Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

Resonance,Parameter Estimation,and Modal Interactions in a Strongly Nonlinear Benchtop Oscillator

We study the vibrations of a strongly nonlinear, electromechanically forced, benchtop experimental oscillator. We consciously avoid first-principles derivations of the governing equations, with an eye towards more complex practical applications where such derivations are difficult. Instead, we spend our effort in using simple insights from the subject of nonlinear oscillations to develop a quantitatively accurate model for the single-mode resonant behavior of our oscillator. In particular, we assume an SDOF model for the oscillator; and develop a structure for, and estimate the parameters of, this model. We validate the model thus obtained against experimental free and forced vibration data. We find that, although the qualitative dynamics is simple, some effort in the modeling is needed to quantitatively capture the dynamic response well. We also briefly study the higher dimensional dynamics of the oscillator, and present some experimental results showing modal interactions through a 0:1 internal resonance, which has been studied elsewhere. The novelty here lies in the strong nonlinearity of the slow mode.     

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

     

Some pansystems studies on panchaos and strange panattractor

Paper [1] discussed the relations among panchaos and strange panattractor and pansystems operators. Paper [2] gave the applications of the fixed-point pansystems theorems to these typical nonlinear problems. In this paper, we firstly present several concepts: increasing relation, maximal panchaos, etc., and discuss the relations among them. We also discuss the problems when two increasing relations have the same panchaos and when panchaos of g is panchaos of g^t as well.     

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

     

On crack propagation in a coupled thermomechanical system of nonlinear media

In some engineering problems.thermo-mechanical coupling is im-portant and may not be ignored.This paper deals With the crackpropagation problem in a coupled thermo-mechanical system of non-linear media.Various nonlinear media.including nonlinear e-lastic and elastic-plastic cases,have been considered and therelated path-independent integrals are given.To explain thephysical meaning of these integrals,a notched specimen hasbeen considered.and the dynamical crack extension force in acoupled thermo-mechanical system is shown to be equal to thisintegral.Thus.we could consider such integrals as some non-linear criteria for coupled thermo-mechanical fracture dynamics.     

On the singular perturbation of a nonlinear ordinary differential equation with two parameters

In this paper, the method of differential inequalities has been applied to study the boundary value problems of nonlinear ordinary differential equation with two parameters. The asymptotic solutions have been found and the remainders have been estimated.Project Supported by the Science Fund of the Chinese Academy of Sciences     

Asymptotic stability criterion for nonlinear monotonic systems and its applications (review)

This paper discusses uniform asymptotic stability criteria for nonlinear monotonic systems and their applications in various problems of nonlinear dynamics and population dynamics     

Double-Diffusive Convection in a Saturated Anisotropic Porous Layer with Internal Heat Source

In the present study, double-diffusive convection in an anisotropic porous layer with an internal heat source, heated and salted from below, has been investigated. The generalized Darcy model is employed for the momentum equation. The fluid and solid phases are considered to be in equilibrium. Linear and nonlinear stability analyses have been performed. For linear theory normal mode technique has been used, while nonlinear analysis is based on a minimal representation of truncated Fourier series. Heat and mass transfers across the porous layer have been obtained in terms of Nusselt number Nu and Sherwood number Sh, respectively. The effects of internal Rayleigh number, anisotropy parameters, concentration Rayleigh number, and Vadasz number on stationary, oscillatory, and weak nonlinear convection are shown graphically. The transient behaviors of Nusselt number and Sherwood number have been investigated by solving the finite amplitude equations using a numerical method. Streamlines, isotherms, and isohalines are drawn for both steady and unsteady (time-dependent) cases. The results obtained, during the above analyses, have been presented graphically, and the effects of various parameters on heat and mass transfers have been discussed.     

绳索系统的建模、动力学和控制

绳索系统具有无限自由度,当计入非线性因素的作用时,其面内和面外的振动相互耦合,呈现非常丰富的非线性动力学行为.另外,绳索系统经常工作在风、流体、微重力、电磁力等作用下,进一步加剧了其动力学的复杂性.绳索系统的动力学现象引起了工程界和力学界的关注.本文对绳索在重要工程系统中的应用及相应的动力学现象进行概述,给出了柔索的动力学建模过程,对绳索系统的动力学和控制研究进行了总结,并指出了值得进一步关注的若干问题.      

