelwert - o Düsenaustrittsgrö\en - Umgebung - Vektor - * dimensionslose Grö\en Kenngrö\en Re =u_o·R/v Reynoldszahl (Radius) - Reynoldszahl (Durchmesser) - Pr=/a Prandtlzahl - Pe =u_o·R/a Pécletzahl - Grashofzahl     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

On Lyapunov Exponents for Non-Smooth Dynamical Systems with an Application to a Pendulum with Dry Friction

We consider dynamical systems from mechanics for which, due to some non-smooth friction effects, Oseledets' Multiplicative Ergodic Theorem cannot be applied canonically to define Lyapunov exponents. For general non-smooth systems which fit into a natural formal framework, we construct a suitable cocycle which lives on a good invariant set of full Lebesgue measure. Afterwards, this construction is applied to investigate a pendulum with dry friction, described through the equation . The Lyapunov exponents obtained by our construction show a good agreement with the dynamical behaviour of the system, and since we will prove that these Lyapunov exponents are always non-positive, we conclude that the system does not show chaotic behaviour.     

2:1 Resonance in the delayed nonlinear Mathieu equation

We investigate the dynamics of a delayed nonlinear Mathieu equation: $$\ddot{x}+(\delta+\varepsilon\alpha\,\cos t)x +\varepsilon\gamma x^3=\varepsilon\beta x(t-T)$$ in the neighborhood of δ = 1/4. Three different phenomena are combined in this system: 2:1 parametric resonance, cubic nonlinearity, and delay. The method of averaging (valid for small ?) is used to obtain a slow flow that is analyzed for stability and bifurcations. We show that the 2:1 instability region associated with parametric excitation can be eliminated for sufficiently large delay amplitudes β, and for appropriately chosen time delays T. We also show that adding delay to an undamped parametrically excited system may introduce effective damping.     

Almost Periodicity in an Ecological Model with <Emphasis Type="Italic">M</Emphasis>-Predators and <Emphasis Type="Italic">N</Emphasis>-Preys by “Pure-Delay Typ” System

By using comparison theorem and constructing suitable Lyapunov functional, we study the following periodic Lotka–Volterra model with M-predators and N-preys by pure-delay type

An Elliptic Lindstedt--Poincaré Method for Certain Strongly Non-Linear Oscillators

An elliptic Lindstedt--Poincaré (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form , in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with are studied in detail.     

Homoclinic Connections in Strongly Self-Excited Nonlinear Oscillators: The Melnikov Function and the Elliptic Lindstedt–Poincaré Method

A criterion to predict bifurcation of homoclinic orbits instrongly nonlinear self-excited one-degree-of-freedom oscillator

On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space

We prove the following statement: Theorem 1. Let E and be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every and > 0, there exists a continuous mapping such that

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRⁿ\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.     

Steady States of Anisotropic Generalized Newtonian Fluids

We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotropic dissipative potential f. More precisely, we are looking for a solution of the following system of nonlinear partial differential equations