Transition to Chaos and Flow Dynamics of Thermochemical Porous Medium Convection

In a fluid-saturated porous medium, dissolved species advect at the pore velocity, while thermal retardation causes heat to move at the Darcy velocity. The Darcy model with the Boussinesq approximation in a square medium with a porosity of = 0.01 subject to two sources of buoyancy is used, to study numerically the dynamics of this so-called double-advective instability. The vertical walls of the medium are impermeable and adiabatic, while Dirichlet boundary conditions are imposed on the horizontal walls such that the medium is heated and salted from below. For an increasing ratio between chemical and thermal buoyancy, while keeping the thermal buoyancy fixed, a transition from a steady to a chaotic convective solution is observed. At the transition a stable limit cycle is found, suggesting that the transition takes the form of a Hopf bifurcation. The dynamics of the chaotic flow is characterized by irregular transitions between nonlayered and layered flow patterns, as a result of the spontaneous formation and disappearance of gravitationally stable interfaces. These interfaces temporarily divide the domain in layers of distinct solute concentration and lead to a significant reduction of kinetic energy and vertical heat and solute fluxes. The stability of an interface is described by a balance between the viscous drag forces in the convective layers and the buoyancy force associated with the density interface.     

A Nonlinear Temporal Headway Model of Traffic Dynamics

In order to describe the dynamics of a group of road vehicles travelling in a single lane, car-following models attempt to mimic the interactions between individual vehicles where the behaviour of each vehicle is dependent upon the motion of the vehicle immediately ahead. In this paper we investigate a modified car-following model which features a new nonlinear term which attempts to adjust the inter-vehicle spacing to a certain desired value. In contrast to our earlier work, a desired time separation between vehicles is used rather than simply being a constant desired distance. In addition, we extend our previous work to include a non-zero driver vehicle reaction time, thus producing a more realistic mathematical model of congested road traffic. Numerical solution of the resulting coupled system of nonlinear delay differential equations is used to analyse the stability of the equilibrium solution to a periodic perturbation. For certain parameter values the post-transient response is a chaotic (non-periodic) oscillations consisting of a broad spectrum of frequency components. Such chaotic motion leads to highly complex dynamical behaviour which is inherently unpredictable. The model is analysed over a range of parameter values and, in each case, the nature of the response is indicated. In the case of a chaotic solution, the degree of chaos is estimated.     

Dynamics of Nonlinear Oscillators under Simultaneous Internal and External Resonances

An analysis is presented for a class of two degree of freedom weakly nonlinear oscillators, with symmetric restoring force. Conditions of one-to-three internal resonance and subharmonic external resonance of the lower vibration mode are assumed to be satisfied simultaneously. As a consequence, the second vibration mode may also be under the action of external primary resonance. Initially, a set of slow-flow equations is derived, governing the amplitudes and phases of approximate long time response of these oscillators, by applying an asymptotic analytical method. Determination of several possible types of steady-state motions is then reduced to solution of sets of algebraic equations. For all these solution types, appropriate stability analysis is also performed. In the second part of the study, this analysis is applied to an example mechanical system. First, a systematic search is performed, revealing effects of system parameters on the existence and stability properties of periodic motions. Frequency-response diagrams are presented and attention is focused on understanding the evolution and interaction of the various solution branches as the external forcing and nonlinearity parameters are varied. Finally, numerical integration of the equations of motion demonstrates that the system exhibits quasiperiodic or chaotic response for some parameter combinations.     

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.     

The Application of Nonlinear Dynamics in Nursing Research

In this article, the authors present an overview of the applications of chaos theory and nonlinear dynamics to problems of relevance not only to nurses, but to anyone dealing with human functioning and interaction. These applications have been in the areas of epidemiology, nursing management and physiological functioning. In some cases, the applications were successful in identifying information that would have been overlooked using traditional methods. In other cases, the problems of data collection and analysis, unique to nonlinear dynamics, are still being developed and tested.     

Nonlinear Dynamics of Turbulent Thermals in Shear Flow

The nonlinear integral model of a turbulent thermal is extended to the case of the horizontal component of its motion relative to the medium (e.g., thermal floating-up in shear flow). In contrast to traditional models, the possibility of a heat source in the thermal is taken into account. For a piecewise constant vertical profile of the horizontal velocity of the medium and a constant vertical velocity shear, analytical solutions are obtained which describe different modes of dynamics of thermals. The nonlinear interaction between the horizontal and vertical components of thermal motion is studied because each of the components influences the rate of entrainment of the surrounding medium, i.e., the growth rate of the thermal size and, hence, its mobility. It is shown that the enhancement of the entrainment of the medium due to the interaction between the thermal and the cross flow can lead to a significant decrease in the mobility of the thermal.     