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

The Post-Hopf-Bifurcation Response of a Loosely Supported Cylinder in an Array Subjected to Cross-Flow. Part II: Theoretical Model and Comparison with Experiments

A nonlinear quasi-steady model for the analysis of the dynamics of a loosely supported cylinder, which takes into account position-dependent nonlinear fluid forces as well as nonuniform flow, is formulated. The model includes an approximation for the equivalent viscous damping associated with energy dissipation on impact at the support. The nonlinear model shows reasonably good agreement with experiments, in predicting the observed bifurcations in the cylinder response. Comparison criteria include the standard orbital plots, time traces and response spectra. A borderline chaotic response is found to be predominant over the test velocity range. In this chaotic regime, the theoretical results were verified via attractor fractal-dimension calculations and saddle orbit distributions; theoretical values of these invariant measures compare reasonably well with their experimental counterparts. Two mechanisms leading to chaos have been identified for this system. The first is a switching mechanism , at the onset of impacting. The second, and more prevalent, is the type I intermittency route to chaos.     

A time-domain nonlinear system identification method based on multiscale dynamic partitions

Based on a theoretical foundation for empirical mode decomposition, which dictates the correspondence between the analytical and empirical slow-flow analyses, we develop a time-domain nonlinear system identification (NSI) technique. This NSI method is based on multiscale dynamic partitions and direct analysis of measured time series, and makes no presumptions regarding the type and strength of the system nonlinearity. Hence, the method is expected to be applicable to broad classes of applications involving time-variant/time-invariant, linear/nonlinear, and smooth/non-smooth dynamical systems. The method leads to nonparametric reduced order models of simple form; i.e., in the form of coupled or uncoupled oscillators with time-varying or time-invariant coefficients forced by nonhomogeneous terms representing nonlinear modal interactions. Key to our method is a slow/fast partition of transient dynamics which leads to the identification of the basic fast frequencies of the dynamics, and the subsequent development of slow-flow models governing the essential dynamics of the system. We provide examples of application of the NSI method by analyzing strongly nonlinear modal interactions in two dynamical systems with essentially nonlinear attachments.     

Linear <Emphasis Type="Italic">Γ</Emphasis>-limits of multiwell energies in nonlinear elasticity theory

We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.      

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.     

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole–Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pru?  equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.     

Energy Transfers in a System of Two Coupled Oscillators with Essential Nonlinearity: 1:1 Resonance Manifold and Transient Bridging Orbits

The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linear oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchanges are encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly in terms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almost completely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energy pumping is triggered by the excitation of transient bridging orbits compatible with the nonlinear attachment being initially at rest, a common feature in most practical applications. A parametric study of the energy exchanges is also performed to understand the influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting the theoretical developments are discussed. This study was carried out while the author was a postdoctoral fellow at the National Technical University of Athens and at the University of Illinois at Urbana-Champaign.     

Nonlinear buckling analysis of hyperbolic cooling tower shell with ring-stiffeners

This paper is concerned with a numerical solution of hyperbolic cooling tower shell, a class of full nonlinear problems in solid mechanics of considerable interest in engineering applications. In this analysis, the post-buckling analysis of cooling tower shell with discrete fixed support and under the action of wind loads and dead load is studied. The influences of ring-stiffener on instability load are also discussed. In addition, a new solution procedure for nonlinear problems which is the combination of load increment iteration with modified R-C are- length method is suggested. Finally, some conclusions having important significance for practice engineering are given.The Project Supported by National Natural Science Foundation of China.     

A brief survey on discontinuous Galerkin methods in computational fluid dynamics

间断Galerkin （DG）方法结合了有限元法（具有弱形式、有限维解和试验函数空间）和有限体积法（具有数值通量、非线性限制器）的优点,特别适合对流占优问题（如激波等线性和非线性波）的模拟研究,本文述评DG 方法,强调其在计算流体力学（CFD）中的应用,文中讨论了DG 方法的必要构成要素和性能特点,并介绍了该方法的一些最近研究进展,相关工作促进了DG 方法在CFD 领域的应用,