The attractor of a Navier-Stokes system in an unbounded channel-like domain

The Navier-Stokes system describes a flow of a fluid in an unbounded planar channel-like domain. It is proved that in the case when an external force decays at infinity, the semigroup generated by this system has a global attractor and its Hausdorff dimension is finite. Estimates in weighted Sobolev spaces are used as a main tool. Asymptotics, as the distance from the origin in the plane tends to infinity, of functions on the attactor is found. This asymptotics show that all dynamics on the attractor decays at infinity and the turbulence generated by the force does not propagate to infinity.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden attractor in the case of multistability as well as a classical self-excited attractor. The hidden attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden attractor and hidden transient chaotic set in the case of multistability are given.     

Existence of attractor and control of a 3D differential system

This paper analyzes the orbit of a three-dimensional differential system based on the Shilnikov criterion. It also applies the Shilnikov method of constructing a heteroclinic connection between saddle focus equilibrium points of the system, proving that the system possesses “horseshoe” chaos. In addition, adaptive backstepping design is used to control this system with three unknown key parameters, and an algorithm of this controller is presented. Finally, we give some numerical simulation studies of the system in order to verify the analytic results.     

Generation of amplitude death and rhythmogenesis in coupled hidden attractor system with experimental demonstration

Nonlinear Dynamics - Third-order nonlinear dynamical systems with attractors (one with no fixed point and the other with a stable fixed point) are conjugately coupled. It is observed that the...     

聊聊动态强度和损伤演化

材料强度在传统上常理解为材料在外载荷下抵抗流动/变形和破断的能力。由流变阶段到貌似突发的破断,其实源于一个隐含的应变率/时间相关的损伤演化过程。动态损伤演化研究的难点在于损伤与流变总是耦合在一起发展的。研究发现,热激活损伤演化模型可成功描述材料宏观损伤的动态演化。在此基础上,从实测的含损伤演化的表观应力应变曲线,可将两者解耦分开,并可确定各自相关的材料参数。这一思路可推广到中医脉诊的客观化研究,通过脉搏波信息定量反演脉搏波系统的正常及病态本构关系,可诊断生命体偏离正常状态的程度(病情),这可视为一种广义的损伤演化和强度问题。上述思路还可推广到地震预报研究,即“对地球把脉”。与加卸载响应比理论相结合,通过相邻3处的地震波信息来反演地球相关板块含损伤演化的非线性载荷-响应曲线,再区分出损伤演化程度,将有利于改进地震预报,这可视为另一种广义的损伤演化和强度问题。     

Generation and evolution of cavitation in magma under dynamic unloading

The full system of equations for the problem of rarefaction-wave passage over the magma-melt column in the gravity field is derived with the use of the kinetic theory of phase transformations, and the problem is numerically solved. With allowance for diffusion zones and nucleation frequency as a function of supersaturation, the dependence of the number of cavitation nuclei formed in the course of phase transformations behind the rarefaction-wave front is found. The dynamics of the size distribution of cavitation bubbles along the magma-melt column (1 km) whose viscosity varies dynamically as a function of the concentration of dissolved water is studied.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 71–80, March–April, 2005.     

Bifurcation of attractor in the falling film problem

Film flows down a vertical plane, the so-called falling films, are covered by waves whose lengths are longer than a threshold value corresponding to neutral stability of the waveless flow. On the other hand, experiments also demonstrate existence of the upper threshold wavelength in unstable interval. This phenomenon is explained through careful analysis of dynamical system modelling the wave dynamics in falling films.     

Dynamics analysis and Hamilton energy control of a generalized Lorenz system with hidden attractor

Nonlinear Dynamics - Hidden attractor can be found in some dynamic systems. More commonly, it can be excited by the stabilized equilibria, or be generated from the systems without equilibria. The...     

Nonlinear dynamic response of an isolation system with superelastic hysteresis and negative stiffness

Nonlinear Dynamics - The negative stiffness exhibited by bi-stable mechanisms together with the tunable superelasticity offered by shape memory alloy (SMA) wires can enhance the dynamic resilience...     

Nonlinear dynamic analysis on rigid-flexible coupling system of an elastic beam

Previous work examined the effect of the attached stiffness matrix terms on stability of an elastic beam undergoing prescribed large overall motion. The aim of the present work is to extend the nonlinear formulations to an elastic beam with free large overall motion. Based on initial stress method, the nonlinear coupling equations of elastic beams are obtained with free large overall motion and the attached stiffness matrix is derived by solving sub-static formulation. The angular velocity and the tip deformation of the elastic pendulum are calculated. The analytical results show that the simulation results of the present model are tabled and coincide with the one-order approximate model. It is shown that the simulation results accord with energy conservation principle.     