The Effect of Downsizing on Hierarchical Work Flow Dynamics in Organizations

The objective was to determine the impact of the downsizing of management personnel on the flow of work throughout the organization, given that the quantity of workers (or technicians) must remain the same. In particular the objective of study was to assess the impact of downsizing on performance dynamics, where social factors such as power objectives of the management groups, job insecurity, and perceptions of fairness were not active variables in the process. Three groups of university students (19-33 in a group) were formed to play the Chaos Exercise in a three-level hierarchy. In the three experimental conditions, management groups were downsized during the game from 3 to 2 players, from 2 players to 1, or from 3 players to 1. Nonlinear dynamical structures were explored for each condition and at all three levels of hierarchy using production records generated by each group. Results showed chaos for two out of three conditions at the lowest level of the hierarchy, and usually a tendency toward asymptotic stabilization in middle management, although the functions themselves were markedly varied. The 2-to-1 condition, however, showed a tendency toward performance stabilization at the worker and middle management levels, but chaos at the top management level.     

Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow

In this paper chaotic behavior of nonlinear viscoelastic panels in asupersonic flow is investigated. The governing equations, based on vonKàarmàn's large deflection theory of isotropic flat plates, areconsidered with viscoelastic structural damping of Kelvin's modelincluded. Quasi-steady aerodynamic panel loadings are determined usingpiston theory. The effect of constant axial loading in the panel middlesurface and static pressure differential have also been included in thegoverning equation. The panel nonlinear partial differential equation istransformed into a set of nonlinear ordinary differential equationsthrough a Galerkin approach. The resulting system of equations is solvedthrough the fourth and fifth-order Runge–Kutta–Fehlberg (RKF-45)integration method. Static (divergence) and Hopf (flutter) bifurcationboundaries are presented for various levels of viscoelastic structuraldamping. Despite the deterministic nature of the system of equations,the dynamic panel response can become random-like. Chaotic analysis isperformed using several conventional criteria. Results are indicative ofthe important influence of structural damping on the domain of chaoticregion.     

非线性刚度不平衡转子径向碰摩动力学研究

以线性项和立方项之和来表示转轴材料的物理非线性因素,建立了考虑非线性油膜力和非线性刚度的轴转子系统的动力学模型,利用数值积分法对转子系统由于局部碰摩故障导致的非线性动力学行为进行了研究,发现此类非线性振动系统具有倍周期分岔、拟周期和混沌等复杂的动力学行为,为此类系统的安全运行和有效识别转子故障提供了理论参考。     

Nonlinear Dynamics of Shells: Theory,Finite Element Formulation,and Integration Schemes

The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.     

A Framework and Methodology for the Study of Nonlinear,Self-Organizing Family Dynamics

Family systems theories have emerged over the past 30 to 40 years primarily through clinical observations, resulting in diverse and internally inconsistent views of family structures, development, dynamics, and pathology; as well as a separation from more empirically based small group research. The 5-R's model is intended to unify the various family systems theories and render them more empirically testable using concepts and methodologies from non-linear dynamical systems theory. The conversation of one family was analyzed using orbital decomposition as a pilot test of the most basic assumptions of the 5-R's model. An optimal string length of three was found along with evidence of coherent complexity (chaos), with Lyapunov dimensionality equal to 1.7 and Shannon's entropy equal to 8.68. Results are discussed with respect to further empirical validation of the 5-R's model and clinical uses of the model and orbital decomposition methodology in conjoint therapy.     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

Adaptive Feedback Linearizing Control of Proper Orthogonal Decomposition Nonlinear Flow Models

This paper treats the question of feedback linearizing control oftwo-dimensional incompressible, unsteady wake flow. For definiteness,flow past a circular cylinder is considered, but the design approachpresented here is applicable to other flow control problems. Twofinite-dimensional lower-order models based on Proper OrthogonalDecomposition (POD) of dimension N with N actuators are considered.Models I and II are obtained using control function and penalty functionmethods, respectively. Control action can be achieved by a combinationof suction, injection, and synthetic jets. For the design ofcontrollers, it is assumed that the system matrices of the POD modelsare unknown. Nonlinear adaptive control systems for the two models arederived. For model I, nontrivial zero-error dynamics exists, which playa key role in the stability of the closed-loop system. But for model II,global adaptive trajectory control is achieved. In the closed-loopsystem, the mode amplitudes asymptotically follow the referencetrajectories. Simulation results for a 4-mode POD model obtained usingthe penalty function method are presented. These results show that inthe closed-loop system, unsteadiness in the mode amplitudes can besuppressed in spite of large uncertainties in the flow model.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part II: Experimental Analysis

