A consistent numerical scheme for the von Mises mixed-hardening constitutive equations

Analytical solutions are derived for the von Mises mixed-hardening elastoplastic model under rectilinear strain paths, and the concept of response subspace is introduced such that the original five-dimensional problem in deviatoric stress space is reduced to a more economic two-dimensional problem, of which two coordinates (x,y) suffice to determine normalized active stress. Furthermore, in this subspace a Minkowski spacetime can be endowed, on which the group action is found to be a proper orthochronous Lorentz group SO_o(2,1). The existence of a fixed point attractor in the normalized active stress space is demonstrated by the long-term behavior deduced from the analytical solutions, which together with the response stability is further verified by Lyapunov's direct method. Two numerical schemes based on a nonlinear Volterra integral equation and on a group symmetry are derived, the latter of which exactly preserves the consistency condition for every time step. The consistent scheme is stable, accurate and efficient, because it updates the stress point automatically on the yield surface at each time step without any iteration. For the purpose of comparison and contrast, numerical results calculated by the above two schemes as well as by the radial return method were displayed for several loading examples.     

一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

Rheometrical and molecular dynamics simulation studies of incipient clot formation in fibrin-thrombin gels: An activation limited aggregation approach

A rheometrical investigation of incipient clots formed in fibrin-thrombin gels is reported in which the Gel Point (GP) is characterised by frequency independence of the loss tangent in small amplitude oscillatory shear measurements over a wide range of thrombin concentration. Values of the fractal dimension (D_f) of the GP network calculated from measurements are consistent with those reported in simulations of diffusion limited cluster-cluster aggregation (DLCCA) and reaction limited cluster-cluster aggregation (RLCCA), but differ insofar as the values of D_f calculated from the present experiments increase progressively with a reduction in gel formation time. A molecular dynamics simulation (MDS) of systems of rod-like particles was designed to (i) test the hypothesis that the presence of an activation profile in a cluster aggregation model could account for the trend of D_f as a function of gel formation time observed experimentally in fibrin-thrombin gels and whole heparinised blood without recourse to the inclusion of fibrinogen-specific interactions; and (ii) to explore the effect of monomer activation kinetics on the microstructure of fractal clusters formed in systems of rigid rod-like particles. The results identify two possible mechanisms for the increase in D_f as the gel formation time decreases, both being a consequence of altering the evolution of the clustering dynamics by a process referred to herein as activation limited aggregation (ALA). This ALA-based MDS substantiates the experimental findings by confirming the trend evident in the formation of incipient clots in fibrin-thrombin gels and in whole heparinised blood. A mechanism for ALA involving the aggregation of pre-GP sub-clusters is proposed.     

Flapping dynamics of a piezoelectric membrane behind a circular cylinder

The flapping dynamics of a piezoelectric membrane placed behind a circular cylinder, which are closely related to its energy harvesting performance, were extensively studied near the critical regime by varying the distance between the cylinder and the membrane. A total of four configurations were used for the comparative study: the baseline configuration in the absence of the upstream circular cylinder, and three configurations with different distances (S) between the cylinder and the membrane (S/D=0, 1, and 2). A polyvinylidene fluoride (PVDF) membrane was configured to flutter at its second mode in these experiments. The Reynolds number based on the membrane’s length was 6.35×10⁴ to 1.28×10⁵, resulting in a full view of membrane dynamics in the subcritical, critical, and postcritical regimes. The membrane shape and the terminal voltage were simultaneously measured with a high-speed camera and an oscilloscope, respectively. The influence of the upstream cylinder on the membrane dynamics was discussed in terms of time-mean electricity, instantaneous variations and power spectra of terminal voltage and membrane shape, fluctuating voltage amplitude, and flapping frequency. The experimental results overwhelmingly demonstrated that the terminal voltage faithfully reflected various unsteady events embedded in the membrane’s flapping motion. For all configurations, dependency of the captured electricity on a flow speed beyond the critical status was found to follow the parabolic relationship. In the two configurations in which S/D=0 and 1, the extraneously induced excitation by the Kármán vortex street behind the circular cylinder substantially reduced the critical flow speed, giving rise to effective energy capture at a lower flow speed and a relatively high gain in power output. However, in the configuration in which S/D=2, the intensified excitation by the Kármán vortex street on the membrane considerably reduced the captured energy. Finally, a transient analysis of the membrane’s flapping dynamics in the configuration in which S/D=0 was performed in terms of phase-dependent variations of the membrane segment’s moving speed, membrane curvature, and terminal voltage; the analysis resulted in a full understanding of the energy harvesting process with consecutive inter transfer of elastic, kinetic, and electric energies.     

Finite Dimensional Attractor for a One-Phase Stefan Problem with Kinetics

For a one-phase free-boundary problem with kinetics, which is known to generate a rich dynamics, we study evolution of the infinitesimal volume along the trajectories in the attractor. We demonstrate that for sufficiently large m that is defined solely by the properties of the kinetics function the m-dimensional volume decays exponentially. This property combined with the uniform differentiability of the semigroup leads to the conclusion that the Hausdorff dimension of the attractor is finite.     

