Andrzej Stefanski

Hysteretic effects of dry friction: modelling and experimental studies.

In this paper, the phenomena of hysteretic behaviour of friction force observed during experiments are discussed. On the basis of experimental and theoretical analyses, we argue that such behaviour can be considered as a representation of the system dynamics. According to this approach, a classification of friction models, with respect to their sensitivity on the system motion characteristic, is introduced. General friction modelling of the phenomena accompanying dry friction and a simple yet effective approach to capture the hysteretic effect are proposed. Finally, the experimental results are compared with the numerical simulations for the proposed friction model.

Estimation of the largest Lyapunov exponent in systems with impacts

Abstract The method of estimation of the largest Lyapunov exponent for mechanical systems with impacts using the properties of synchronization phenomenon is demonstrated. The presented method is based on the coupling of two identical dynamical systems and is tested on the classical Duffing oscillator with impacts.

Routes to complex dynamics in a ring of unidirectionally coupled systems.

We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization

A novel method of estimation of the largest Lyapunov exponent for discrete maps is introduced and evaluated for chosen examples of maps described by difference equations or generated from non-smooth dynamical systems. The method exploits the phenomenon of full synchronization of two identical discrete maps when one of them is disturbed. The presented results show that this method can be successfully applied both for discrete dynamical systems described by known difference equations and for discrete maps reconstructed from actual time series. Applications of the method for mechanical systems with discontinuities and examples of classical maps are presented. The comparison between the results obtained by means of the known algorithms and novel method is discussed. 2002 Elsevier Science Ltd. All rights reserved.

Using chaos synchronization to estimate the largest lyapunov exponent of nonsmooth systems

We describe the method of estimation of the largest Lyapunov exponent of nonsmooth dynamical systems using the properties of chaos synchronization. The method is based on the coupling of two identical dynamical systems and is tested on two examples of Duffing oscillator: (i) with added dry friction, (ii) with impacts.

Clustering of Huygens' Clocks

We study synchronization of a number of pendulum clocks hanging from an elastically fixed horizontal beam. It has been shown that after a transient, different types of synchronization between pendulums can be observed; (i) the complete synchronization in which all pendulums behave identically, (ii) pendulums create three or five clusters of synchronized pendulums, (iii) anti-phase synchronization in pairs (for even n). We give evidence why the configurations with a different number of clusters are not observed. Subject Index: 000, 034 In the 17th century the Dutch researcher Christian Huygens showed that a couple of mechanical clocks hanging from a common support were synchronized. 1) Over the last three decades the subject of the synchronization has attracted the increasing attention from different fields. 2)−4) In Huygens experiment 1),5)−9) clocks (subsystems) are coupled thought elastic structure. Generally, this type of coupling allows investigating how the dynamics of the particular subsystem is influenced by the dynamics of other subsystems. 10)−12) However, the precise dynamics of the n clocks hanging from the common support is unknown. Here, we study a synchronization problem for n pendulum clocks hanging from an elastically fixed horizontal beam. Each pendulum performs a periodic motion which starts from different initial conditions. We show that after a transient, different types of synchronization between pendulums can be observed; (i) the complete synchronization in which all pendulums behave identically, (ii) pendulums create three or five clusters of synchronized pendulums, (iii) anti-phase synchronization in pairs (for even n). Our results demonstrate that other stable cluster configurations do not exist. We anticipate our assay to be a starting point for further studies of the synchronization and creation of the small-worlds 13)−16) in the systems coupled by an elastic medium. For example, the behavior of the biological systems (groups of humans or animals) located on elastic structure could be investigated. In particular, a general mechanism for crowd synchrony can be identified. The large oscillations of London’s Millennium Bridge on the day it was opened have restarted the interest in be dynamical behavior of the systems coupled by elastic structure. The detailed theoretical and experimental explanation of the phenomena observed by Huygens for two pendulum clocks has been presented. 5)−9) In our pre

Chaos caused by non-reversible dry friction

Abstract In this short communication we investigate how the non-reversible dry friction characteristics will alter the non-linear responses of a simple mechanical oscillator. The presented approach is based on a certain mathematical description of the experimentally determined non-reversible dry friction characteristics, which causes chaotic and irregular motion of the studied system. A novelty of our idea is an introduction of the relative acceleration in description of the dry friction model.

SYNCHRONIZATION OF SLOWLY ROTATING PENDULUMS

We study synchronization of a number of rotating pendulums mounted on a horizontal beam which can roll on the parallel surface. It has been shown that after the initial transient, different states of pendulums synchronization occur. We derive the analytical equations for the estimation of the phase differences between phase synchronized pendulums. After the study of the basins of attraction of different synchronization states, we argue that the observed phenomena are robust as they occur for a wide range of both initial conditions and system parameters.

Synchronization of mechanical systems driven by chaotic or random excitation

The phenomenon of synchronization in dynamical and, in particular, mechanical systems has been known for a long time. Recently, the idea of synchronization has been also adopted for chaotic systems. It has been demonstrated that two or more chaotic systems can synchronize by linking them with mutual coupling or with a common signal or signals [1–5,15]. In the case of linking a set of identical chaotic systems (the same set of ODEs and values of the system parameters) ideal synchronization can be obtained. The ideal synchronization takes place when all trajectories converge to the same value and remain in step with each other during further evolution (i.e., limt-N jxðtÞ yðtÞj 1⁄4 0 for two arbitrarily chosen trajectories xðtÞ and yðtÞ). In such a situation all subsystems of the augmented system evolve on the same attractor on which one of these subsystems evolves (the phase space is reduced to the synchronization manifold). Linking homochaotic systems (i.e., systems given by the same set of ODEs but with different values of the system parameters) can lead to practical synchronization (i.e., limt-N jxðtÞ yðtÞjpe; where e is a vector of small parameters) [6,7]. In such linked systems it can also be observed that there is a significant change of the chaotic behaviour of one or more systems. This so-called ‘‘controlling chaos by chaos’’ procedure has some potential importance for mechanical and electrical systems. An attractor of such two systems coupled by a negative feedback mechanism can be even reduced to the fixed point [8]. This paper concerns the ideal synchronization of a set of identical uncoupled mechanical oscillators (with one or more degrees of freedom) linked by common external excitation only. This problem has been described widely for oscillators with periodic excitation because such kind of driving is often met in real oscillators [1]. However, from a viewpoint of practical considerations, a non-periodic external excitation can also occur in mechanical systems. For that reason, this paper concentrates on the analysis of the ideal synchronization for non-linear oscillators forced by chaotic and stochastic external driving. The analysis presented is based on the connections

Steady state locking in coupled chaotic systems

Abstract Two Lorenz systems working in different chaotic ranges can be stabilized simultaneously in the same steady state by coupling them through the negative feedback mechanism. This kind of locking is robust as it can be realized for a wide range of parameters.