Exponential attractors for a generalized ginzburg-landau equation

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Convergence of attractors

I.IntroductionConsidersystemofcoupledoscillatorsi~Ax j(x)(l.;l)inwhichx~(xl,x',..',x")CRe(m>3),Aisnlxinmatrix,fisanonlinearmapfromR-toRewithconstantb>0.(l.l)isdiscretizedbyone-stepmethod:xo I~x. h(Ax" j(x.))(l'2)withconstantstepsizeh>0.LetF4denotethemapfr…     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

Lower bounds for the Hausdorff dimension of attractors

The existence of certainm-dimensional structures in a dynamical system implies that the Hausdorff dimension of its attractor is at leastm+1. A Bendixson criterion for the nonexistence of periodic orbits for systems in Hilbert spaces is found.     

H 2-regularity random attractors of stochastic non-Newtonian fluids with multiplicative noise

In this paper, the authors study the long time behavior of solutions to stochastic non-Newtonian fluids in a two-dimensional bounded domain, and prove the existence of H2-regularity random attractor.     

Dimensions of attractors for discretizations for Navier-Stokes equations

In this paper, we discretize the 2-D incompressible Navier-Stokes equations with the periodic boundary condition by the finite difference method. We prove that with a shift for discretization, the global solutions exist. After proving some discrete Sobolev inequalities in the sense of finite differences, we prove the existence of the global attractors of the discretized system, and we estimate the upper bounds for the Hausdorff and the fractal dimensions of the attractors. These bounds are indepent of the mesh sizes and are considerably close to those of the continuous version.     

Impulsive control of chaotic attractors in nonlinear chaotic systems

Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented. By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, andwith it, the upper bond of the impulse interval for asymptotically stable control was given.Numerical results are presented, which are considered with important reference value for control of chaotic attractors.     

Practical Evaluation of Invariant Measures for the Chaotic Response of a Two-Frequency Excited Mechanical Oscillator

This paper presents results which characterize the chaotic response of alow-dimensional mechanical oscillator. An experimental system based on acart rolling on a two-well potential surface has been shown to closelyapproximate a modified form of Duffing's equation. Two-frequency forcingis applied, providing a useful means of varying the dimension of theresponse. Computation of correlation dimension and Lyapunov spectra areperformed on both experimental and numerical data in order to assess theutility of these measures in a practical setting. A specific focus isthe distinction between subharmonic and quasi-periodic forcing, sincethis has a subtle, and interesting, effect on the subsequent dynamics.The results tend to highlight the statistical nature of the measures andthe caution that should be used in their interpretation.     

Limit cycles,tori, and complex dynamics in a second-order differential equation with delayed negative feedback

We analyze a second-order, nonlinear delay-differential equation with negative feedback. The characteristic equation for the linear stability of the equilibrium is completely solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The bifurcations occurring as the linear stability is lost are investigated by the construction of a center manifold: The nature of Hopf bifurcations and more degenerate, higher-codimension bifurcations are explicitly determined.     

An infinite 2-D lattice of strange attractors

Nonlinear Dynamics - We investigate a (2+1)-dimensional coupled nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation, and derive analytical vector...     

Invariant Set and Attractor of Nonautonomous Functional Differential Systems: A Decomposition Approach

In this paper, the invariant set and attractor are addressed for the nonautonomous functional differential systems. An estimation of the existence range of the invariant set and attractor are given by using a decomposition approach and the properties of nonnegative matrices. In addition, an example is given to denote the application of the new results.     

An example of a quasiconvex function that is not polyconvex in two dimensions

We study the different notions of convexity for the function f () = ||² (||² – 2 det ) where ^2×2, introduced by Dacorogna & Marcellini. We show that f is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦¦ 2/3 2, 1, 1+ (for some >0), 2/3, respectively.     

Singularity structure of the Hopf bifurcation surface of a differential equation with two delays

We investigated the structure of the so-called first Hopf bifurcation surface associated to a differential equation with two time delays. A geometrical approach leading naturally to a number theoretic approach provides rigourous results which are corroborated by previous numerical and experimental (optical compound resonator) results.     

An exact numerical method of calculating inviscid heated flows in two dimensions with an example of duct flow

Summary A twodimensional flow problem with heat addition can be expressed in terms of five parameters (pressure p, density , flow speed u, flow direction , rate of heating q) which must satisfy four equations (continuity, two components of momentum, and energy). It is shown how the equations become particularly simple, being linear and hyperbolic, if is specified and solutions obtained for the other four variables. An example is given of the flow through a supersonic combustion chamber.

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Übersicht Zweidimensionale Strömungsprobleme mit Wärmezufuhr können mit Hilfe von 5 Größen (Druck p, Dichte , Strömungsgeschwindigkeit u, Strömungsneigung , Wärmezufuhr q) formuliert werden, die 4 Gleichungen erfüllen müssen (Erhaltungssätze für Masse, Energie und zwei Komponenten des Impulses). Es wird gezeigt, daß die Gleichungen besonders einfach werden, nämlich linear und hyperbolisch, wenn vorgegeben wird und Lösungen für die andern 4 Veränderlichen bestimmt werden. Als Beispiel wird die Überschallströmung in einer Brennkammer behandelt.

     

An invariant-manifold analysis of electrophoretic traveling waves

A new, geometric proof of a theorem of Fife, Palusinski, and Su on electrophoretic traveling waves is presented. The proof is based upon the perturbation theory for invariant manifolds due to Fenichel. The results proved here reproduce the existence, uniqueness, and asymptotic approximation theorem proved by Fifeet al. The proof given here is substantially simpler, and in addition, it provides additional insight into the geometric structure of the phase space of the traveling wave equations for this system.     

Crack in a 2D beam lattice: Analytical solutions for two bending modes

We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass-spring lattices. Considering the static problems we use the discrete Fourier transform and the Wiener-Hopf technique. Monotonically distributed bending moments ahead of the crack are determined for the symmetric mode, and a self-equilibrated transverse force distribution is found for the antisymmetric mode. It is shown that in the latter case only the crack-front beam resists to the fracture development, whereas the forces in the other beams facilitate the fracture. In this way, the macrolevel fracture energy is determined in terms of the material strength. The macrolevel energy release is found to be much greater than the critical strain energy of the beam, especially in the hyper-singular mode. In both problems, it is found that among the beams surrounding the crack the crack-front beam is maximally stressed, and hence its strength defines the strength of the structure.     

两系非线性悬挂车辆的运行稳定性与分叉

本文选取两系具有滞后非线性悬挂的车辆为目标，建立其数学模型和运动微分方程，用常微分方程稳定性理论对车辆蛇行运动进行理论分析，并应用分叉理论研究了整车在蛇行失稳后的动力学行为，得出蛇行运动的分叉解及稳定判据，得到防止车辆蛇行运动的充分条件，并研究了系统参数对临界速度的影响、分叉解振幅及稳定性的影响，为车辆设计和参数选取提供依据。     

Codimension two bifurcation of a vibro-bounce system

A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.The project supported by the National Natural Science Foundation of China (10172042, 50475109) and the Natural Science Foundation of Gansu Province Government of China (ZS-031-A25-007-Z (key item))     

Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.     

A Semilinear Reaction–Diffusion System of Equations and Large Diffusion

The long time dynamics for a semilinear system of reaction and diffusion equations with nonlinear boundary conditions in which large diffusion is assumed on all parts of the domain is studied. We show in both local and global dynamics of the system that flows on attractors are essentially close to the ones of a finite dimensional system of equations, which turn out to be the natural limit of the process for large diffusivity.