Smart Baffle Placement for Chaotic Mixing

It is well known that fluid mixing can often be improved by the introduction of ‘baffles’ into the flow – the problem of baffle placement is examined here for chaotic fluid mixing of a highly viscous fluid. A simple model for a planetary mixer, with one stirring element, is modified by the introduction of one or more stationary baffles. Regular regions of poor mixing in the unbaffled flow are shown to be significantly reduced in size if the location of the baffles is chosen so that the flow necessarily generates ‘topological chaos’. By contrast, the positioning of baffles in superficially similar ways that do not generate such ‘topological chaos’ fails to provide a similar improvement.     

Adaptive <Emphasis Type="Italic">Q</Emphasis>–<Emphasis Type="Italic">S</Emphasis> synchronization of non-identical chaotic systems with unknown parameters

This work investigates the adaptive Q–S synchronization of non-identical chaotic systems with unknown parameters. The sufficient conditions for achieving Q–S synchronization of two different chaotic systems (including different dimensional systems) are derived, based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the non-identical chaotic systems are to have Q–S synchronization. Finally, four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China     

ANALYSIS OF THE CHAOTIC MOTION FOR A ROTOR SYSTEM WITH A TRANSVERSE CRACK

The chaotic motion of an elastic shaft-disk rotor system including the geometricnonlinearity of the shaft with a transverse crack is investigated.The critical condition for thesystem to enter chaotic state is given by the Melnikov method.Meanwhile,whether the chaosoccurs is determined by the Poincaré map,phase portrait and time-displacement history diagram.     

Chaos in the softening duffing system under multi-frequency periodic forces

The chaotic dynamics of the softening-spring Duffing system with multi-frequency external periodic forces is studied. It is found that the mechanism for chaos is the transverse heteroclinic tori. The Poincare map, the stable and the unstable manifolds of the system under two incommensurate periodic forces were set up on a two-dimensional torus. Utilizing a global perturbation technique of Melnikov the criterion for the transverse interaction of the stable and the unstable manifolds was given. The system under more but finite incommensurate periodic forces was also studied. The Melnikov‘s global perturbation technique was therefore generalized to higher dimensional systems. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies will increase the area in parameter space where chaotic behavior can occur.     

Lag synchronization between discrete chaotic systems with diverse structure

A lag synchronization controller is designed in studying discrete chaotic systems with diverse structures to realize synchronization between Henon and Ikeda sys- terns. The structure of the lag synchronization controller and the error equations of state variables between discrete chaotic systems are presented based on the stability theory. The designed controller has unique structures for different chaotic systems. Lag synchro- nization between any discrete chaotic systems with diverse structures can be achieved. Simulation results show that this control method is effective and feasible.     

Chaotic behaviour of the general symbolic dynamics

This paper extends symbolic dynamics to general cases. Some chaotic properties and applications of the general symbolic dynamics (∑(X),σ) and its special cases are discussed, where X is a separable metric space.     

Q–S synchronization between chaotic systems with double scaling functions

A double function Q–S synchronization (DFQSS) scheme of non-identical chaotic systems is proposed and analyzed with the assumption that all of the parameters are unknown. The sufficient conditions for achieving the double function Q–S synchronization with the desired scaling functions of two different chaotic systems (including the systems of non-identical dimension) are derived based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are presented such that the DFQSS of non-identical chaotic systems is to be achieved. Numerical simulations and a brief discussion conclude the paper.     

Optimal nonlinear observers for chaotic synchronization with message embedded

This paper presents an optimal nonlinear observer for synchronizing the transmitter-receiver pair with guaranteed optimal performance. In the proposed scheme, a generalized nonlinear state-space observer via uniform matrix transformations is constructed to estimate the transmitter state and the information signal, simultaneously. A nonlinear optimal design approach is used to synchronize chaotic systems. Solving the Hamilton–Jacobi–Bellman (H–J–B) equations we can obtain a linear optimal feedback scheme for piecewise-linear chaotic systems. Moreover, a robust scheme derived from the H _∞ optimization theory improves the synchronization performance of general nonlinear chaotic systems by suppressing the influence of their high order residual terms. Finally, two numerical simulation examples are illustrated by the chaotic Chua’s circuit system and the Lorenz chaotic system to demonstrate the effectiveness of our scheme.     

Numerical simulations of chaotic dynamics in a model of an elastic cable

The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities.The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.Part of this work was presented at the XVIIth Int. Congr. of Theor. and Appl. Mech., Grenoble, August 1988.     

Asymptotic expansions by Γ-convergence

Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.      

Finite-time generalized synchronization of chaotic systems with different order

In this paper, the generalized synchronization of chaotic systems with different order is studied. The definition of finite-time generalized synchronization is put forward for the first time. Based on the finite-time stability theory, two control strategies are proposed to realize the generalized synchronization of chaotic systems with different order in finite time. Besides the relation between the parameter β, the initial states of systems and the convergent time were obtained. The corresponding numerical simulations are presented to demonstrate the effectiveness of proposed schemes.     