Reconstruction and prediction of the in-cylinder pressure attractor of an internal combustion engine using locally constant models

In this research, the auto-mutual information function and correlation dimension are used for determination of lag and embedding dimension needed for reconstruction of the attractor of in-cylinder pressure of an internal combustion engine. Subsequently, a locally constant model is used for a 20-step prediction. The time series are acquired at three different test conditions and consisting of succeeding pressures at compression Top Dead Center (TDC) position of one of four cylinders. We have concluded that at least at relatively low engine speeds (below 2000 rpm) this method produces acceptable results.     

Nonlinear dynamic evolution and control in a new scale-free networks modeling

The nonlinear evolving and controlling in complex networks are an important way to understand the dynamic mechanism for real networks. In order to explore universality of scale-free systems, we propose an extended network model based on Barabási–Albert model by developing and decaying networks. The novel network evolves by growing and optimizing processes, such as the addition of new nodes and edges, or deletion of edges at every time step. Meanwhile, in order to describe more realistic phenomena of reality, we introduce the fitness to reflect the competition and local event of inner anti-preferential mechanism to delete the edges. We calculate analytically the degree distribution and find that the Barabási–Albert model is only one of its special cases and the model self-organizes into scale-free networks, moreover, the numerical simulations are in good agreement with the analytical conclusions. The results imply that this extended model has more comprehensive and universal simulation and reflection in complex network topology characters and evolution with practices and applications.     

Linear response of an elastic plate to actual random load

Summary Generalized harmonic analysis was applied to solve the linear response problem of vibrating elastic plates with damping under stochastic pressure. The power spectral density of the external pressure was taken to be of a typical peaked shape resembling actual physical conditions. The spatial correlation of pressure over the plate area was taken to be unity. Results give the cross-correlation functions for the acceleration and stresses in terms of the mean-square value of the load and the characteristic constants used to describe the power spectral density curve. For the centres of rectangular plates with simply-supported and built-in edges dimensionless response autocorrelation functions are plotted.

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Übersicht Mit Hilfe der verallgemeinerten Fourier-Analyse werden lineare Plattenschwingungen mit Dämpfung bei stochastischer Erregung behandelt. Die Spektraldichte der Druckerregung wird dabei in einer besonderen Form (mit ausgeprägtem Maximum) angenommen, wie sie oft bei tatsächlichen technischen Problemen vorhanden ist. Die Raumkorrelation des Drucks über die Plattenoberfläche sei Eins. Lösungen des Problems werden numerisch ausgearbeitet bis zu den Kreuzkorrelationsfunktionen für die Beschleunigung und die Biegespannungen, wobei der quadratische Mittelwert des Erregerdrucks und einige zur Beschreibung der speziellen Spektraldichte nötigen Kennwerte verwendet werden. Für die Mittelpunkte von Rechteckplatten mit allseitig-drehbaren und allseitig-eingespannten Rändern sind die Autokorrelationsfunktionen dimensionslos in Diagrammen dargestellt.

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On leave from Staatliche Ingenieurschule Karlsruhe, Germany.

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Paper presented at the 21st Annual General Meeting of the Aeronautical Society of India, Madras, April 1969.     

Homoclinic bifurcation and chaos attractor in elastic two-bar truss

In this paper, we present a dynamic bifurcation analysis of the non-linear Duffing's equation on a simple elastic structure. The structure is a two-bar elastic truss with a damper, and possesses geometrical non-linear stiffness. We consider the dynamic instability of its structure based on Duffing's oscillation, which shows bifurcation behavior of the homoclinic orbit. We could numerically forecast the trajectory near the invariant saddle point of homoclinic bifurcation on this model, and we found that it is possible to solve dynamic bifurcation and strange attractors (chaos) on this non-linear structure. On this truss, we could investigate the dynamic stability of the strange attractor using Lyapunov exponents under the frequency and/or the amplitude parameter of periodic load.     

Simulation of damage evolution in ductile metals undergoing dynamic loading conditions

A set of constitutive equations for large rate-dependent elastic-plastic-damage materials at elevated temperatures is presented to be able to analyze adiabatic high strain rate deformation processes for a wide range of stress triaxialities. The model is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative approach is proposed. It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. In addition, the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure is discussed.Furthermore, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor-elastic corrector technique. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. Numerical simulations of dynamic experiments with different specimens are performed and good correlation of numerical results and published experimental data is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson-Bar experiments.     

TURBULENT COHERENT STRUCTURE AND DYNAMIC SYSTEM

The main difficulties in the study of turbulence via dynamic system lie in how to relatecontinuum systems of infinite dimension with dynamic system in low dimension space andhow to depict its special structure.In this paper,we’ll give a comprehensive review onvarious methods to describe complex systems in low dimension space and new approaches tothe resolution of turbulence problems.