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Dec 1998

Volume 8, Issue 4, pp. 739-879

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Ordering of the Mandelbrot-like set of the exponential map

M. Romera, G. Pastor, G. Álvarez, and F. Montoya

Chaos 8, 739 (1998); http://dx.doi.org/10.1063/1.166359 (2 pages) | Cited 1 time

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We graphically study the Mandelbrot-like set of the complex exponential family of maps Eλ(z) = λez, which we call the Baker–Rippon–Devaney (BRD) set. We observe that the period of every hyperbolic component can be deduced with the naked eye by using two simple rules. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.30.-f Function theory, analysis
02.10.Ab Logic and set theory

Spectral decomposition of the tent map with varying height

Suresh Subbiah and Dean J. Driebe

Chaos 8, 741 (1998); http://dx.doi.org/10.1063/1.166360 (16 pages) | Cited 1 time

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The generalized spectral decomposition of the Frobenius–Perron operator of the tent map with varying height is determined at the band-splitting points. The decomposition includes both decay onto the attracting set and the approach to the asymptotically periodic state on the attractor. Explicit compact expressions for the polynomial eigenstates are obtained using algebraic techniques. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.30.Tb Operator theory

Near threshold anomalous transport in the standard map

R. B. White, S. Benkadda, S. Kassibrakis, and G. M. Zaslavsky

Chaos 8, 757 (1998); http://dx.doi.org/10.1063/1.166361 (11 pages) | Cited 21 times

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Anomalous transport is investigated near threshold in the standard map. Very long time flights, and a large anomaly in the transport, are shown to be associated with a new form of multi-island structures causing orbit sticking. The phase space structure of these traps, and the exponents of the characteristic long time tails associated with them are determined. In general these structures are very complex, but some cases, consisting of layers of islands, allow simple modeling. © 1998 American Institute of Physics.    
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05.60.-k Transport processes

Aperiodic stochastic resonance in chaotic maps

A. Krawiecki and A. Sukiennicki

Chaos 8, 768 (1998); http://dx.doi.org/10.1063/1.166362 (7 pages) | Cited 4 times

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It is shown by means of numerical simulations that aperiodic stochastic resonance occurs in chaotic one-dimensional maps with various kinds of intermittency. The effect appears in the absence of external noise, as the system control parameter is varied. In the case of input signals slowly varying in time the analytic treatment, using the adiabatic approximation based on the expressions for the mean laminar phase duration, yields the input-output covariance function comparable with numerical results. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.60.Cb Numerical simulation; solution of equations
02.50.Ey Stochastic processes
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Nonlinear noise reduction through Monte Carlo sampling

M. E. Davies

Chaos 8, 775 (1998); http://dx.doi.org/10.1063/1.166363 (7 pages) | Cited 7 times

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We consider the problem of nonlinear noise reduction within the framework of Bayesian Theory. This enables us to place appropriate weights on the measurement and dynamic errors and thereby avoid over cleaning the data. Using a Metropolis–Hastings sampler, we are able to achieve robust noise reduction without the introduction of ad hoc parameters but at the expense of higher computational complexity. Such an algorithm should also allow us to explore the potential and limitations of other noise reduction methods. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.50.Ng Distribution theory and Monte Carlo studies
02.50.Ga Markov processes
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.70.Rr General statistical methods

Stabilizing unstable steady states using extended time-delay autosynchronization

Austin Chang, Joshua C. Bienfang, G. Martin Hall, Jeff R. Gardner, and Daniel J. Gauthier

Chaos 8, 782 (1998); http://dx.doi.org/10.1063/1.166357 (9 pages) | Cited 10 times

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We describe a method for stabilizing unstable steady states in nonlinear dynamical systems using a form of extended time-delay autosynchronization. Specifically, stabilization is achieved by applying a feedback signal generated by high-pass-filtering in real time the dynamical state of the system to an accessible system parameter or variables. Our technique is easy to implement, does not require knowledge of the unstable steady state coordinates in phase space, automatically tracks changes in the system parameters, and is more robust to broadband noise than previous schemes. We demonstrate the controller’s efficacy by stabilizing unstable steady states in an electronic circuit exhibiting low-dimensional temporal chaos. The simplicity and robustness of the scheme suggests that it is ideally suited for stabilizing unstable steady states in ultra-high-speed systems. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

A novel nonlinear car-following model

Paul S. Addison and David J. Low

Chaos 8, 791 (1998); http://dx.doi.org/10.1063/1.166364 (9 pages) | Cited 11 times