In Part I, theoretical analysis of the dynamic behaviour of a rigid rotor with nonlinear elastic restoring forces was carried out. In this part (Part II), an experimental confirmation of the theoretical data from that analysis was sought. With this aim, an experimental model was set up consisting mainly of a practically rigid rotor clamped onto a small diameter piano wire symmetrical to the wire supports. These supports were rigid and equipped with roller bearings and a device that made it possible to adjust the initial tension in the wire so as to make the elastic restoring forces less or more linear. The rotor was dynamically unbalanced and was driven by an asynchronous motor regulated by means of an inverter in order to adjust the rotor speed. A series of tests was performed on this rig with different values of the initial tension in the wire, and the trajectories of two points on the rotor axis were recorded in the course of the tests. These trajectories were obtained, under the hypothesis of similarity, from the orbits covered by two given sections of the wire and detected with two pairs of capacitive transducers. The collected data was compared with the theoretical results from Part I of the present investigation. Comparison of the collected data with the corresponding theoretical results made it possible to infer that system nonlinearity in the presence of small damping can give rise to motions that are periodic, whether synchronous or not, or quasi-periodic, but never chaotic.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis

In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.     

Nonlinear Analysis of a Cantilever Pipe Containing Pulsatile Flow

In this paper we investigate the nonlinear dynamics of a cantilever elastic pipe that contains pulsatile flow. The equation of motion was derived by using Hamiltonian action function. We use Galerkin's technique to include only finite number of spatial modes in the solution.The stability chart of the time-varying system was computed in the space of the relative perturbation amplitude of the flow velocity and dimensionless forcing frequency using an efficient numerical method based on Chebyshev polynomials. In the near of some critical regions bifurcation diagrams were also computed which show secondary Hopf bifurcations and phase locking followed by chaotic motion.     

Speculative Behavior and Asset Price Dynamics

This paper deals with speculative trading. Guided by empirical observations, a nonlinear deterministic asset pricing model is developed in which traders repeatedly choose between technical and fundamental analysis to determine their orders. The interaction between the trading rules produces complex dynamics. The model endogenously replicates the stylized facts of excess volatility, high trading volumes, shifts in the level of asset prices, and volatility clustering.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part II: Experimental Analysis

In the first part of the present investigation [9], the dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. In the present paper an experimental confirmation of the theoretical results is sought. The steel rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the parameter assigned in [9]. Two steel rings, each one with a series of holes and a clamping screw, were mounted onto the rotor with a small clearance. This arrangement made it possible to fix the positions of the rings and their holes respect to the rotor, so as to realize a pre-estabilished unbalance. The two bronze journal bearings were characterised by a relatively low length/diameter ratio, and a relatively high value of the radial clearance and were lubricated with oil delivered from a thermostatic tank. In this way, despite the relative lightness of the rotor, the dimensionless static eccentricity _s was given the high values that were apt to realize the operating conditions assumed in the theoretical analysis. The rotor was driven by means of a d.c. motor connected to a toothed belt-drive. Varying the rotor speed in the range 1000 ÷ 10000 r.p.m., made it possible to assign the values of the modified Sommerfeld number assumed in the theoretical analysis. Three pairs of eddy-current probes were mounted in order to detect the trajectories of three points (C₁, C and C₂) suitably fixed along the rotor axis. These orbits were finally put in comparison with the corresponding ones previously obtained through numerical analysis. The comparison pointed out that the experimental data were in good agreement with the theoretical predictions, despite the approximations that characterise the theoretical model and the unavoidable errors affecting measures in the course of the experimental test.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis

The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the -film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.     

Chen系统的非线性动力学究研

利用受控Chen系统，并基于镜像操作方法，发现Chen吸引子是由左、右两个吸引子所组成的复合结构，且左、右吸引子均可由极限环生成。采用一维时间序列相空间重构技术和系统混沌的定量判据准则，揭示出Chen系统从规则运动转化到混沌运动所具有的普适特征：Chen系统可通过Pomeau-Manneville途径走向混沌，且其间歇性与Hopf分岔和倍周期分岔有关、在这些途径上既可观察到锁相和准周期运动，也可观察到类Chen吸引子、Chen系统和Lorenz系统之间的过渡吸引子和类Lorenz吸引子。