Synchronization of spatiotemporal chemical chaos using random signals

We report the synchronization of two uncoupled spatially extended chemical systems by superimposing identical external random signals to both of them. In one spatial dimension, under appropriate parameter conditions the model systems exhibits a transition to turbulence via backfiring of pulses. Implementing the non-vanishing random signal control to the underlying partial differential equations, synchronization is achieved not only for identical systems, but also for systems operating under unequal parameter values exhibiting a different dynamical behavior (generalized synchronization). Finally, synchronization is also achieved under the influence of a random signal superimposed globally, thus making it relevant to experimental situations.     

Wave-induced oscillations in large-mouthed harbours

Summary The response of rectangular large-mouthed harbours to exciting monochromatic plane waves, particularly to waves with rays not orthogonal to the straight coastline, has been numerically investigated. Some of the transversal modes are found to be strongly excited, though in narrow frequency bands. The first transversal mode is quite relevant, especially in harbours with transversal size greater than the length. It can be reasonably suspected that the mechanism of excitation may be rather different from that prescribed by the theory in the case of narrow mouth. Particularly, it has been found that the eigenmodes of the basin with zero elevation on the entrance line play a primary role in determining the harbour response, especially in the excitation of the pure transversal modes.

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Riassunto Si studia, mediante un approccio di tipo numerico, la risposta di un porto rettangolare con ampia imboccatura ad un’onda piana monocromatica. Soprattutto si studia l’effetto di onde incidenti obliquamente rispetto alla linea di costa. Si trova che alcuni modi trasversali del bacino sono fortemente eccitati, anche se in bande ristrette dello spettro. Il primo modo trasversale à particolarmente rilevante in porti più larghi che lunghi. è ragionevole sospettare che il meccanismo di eccitazione sia alquanto differente da quanto previsto dalla teoria relativa a porti con bocca stretta. In particolare, si trova che le autofunzioni del bacino con elevazione nulla sull’im boccatura giocano un ruolo importante agli effetti della risposta, specialmente per quel che riguarda l’eccitazione dei modi trasversali.

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Резюме Численно исследовах отклик прямоугольных гаваней с большим входом на возбуждение монохроматических плоских волн. В частности, исследуется случай не ортогонального падения на прямую береговую линию. Обнаружено, что некоторые из поперечных мод оказываются сильно возбуждены. Первая поперечная мода соответствует гаваням с поперечным размером, превалирующим над длиной. Можно предположить, что механизм возбуждения может существенно отличаться от механизма, описанного теорией в случае узкого входа. В частности, обнаружено, что собственные моды водоема с нулевым возвышением на линии входа играют существенную роль при определении отклика особенно при возбуждении чистых поперечных мод.

     

Communication with chemical chaos in the presence of noise

We use control of chaos to encode information into the oscillations of the Belousov-Zhabotinsky reaction. An arbitrary binary message is encoded by forcing the chaotic oscillations to follow a specified trajectory. The information manipulating control requires only small perturbations to vary the binary message. In this paper we extend our recent theoretical work [Bollt and Dolnik, Phys. Rev. E 64, 1196 (1990)] by introducing a new and simplified encoding technique which can be utilized in the presence of experimental noise. We numerically and theoretically study several practical aspects of controlling symbol dynamics including: modeling noisy time-series, learning underlying symbol dynamics, and evaluation of derivatives for control by observing system responses to an intelligent and deliberate sequence of input parameter variations. All of the modeling techniques incorporated here are ultimately designed to learn and control symbol dynamics of experimental data known only as an observed time-series; the simulation assumes no global model. We find that noise affects reliability of encoding information and may cause coding errors. But, if the level of noise is confined to relatively small values, which are achievable in experiments, the control mechanism is robust to the noise. Thus we can still produce a desired symbolic code. However, scarce errors in encoding may occur due to rare but large fluctuations. These errors may be corrected during the decoding process by a variation of the filtering technique suggested by Rosa et al. [Phys. Rev. Lett. 78, 1247 (1997)]. (c) 1998 American Institute of Physics.     

Spatiotemporal chaos in the dynamics of buoyantly and diffusively unstable chemical fronts

Nonlinear dynamics resulting from the interplay between diffusive and buoyancy-driven Rayleigh-Taylor (RT) instabilities of autocatalytic traveling fronts are analyzed numerically for various values of the relevant parameters. These are the Rayleigh numbers of the reactant A and autocatalytic product B solutions as well as the ratio D=D(B)/D(A) between the diffusion coefficients of the two key chemical species. The interplay between the coarsening dynamics characteristic of the RT instability and the constant short wavelength modulation of the diffusive instability can lead in some regimes to complex dynamics dominated by irregular succession of birth and death of fingers. By using spectral entropy measurements, we characterize the transition between order and spatial disorder in this system. The analysis of the power spectrum and autocorrelation function, moreover, identifies similarities between the various spatial patterns. The contribution of the diffusive instability to the complex dynamics is discussed.     