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.     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

A study of the chaotic behaviour of the elastic-plastic Euler arch

This paper deals with the dynamics of a truss structure, the Euler arch. The bars are made of elastic-plastic material, and the structure can exhibit large displacements. The aim of this paper is to give evidence of the possible chaotic behavior of this structure, even in the presence of a hardening plastic branch. The tools used are the diagrams of bifurcation, the measure of the dimension of the attractor, the Kolmogorov entropy, and the maximum Lyapunov exponent. This study emphasizes the sensitivity to the initial conditions by means of generalized basins of attraction.     

Nonlinear Coherence in Multivariate Research: Invariants and the Reconstruction of Attractors

First, some linear techniques in multivariate time-series analysis in EEG research are reviewed to highlight the problem of estimating the dimensionality of the state space (embedding dimension), the reconstruction of an attractor, and the evaluation of invariant properties of the attractor. The traditional linear techniques included the usual spectral and cospectral measures of power, phase, and coherence to which stepwise discriminant analysis was applied for canonical representation of the attractor. Then, some traditional nonlinear techniques of attractor reconstruction and dimensional analysis which use the time-lagged univariate approach of Ruelle and Takens (Takens, 1981) are reviewed. Next, updates and multivariate generalizations that use singular-value decomposition (Broomhead & King, 1986) are reviewed. Finally, Stewart's (1995, 1996) multivariate generalization of the method of false nearest neighbors (Abarbanel, Brown, Sidorowich, & Tsimring, 1993; Kennel, Brown, & Abarbanel, 1992) is reviewed. These are particularly relevant for evaluating multivariate coherence in research on the complex cooperative dynamical systems found in neuroscience, psychology, and social science when time series of sufficient length are investigated.     

Estimation of Lyapunov exponents for a system with sensitive friction model

Mathematical modelling and numerical analysis of a vibrating system with dry friction is presented. Three qualitatively different friction characteristics are considered. One of them is an example of so-called sensitive friction characteristic. Their influence on the dynamics on the attractor of the friction oscillator is investigated through bifurcational analysis. This analysis is supported by Lyapunov exponents estimated using approach for the systems with discontinuities. Theoretical background for such a synchronisation-based method of determining the largest Lyapunov exponent is explained. The results obtained through the proposed approach approximate the LLE with a good precision.     

Analysis of complex parametric vibrations of plates and shells using Bubnov-Galerkin approach

Summary The Bubnov-Galerkin method is applied to reduce partial differential equations governing the dynamics of flexible plates and shells to a discrete system with finite degrees of freedom. Chaotic behaviour of systems with various degrees of freedom is analysed. It is shown that the attractor dimension of a system has no relationship with the attractor dimension of any of its subsystems.This work has been partially supported by Department of Mathematics of the Central European University in Budapest.     

On a New Route to Chaos in Railway Dynamics

Cooperrider's mathematical model of a railway bogie running on a straight track has been thoroughly investigated due to its interesting nonlinear dynamics (see True [1] for a survey). In this article a detailed numerical investigation is made of the dynamics in a speed range, where many solutions exist, but only a couple of which are stable. One of them is a chaotic attractor.Cooperrider's bogie model is described in Section 2, and in Section 3 we explain the method of numerical investigation. In Section 4 the results are shown. The main result is that the chaotic attractor is created through a period-doubling cascade of the secondary period in an asymptotically stable quasiperiodic oscillation at decreasing speed. Several quasiperiodic windows were found in the chaotic motion.This route to chaos was first described by Franceschini [9], who discovered it in a seven-mode truncation of the plane incompressible Navier–Stokes equations. The problem investigated by Franceschini is a smooth dynamical system in contrast to the dynamics of the Cooperrider truck model. The forcing in the Cooperrider model includes a component, which has the form of a very stiff linear spring with a dead band simulating an elastic impact. The dynamics of the Cooperrider truck is therefore non-smooth.The quasiperiodic oscillation is created in a supercritical Neimark bifurcation at higher speeds from an asymmetric unstable periodic oscillation, which gains stability in the bifurcation. The bifurcating quasiperiodic solution is initially unstable, but it gains stability in a saddle-node bifurcation when the branch turns back toward lower speeds.The chaotic attractor disappears abruptly in what is conjectured to be a blue sky catastrophe, when the speed decreases further.     

Localization and approximation of attractors for the Ginzburg-Landau equation

This paper studies the long-term behavior of solutions to the Ginzburg-Landau partial differential equation. For each positive integerm we explicitly produce a sequence of approximate inertial manifolds _m,j ,j = 1, 2,..., of dimensionm and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time. Of course this neighborhood contains the universal attractor which embodies the large time dynamics of the equations. The thickness of these neighborhoods decreases with increasingm for a fixed orderj; however, for a fixedm no conclusion can be made about the thickness of the neighborhoods associated to two differentj's. The neighborhoods associated to the manifolds localize the universal attractor and provide computabie large time approximations to solutions of the Ginzburg-Landau equation.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics.Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.     

Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents’ spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan–Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.     

An experimental study of bifurcation,chaos, and dimensionality in a system forced through a bifurcation parameter

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.     

Local and Global Lyapunov exponents

Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.