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The mathematical models used to describe the dynamical behavior of a group of road vehicles traveling in a single lane without overtaking are known as car-following models. These models are widely used in many commercially available microscopic traffic simulation software packages. They attempt to mimic the interactions between individual vehicles that are traveling sufficiently close together for the behavior of each vehicle to be dependent upon the motion of the vehicle immediately in front. In this paper we modify the traditional car-following model by adding a new nonlinear term to take account of the driver attempting to achieve a certain desired intervehicle separation distance as well as the traditional aim of matching the velocity of the vehicle ahead. Numerical solution of the resulting coupled system of nonlinear differential equations is used to analyze the stability of the equilibrium solution to a periodic perturbation. For certain parameter values chaotic oscillations are generated, consisting of a broad spectrum of frequency components. Such chaotic motion produces extremely complicated dynamical behavior that has an inherent lack of predictability associated with it. The results of simulating over a range of parameter values are presented and, where it is present, the degree of chaos is estimated. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.60.Lj Ordinary and partial differential equations; boundary value problems

Using chaos to generate variations on movement sequences

Elizabeth Bradley and Joshua Stuart

Chaos 8, 800 (1998); http://dx.doi.org/10.1063/1.166365 (8 pages) | Cited 2 times

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We describe a method for introducing variations into predefined motion sequences using a chaotic symbol-sequence reordering technique. A progression of symbols representing the body positions in a dance piece, martial arts form, or other motion sequence is mapped onto a chaotic trajectory, establishing a symbolic dynamics that links the movement sequence and the attractor structure. A variation on the original piece is created by generating a trajectory with slightly different initial conditions, inverting the mapping, and using special corpus-based graph-theoretic interpolation schemes to smooth any abrupt transitions. Sensitive dependence guarantees that the variation is different from the original; the attractor structure and the symbolic dynamics guarantee that the two resemble one another in both aesthetic and mathematical senses. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.60.Ed Interpolation; curve fitting
02.30.Lt Sequences, series, and summability
02.10.-v Logic, set theory, and algebra
02.70.-c Computational techniques; simulations

Chaotic evolution of arms races

Masaki Tomochi and Mitsuo Kono

Chaos 8, 808 (1998); http://dx.doi.org/10.1063/1.166366 (6 pages)

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A new set of model equations is proposed to describe the evolution of the arms race, by extending Richardson’s model with special emphases that (1) power dependent defensive reaction or historical enmity could be a motive force to promote armaments, (2) a deterrent would suppress the growth of armaments, and (3) the defense reaction of one nation against the other nation depends nonlinearly on the difference in armaments between two. The set of equations is numerically solved to exhibit stationary, periodic, and chaotic behavior depending on the combinations of parameters involved. The chaotic evolution is realized when the economic situation of each country involved in the arms race is quite different, which is often observed in the real world. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Synchronization of cellular automaton pairs

Jesús Urías, Gelasio Salazar, and Edgardo Ugalde

Chaos 8, 814 (1998); http://dx.doi.org/10.1063/1.166367 (5 pages)

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The phenomenon of synchronization in pairs of cellular automata coupled in a driver–replica mode is studied. Necessary and sufficient conditions for synchronization in linear cellular automaton pairs are given. The couplings that make a pair synchronize are determined for all linear elementary cellular automata. © 1998 American Institute of Physics.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.45.-a Nonlinear dynamics and chaos
02.70.-c Computational techniques; simulations

A cryptosystem based on cellular automata

Jesús Urías, Edgardo Ugalde, and Gelasio Salazar

Chaos 8, 819 (1998); http://dx.doi.org/10.1063/1.166368 (4 pages) | Cited 1 time

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Cryptosystems for binary information are based on two primitives: an indexed family of permutations of binary words and a generator of pseudorandom sequences of indices. A very efficient implementation of the primitives is constructed using the phenomenon of synchronization in cellular automata. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
84.40.Ua Telecommunications: signal transmission and processing; communication satellites

(Global and local) fluctuations of phase space contraction in deterministic stationary nonequilibrium

F. Bonetto, N. I. Chernov, and J. L. Lebowitz

Chaos 8, 823 (1998); http://dx.doi.org/10.1063/1.166369 (11 pages) | Cited 17 times