Jounce and Impact in Cam-Tappet Conjunction Induced by the Elastodynamics of Valve train System

The paper highlights the importance of a multi-physics integrated approach in the design evaluation of complex valve trains. The multi-disciplinarity of the problems requires simultaneous solution to inertial dynamics, structural compliance of system elements and tribological conditions in load bearing conjunctions, all of which are critical to thermodynamic functional assurance of the system as the primary objective, and system efficiency in terms of reduced mechanical and frictional losses. This paper is an extended version of: ‘PREDICTION OF CAM-TO-TAPPET CONTACT LOSS WITH VALVE TRAIN ELASTODYNAMIC ANALYSIS’, 2004 AIMETA International Tribology Conference, September 14–17, 2004, Rome, Italy.     

ℒ<Subscript>2</Subscript>–ℒ<Subscript>∞</Subscript> nonlinear system identification via recurrent neural networks

This paper proposes a new neural network ℋ_∞ synchronization (NNHS) scheme for unknown chaotic systems. In the proposed framework, a dynamic neural network is constructed as an alternative to approximate the chaotic system. Based on this neural network and linear matrix inequality (LMI) formulation, the NNHS controller and the learning law are presented to reduce the effect of disturbance to an ℋ_∞ norm constraint. It is shown that finding the NNHS controller and the learning law can be transformed into the LMI problem and solved using the convex optimization method. A numerical example is presented to demonstrate the validity of the proposed NNHS scheme.     

Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg. 49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg. 58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg. 13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering 32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech. 53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech. 53, 1986, 855–863].     

Achievement of chaotic synchronization trajectories of master–slave manipulators with feedback control strategy

This paper addresses a master-slave synchro- nization strategy for complex dynamic systems based on feedback control. This strategy is applied to 3-DOF pla- nar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors. A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors. The dynamical equations of master-slave manipulators with synchronization controller are derived, and the Lyapunov stability theory is used to determine the stability of this controlled synchronization system. In numer- ical experiments, the synchronous motions of end-effectors as well as three joint angles and torques of master-slave manipulators are studied under the control of the proposed synchronization strategy. It is found that the positive gain matrix affects the implementation of synchronization con- trol strategy. This synchronization control strategy proves the synchronization＇s feasibility and controllability for com- plicated motions generated by master-slave manipulators.     

On the identification of continuum concepts and fields with molecular variables

The need to explore a molecular foundation for continuum mechanics is here motivated by recognition of the scale dependence of mass density and boundaries of solid bodies. Modelling molecules as interacting point masses, continuum fields are defined via local spatial averaging using a scale-dependent weighting function. Local balances of linear and angular momentum, and of energy, are established directly, rather than as localised versions of integral relations. Attention is drawn to the non-uniqueness of stress, couple-stress, and heat flux, and to the physical interpretations thereof. A conservation relation for a local measure of inhomogeneity is derived and related to generalised (i.e. tensor valued) moment of momentum. Remarks are made on the scale dependence of the notions of ‘material point’ and ‘boundary’, choices of weighting function, and how further temporal averaging can be implemented, with particular reference to systems whose molecular content changes with time.     

A new digital communication scheme based on chaotic modulation

Based on parameter identification, a parameter observer is designed for a class of chaotic systems. The digital signals modulated in the parameter will be recovered by the observer. By choosing different frequency signals as “0” and “1,” a practical digital secure communication scheme is proposed by this parameter modulation method. Numerical simulations show the effectiveness of this communication scheme.     

Estimation of Chaotic and Regular (Stick–Slip and Slip–Slip) Oscillations Exhibited by Coupled Oscillators with Dry Friction

In this paper, we present a novel approach to quantify regular or chaotic dynamics of either smooth or non-smooth dynamical systems. The introduced method is applied to trace regular and chaotic stick–slip and slip–slip dynamics. Stick–slip and slip–slip periodic and chaotic trajectories are analyzed (for the investigated parameters, a stick–slip dynamics dominates). Advantages of the proposed numerical technique are given.     

Chaos synchronization on the N-torus and cryptography

A class of chaotic dynamical systems on the N-dimensional torus is proposed for masking some information in secure communications. The information is then recovered thanks to a chaos synchronization process. To cite this article: L. Rosier et al., C. R. Mecanique 332 (2004).

Résumé

Nous proposons une classe de systèmes chaotiques sur le tore N-dimensionnel pour masquer une information à transmettre dans une communication sécurisée. Cette information est ensuite reconstruite à l'aide d'un mécanisme de synchronisation du chaos. Pour citer cet article : L. Rosier et al., C. R. Mecanique 332 (2004).