Wave-induced bed flows by a Lagrangian vortex scheme

Nominally 2-dimensional viscous flow induced by gravity waves over a spatially periodic bed is simulated by a Lagrangian vortex scheme. A vortex sheet is introduced on the surface at each time step to satisfy the zero velocity conditions. The sheet is discretised; the vortex-in-cell method is used to convect vorticity and random walks are added to effect viscous diffusion. Good agreement with analytical theory is obtained for velocity profiles in uniform sinusoidal flow and for mass transport due to linear waves. Mass transport for finite amplitude waves is also obtained. For separated flow over rippled beds, which is still liminar, a vortex decay factor is required to produce agreement with experiment and is thought to compensate for large scale 3-dimensional effects.     

Quantum chaos

A quantum system which is allowed to interact with its boundary in a self-consistent way is shown to exhibit chaos. We conjecture that in general genuine wave chaos (decaying autocorrelation functions, exponential sensitivity of wavefunctions to initial wavefunction configurations) can be obtained whenever a wavefield is allowed to modify its confining boundaries in a self-consistent way. We suggest to test this conjecture in the acoustic regime.     

Quantum chaos

In this paper we present an overview of important recent results in the study of a very controversial topic, the so-called quantum chaos. The theoretical and numerical results are compared with real laboratory experiments with special emphasis on the problem of ionization of hydrogen atoms in external microwave fields. (c) 1996 American Institute of Physics.     

Asymptotic chaos

We provide in table I a list of normal forms of ordinary differential equations describing the dynamics of physical systems in conditions near to the simultaneous onset of up three instabilities. The first (quadratic) terms in the Taylor series for the nonlinear terms in these amplitude equations (as they are called in fluid dynamics) are given in each case. We focus on a particular example involving three instabilities and derive an asymptotic version of the corresponding normal form as a limit of small dissipation is approached. The numerical investigation of this asymptotic normal form strongly suggests that chaotic behavior occurs as close as one wants to the onset of the triple instability. This chaotic behavior is also exhibited by a return map constructed by direct numerical experiments on the amplitude equation. We also derive by analytic methods a model return map that qualitatively reproduces much of the dynamics observed numerically in the solutions of the asymptotic normal form in nearly homoclinic conditions. In the limit of strong contraction, this model map of the plane reduces to a unidimensional map that is valuable in understanding the dynamics of the original system.     

Hyperlabyrinth chaos: from chaotic walks to spatiotemporal chaos

In this paper we examine a very simple and elegant example of high-dimensional chaos in a coupled array of flows in ring architecture that is cyclically symmetric and can also be viewed as an N-dimensional spatially infinite labyrinth (a "hyperlabyrinth"). The scaling laws of the largest Lyapunov exponent, the Kaplan-Yorke dimension, and the metric entropy are investigated in the high-dimensional limit (3相似文献   

Lagrangian chaos and Eulerian chaos in shear flow dynamics

Shear flow dynamics described by the two-dimensional incompressible Navier-Stokes equations is studied for a one-dimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin-Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter lambda specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat's eyes. Lagrangian chaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of lambda, there is Lagrangian chaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of lambda Lagrangian chaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangian chaos and its relation to Eulerian chaos is given.(c) 2001 American Institute of Physics.     

Synchronizing spatiotemporal chaos

We show analytically and numerically that a pair of uni-directionally coupled spatially extended systems can synchronize. For the case of partial differential equations the synchronization can be achieved by applying the scalar driving signals only at finite number of space points. Our approach is very general and can be useful for practical applications since the synchronization is achieved via feeding in the response system only the information from certain (discrete) spatial locations of the drive system. We also stress some open problems in the field of synchronization of spatiotemporal chaos. (c) 1997 American Institute of Physics.     

Characterization of chaos

We give a brief introduction to chaos and its characterization. We examine some standard systems in detail from the perspective of chaos and review their properties. Concepts necessary to understand them, such as dimension, are also reviewed. To illustrate the main ideas, we choose three examples which have served as paradigms for the study of chaos in physical systems, namely, the Hénon discrete mapping, the Lorenz system of coupled ordinary differential equations, and the Mackey-Glass infinite-dimensional delay differential equation.     

Intermittent incommensurate chaos

Near the onset of intermittent chaos from quasiperiodic motion lying on an attracting 2D torus with rotation number ρ=ω₂/ω₁=(√5?1)/2, the power spectrum of the cartesian coordinate of the intersection point on the Poincaré section is studied. The Poincaré section is distorted from the ellipse near the onset of chaos. Then a sequence of spectral lines are excited at frequencies $\text{Ω}{}_{\mathrm{i}}\text{= ρ}{}^{\mathrm{i}}\text{Ω}{}_2\text{, (i=1,2,…)}$. Their intensities are found to obey the power law $\text{Ω}{}^4{}_{\mathrm{i}}\text{or}\text{Ω}{}^2{}_{\mathrm{i}}\text{for}\text{i ? 1}$ according as the Poincaré section has a sharp wrinkle or not. A similar spectrum is obtained also in the chaotic regime ε > 0. The mean value of time intervals of quasiperiodic states between two consecutive bursts and the square root of their variance are found to be inversely proportional to ε near the onset point g3 = 0.