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We studied numerically the validity of the fluctuation relation introduced in Evans et al. [Phys. Rev. Lett. 71, 2401–2404 (1993)] and proved under suitable conditions by Gallavotti and Cohen [J. Stat. Phys. 80, 931–970 (1995)] for a two-dimensional system of particles maintained in a steady shear flow by Maxwell demon boundary conditions [Chernov and Lebowitz, J. Stat. Phys. 86, 953–990 (1997)]. The theorem was found to hold if one considers the total phase space contraction σ occurring at collisions with both walls: σ = σ↑+σ↓. An attempt to extend it to more local quantities σ↑ and σ↓, corresponding to the collisions with the top or bottom wall only, gave negative results. The time decay of the correlations in σ↑,↓ was very slow compared to that of σ. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.70.Ce Thermodynamic functions and equations of state
05.70.Ln Nonequilibrium and irreversible thermodynamics

Systematic derivation of amplitude equations and normal forms for dynamical systems

M. Ipsen, F. Hynne, and P. G. Sørensen

Chaos 8, 834 (1998); http://dx.doi.org/10.1063/1.166370 (19 pages) | Cited 12 times

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We present a systematic approach to deriving normal forms and related amplitude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general, explicit recurrence relation that completely determines the amplitude equation and the associated transformation from amplitudes to physical space. At any order, the relation provides explicit expressions for all the nonvanishing coefficients of the amplitude equation together with straightforward linear equations for the coefficients of the transformation. The recurrence relation therefore provides all the machinery needed to solve a given physical problem in physical terms through an amplitude equation. The new result applies to any local bifurcation of a flow or map for which all the critical eigenvalues are semisimple (i.e., have Riesz index unity). The method is an efficient and rigorous alternative to more intuitive approaches in terms of multiple time scales. We illustrate the use of the method by deriving amplitude equations and associated transformations for the most common simple bifurcations in flows and iterated maps. The results are expressed in tables in a form that can be immediately applied to specific problems. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
02.30.Uu Integral transforms
02.30.Vv Operational calculus
02.60.-x Numerical approximation and analysis

Counting unstable periodic orbits in noisy chaotic systems: A scaling relation connecting experiment with theory

Xing Pei, Kevin Dolan, Frank Moss, and Ying-Cheng Lai

Chaos 8, 853 (1998); http://dx.doi.org/10.1063/1.166371 (8 pages) | Cited 8 times

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The experimental detection of unstable periodic orbits in dynamical systems, especially those which yield short, noisy or nonstationary data sets, is a current topic of interest in many research areas. Unfortunately, for such data sets, only a few of the lowest order periods can be detected with quantifiable statistical accuracy. The primary observable is the number of encounters the general trajectory has with a particular orbit. Here we show that, in the limit of large period, this quantity scales exponentially with the period, and that this scaling is robust to dynamical noise. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Recurrence plots of experimental data: To embed or not to embed?

Joseph S. Iwanski and Elizabeth Bradley

Chaos 8, 861 (1998); http://dx.doi.org/10.1063/1.166372 (11 pages) | Cited 20 times

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A recurrence plot is a visualization tool for analyzing experimental data. These plots often reveal correlations in the data that are not easily detected in the original time series. Existing recurrence plot analysis techniques, which are primarily application oriented and completely quantitative, require that the time-series data first be embedded in a high-dimensional space, where the embedding dimension dE is dictated by the dimension d of the data set, with dE ≥ 2d+1. One such set of recurrence plot analysis tools, recurrence quantification analysis, is particularly useful in finding locations in the data where the underlying dynamics change. We have found that for certain low-dimensional systems the same results can be obtained with no embedding. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.50.-r Probability theory, stochastic processes, and statistics

Formation and evolution of scroll waves in photosensitive excitable media

Takashi Amemiya, Petteri Kettunen, Sándor Kádár, Tomohiko Yamaguchi, and Kenneth Showalter

Chaos 8, 872 (1998); http://dx.doi.org/10.1063/1.166373 (7 pages) | Cited 8 times

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Experimental and computational studies of the formation and evolution of scroll waves in three-dimensional excitable media are presented. Scroll waves are initiated in the photosensitive Belousov–Zhabotinsky reaction by perturbing traveling waves transverse to their direction of propagation. Scroll rings are generated by perturbing circular waves, which expand or contract depending on the strength of an imposed excitability gradient and its direction relative to the rotational direction of the scroll wave. © 1998 American Institute of Physics.
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82.40.Bj Oscillations, chaos, and bifurcations
82.60.Hc Chemical equilibria and equilibrium constants
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Erratum: “Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation” [Chaos 8, 20–47 (1998)]

Flavio Fenton and Alain Karma

Chaos 8, 879 (1998); http://dx.doi.org/10.1063/1.166374 (1 page) | Cited 2 times

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Abstract Unavailable
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99.10.Cd Errata
87.19.-j Properties of higher organisms
05.45.-a Nonlinear dynamics and chaos
02.60.Cb Numerical simulation; solution of equations
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