Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps-Introduction to the Method.

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems' stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.

Interacting plasmon and phonon polaritons in aligned nano- and microwires.

The availability of macroscopic, nearly periodic structures known as eutectics opens a new path for controlling light at wavelength scales determined by the geometrical parameters of these materials and the intrinsic properties of their component phases. Here, we analyze the optical waveguiding properties of eutectic mixtures of alkali halides, formed by close-packed arrangements of aligned cylindrical inclusions. The wavelengths of phonon polaritons in these constituents are conveniently situated in the infrared and are slightly larger than the diameter and separation of the inclusions, typically consisting on single-crystal wires down to submicrometer diameter. We first discuss the gap mode and the guiding properties of metallic cylindrical waveguides in the visible and near-infrared, and in particular we investigate the transition between cylinder touching and non-touching regimes. Then, we demonstrate that these properties can be extended to the mid infrared by means of phonon polaritons. Finally, we analyze the guiding properties of an actual eutectic. For typical eutectic dimensions, we conclude that crosstalk between neighboring cylindrical wires is small, thus providing a promising platform for signal propagation and image analysis in the mid infrared.

Why two clocks synchronize: energy balance of the synchronized clocks.

We consider the synchronization of two clocks which are accurate (show the same time) but have pendulums with different masses. We show that such clocks hanging on the same beam beside the complete (in-phase) and antiphase synchronizations perform the third type of synchronization in which the difference of the pendulums' displacements is a periodic function of time. We identify this period to be a few times larger than the period of pendulums' oscillations in the case when the beam is at rest. Our approximate analytical analysis allows to derive the synchronizations conditions, explains the observed types of synchronizations, and gives the approximate formula for both the pendulums' amplitudes and the phase shift between them. We consider the energy balance in the system and show how the energy is transferred between pendulums via oscillating beam allowing pendulums' synchronization.

Quantifying the synchronizability of externally driven oscillators.

This paper is focused on the problem of complete synchronization in arrays of externally driven identical or slightly different oscillators. These oscillators are coupled by common driving which makes an occurrence of generalized synchronization between a driving signal and response oscillators possible. Therefore, the phenomenon of generalized synchronization is also analyzed here. The research is concentrated on the cases of an irregular (chaotic or stochastic) driving signal acting on continuous-time (Duffing systems) and discrete-time (Henon maps) response oscillators. As a tool for quantifying the robustness of the synchronized state, response (conditional) Lyapunov exponents are applied. The most significant result presented in this paper is a novel method of estimation of the largest response Lyapunov exponent. This approach is based on the complete synchronization of two twin response subsystems via additional master-slave coupling between them. Examples of the method application and its comparison with the classical algorithm for calculation of Lyapunov exponents are widely demonstrated. Finally, the idea of effective response Lyapunov exponents, which allows us to quantify the synchronizability in case of slightly different response oscillators, is introduced.

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.

Ragged synchronizability of coupled oscillators.

We discuss synchronization thresholds in an array of nondiagonally coupled oscillators. We argue that nondiagonal coupling can cause the appearance or disappearance of desynchronous windows in the coupling parameter space. Such a phenomenon is independent of the motion character (periodic or chaotic) of the isolated node system. A mechanism governing this phenomenon is explained and its influence on the global network dynamics is analyzed.

Dynamics of an array of mutually coupled semiconductor lasers.

We consider the dynamics of a linear array of coupled semiconductor lasers. Particular attention is paid to the synchronous states, which are caused by the permutation of two outer lasers. A system of three coupled lasers is studied in more details. We report different types of multistability of synchronous and asynchronous states including chaotic ones. We identify parameter values, for which a synchronous chaos can occur. Moreover, we show that transition to the synchronization occurs via blowup of the synchronous transversely unstable invariant set within the synchronization manifold. Finally, we present numerical analysis of larger arrays of coupled lasers and note some common qualitative features of the synchronization regions, which are independent of the number of lasers.

Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems.

In this paper, we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems (flows or maps) with diagonal diffusive coupling. The condition of chaotic synchronization is determined only by two "parameters of order," i.e., the largest Lyapunov exponent and the coupling coefficient. Our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular, arbitrarily assumed structure of coupling. The main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems. Numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision.