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Oct 1992

Volume 2, Issue 4, pp. 469-602


Billiard in a barrel

George M. Zaslavsky and H. R. Strauss

Chaos 2, 469 (1992); http://dx.doi.org/10.1063/1.165889 (4 pages) | Cited 6 times

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Orbits in the three‐dimensional billiard of the form of a truncated ellipsoid (‘‘barrel’’) are studied both analytically and numerically. A special form of mapping is proposed to get the expression for Kolmogorov–Sinai entropy, and the transition from strong chaos to weak chaos is obtained.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos

A decoding problem in dynamics and in number theory

Ralph M. Siegel, Charles Tresser, and George Zettler

Chaos 2, 473 (1992); http://dx.doi.org/10.1063/1.165890 (21 pages) | Cited 6 times

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Given a homeomorphism f of the circle, any splitting of this circle in two semiopen arcs induces a coding process for the orbits of f, which can be determined by recording the successive arcs visited by the orbit. The problem of describing these codes has a two hundred year history (that we briefly recall) in the particular case when the arcs are limited by a point and its image; in modern language, it is the kneading theory of such maps, and as such is relevant for our understanding of dynamical problems involving oscillations. This paper deals with questions attached to the general case, a problem considered by many mathematicians in the 50’s and 60’s in the case where f is a rotation, and which has recently found some applications in physiology. We show that, except for trivial cases, any code determines the rotation number, up to the orientation, of the homeomorphism which generates it. In the case the code is periodic, we can also determine whether or not it can be generated in this way. An equivalent problem in arithmetic consists of finding ±p, knowing a collection of classes in Z/qZ of the form {m,m+p,...,m+(k−1)p}, where 2≤k≤q−2 and p and q are relatively prime. We describe this equivalence, and give simple solutions of the decoding problem both in the dynamical context and in the number theoretic context.  
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05.45.-a Nonlinear dynamics and chaos
02.10.De Algebraic structures and number theory

Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian

P. Lochak and A. I. Neishtadt

Chaos 2, 495 (1992); http://dx.doi.org/10.1063/1.165891 (5 pages) | Cited 15 times

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A Hamiltonian system differing from an integrable system by a small perturbation ≂ϵ is analyzed. According to the Nekhoroshev theorem, the changes in the perturbed motion of the ‘‘action’’ variables of the unperturbed system are small over a time interval which increases exponentially in length as ϵ decreases linearly. If the unperturbed Hamiltonian is a quasiconvex function of these ‘‘actions,’’ the changes in them remain small (≂ϵ1/2n) over a time interval on the order of exp(const/ϵ1/2n), where n is the number of degrees of freedom of the system.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

Y. X. Li, J. Halloy, J. L. Martiel, and A. Goldbeter

Chaos 2, 501 (1992); http://dx.doi.org/10.1063/1.165892 (12 pages) | Cited 2 times

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The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three‐variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five‐variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three‐variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small‐amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.
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05.45.-a Nonlinear dynamics and chaos
87.10.-e General theory and mathematical aspects

Pattern formation in an N+Q component reaction–diffusion system

John E. Pearson and William J. Bruno

Chaos 2, 513 (1992); http://dx.doi.org/10.1063/1.165893 (12 pages) | Cited 18 times

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A general N+Q component reaction–diffusion system is analyzed with regard to pattern forming instabilities (Turing bifurcations). The system consists of N mobile species and Q immobile species. The Q immobile species form in response to reactions between the N mobile species and an immobile substrate and allow the Turing instability to occur. These results are valid both for bifurcations from a spatially uniform state and for systems with an externally imposed gradient as in the experimental systems in which Turing patterns have been observed. It is shown that the critical wave number and the location of the instability in parameter space are independent of the substrate concentration. It is also found that the system necessarily undergoes a Hopf bifurcation as the total substrate concentration is decreased. Further, in the case that all the mobile species diffuse at identical rates we show that if the full system is at a point of Turing bifurcation then the N component mobile subsystem is at transition from an unstable focus to an unstable node, and the critical wave number is simply related to the degenerate positive eigenvalue of the mobile subsystem. A sequence of bifurcations that occur in the eigenspectra as the total substrate concentration is decreased to zero is also discussed.
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05.45.-a Nonlinear dynamics and chaos
87.10.-e General theory and mathematical aspects

Use of forecasting signatures to help distinguish periodicity, randomness, and chaos in ripples and other spatial patterns

David M. Rubin

Chaos 2, 525 (1992); http://dx.doi.org/10.1063/1.165894 (11 pages) | Cited 21 times

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Forecasting of one‐dimensional time series previously has been used to help distinguish periodicity, chaos, and noise. This paper presents two‐dimensional generalizations for making such distinctions for spatial patterns. The techniques are evaluated using synthetic spatial patterns and then are applied to a natural example: ripples formed in sand by blowing wind. Tests with the synthetic patterns demonstrate that the forecasting techniques can be applied to two‐dimensional spatial patterns, with the same utility and limitations as when applied to one‐dimensional time series. One limitation is that some combinations of periodicity and randomness exhibit forecasting signatures that mimic those of chaos. For example, sine waves distorted with correlated phase noise have forecasting errors that increase with forecasting distance, errors that are minimized using nonlinear models at moderate embedding dimensions, and forecasting properties that differ significantly between the original and surrogates. Ripples formed in sand by flowing air or water typically vary in geometry from one to another, even when formed in a flow that is uniform on a large scale; each ripple modifies the local flow or sand‐transport field, thereby influencing the geometry of the next ripple downcurrent. Spatial forecasting was used to evaluate the hypothesis that such a deterministic process—rather than randomness or quasiperiodicity—is responsible for the variation between successive ripples. This hypothesis is supported by a forecasting error that increases with forecasting distance, a greater accuracy of nonlinear relative to linear models, and significant differences between forecasts made with the original ripples and those made with surrogate patterns. Forecasting signatures cannot be used to distinguish ripple geometry from sine waves with correlated phase noise, but this kind of structure can be ruled out by two geometric properties of the ripples: Successive ripples are highly correlated in wavelength, and ripple crests display dislocations such as branchings and mergers.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.45.-a Nonlinear dynamics and chaos
02.50.Tt Inference methods

Topology of trajectories of the 2D Navier–Stokes equations

Jon Lee

Chaos 2, 537 (1992); http://dx.doi.org/10.1063/1.165861 (27 pages) | Cited 7 times

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In spectral form the 2D incompressible Navier–Stokes equations in a square periodic region will be represented by 430 complex Fourier amplitudes which correspond to isotropic truncation of the upper wave number 16. For small viscosity, we have found five equilibrium states I–V in the entire range of forcing; I—fixed point, II—circle, III—closed orbit, IV—torus, and V—chaos. The fixed‐point equilibrium state is the laminar flow. As the forcing passes through a critical value, the fixed point evolves directly to equilibrium state III under a typical multimode forcing. The chaotic transition takes place on a 2‐torus‐like manifold (equilibrium state IV) which is the product space of a circle and the closed orbit of equilibrium state III, similar to the quasiperiodic 2‐torus of Ruelle and Takens. For sufficiently large forcing, the evolution of equilibrium state V is nothing but a simulation of quasistationary 2D turbulence. From the Lyapunov exponents of turbulent flows, we have evaluated the constants in the theoretical results of Foias and his colleagues, which relate the determining mode and fractal dimension with the enstrophy dissipation wave number of 2D turbulence.
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47.52.+j Chaos in fluid dynamics
47.27.Cn Transition to turbulence
05.45.-a Nonlinear dynamics and chaos

Dynamical systems in the theory of solitons in the presence of nonlocal interactions

G. L. Alfimov, V. M. Eleonsky, and N. E. Kulagin

Chaos 2, 565 (1992); http://dx.doi.org/10.1063/1.165862 (6 pages) | Cited 9 times

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The behavior of solitons in models which take into account complex dispersion or nonlocal interaction of nonlinear waves is examined. A method is proposed to reduce this problem to one involving special trajectories (homoclinic and heteroclinic) of the dynamic system. This method involves replacing the nonlinear integrodifferential equation with the differential equations which link the original nonlinear field with the auxiliary linear fields. The interaction of fields in such a model is a local interaction. The number of introduced linear fields is determined by the Laplace transform of the integral operator kernel of the basic integrodifferential equation. The problem involving topological solitons for the nonlocal generalization of the Klein–Gordon equation is considered. Nonlocal interactions are found to lead to a number of singularities (unrestricted increase in the slope of the topological soliton front, break in the solutions, and other singularities).
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.30.Sa Functional analysis

Bifurcations of the trajectories at the saddle level in a Hamiltonian system generated by two coupled Schrödinger equations

V. M. Eleonsky, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov

Chaos 2, 571 (1992); http://dx.doi.org/10.1063/1.165863 (9 pages) | Cited 1 time

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Bifurcations of the complex homoclinic loops of an equilibrium saddle point in a Hamiltonian dynamical system with two degrees of freedom are studied. It arises to pick out the stationary solutions in a system of two coupled nonlinear Schrödinger equations. Their relation to bifurcations of hyperbolic and elliptic periodic orbits at the saddle level is studied for varying structural parameters of the system. Series of complex loops are described whose existence is related to periodic orbits.  
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos

Some new systems that generate a uniform stochastic web

L. Y. Yu and R. H. Parmenter

Chaos 2, 581 (1992); http://dx.doi.org/10.1063/1.165864 (8 pages) | Cited 6 times

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Evidence is given that many classes of periodically kicked Hamiltonian system with 1.5 degree of freedom generate infinite, uniform stochastic webs. The kick term in the Hamiltonian or the equation of motion need not be purely sinusoidal or some small perturbation of a sinusoidal function. For the resonance condition q=4 the structure of the web can be different from a square lattice; However, remarkably symmetric patterns of chaos are still present throughout the whole phase space. Examples are given for the square wave function and sawtooth function in the kick term of the equation of motion. The sensitive dependence on initial conditions of those systems is investigated.
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05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

The periodically kicked quantum spin

R. H. Parmenter and L. Y. Yu

Chaos 2, 589 (1992); http://dx.doi.org/10.1063/1.165865 (10 pages) | Cited 3 times

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It is shown that the same kind of deterministic chaos that occurs in classical systems can occur in certain quantum mechanical, many‐body systems. The example of the physical realization of the periodically kicked quantum spin (PKQS) is considered in detail. The quantum mechanical equations of motion for this system can be converted into the three‐dimensional PKQS map, which exhibits deterministic chaos and Arnold diffusion. Although the case of quantum spin s= 1/2 is assumed, it is shown that the same map results for s=1 (but not for s≥3/2), and for a suitably chosen classical particle with orbital angular momentum. A simple generalization of the PKQS model gives rise to stochastic webs on the surface of the unit sphere very similar to the Zaslavsky stochastic webs in a plane.  
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05.45.-a Nonlinear dynamics and chaos
03.65.Ta Foundations of quantum mechanics; measurement theory

Viscous attractor for the Galton board

W. G. Hoover and B. Moran

Chaos 2, 599 (1992); http://dx.doi.org/10.1063/1.165866 (4 pages) | Cited 10 times

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We analyze the Galton Board [or periodic ‘‘Lorentz Gas’’] with a point mass scattered by elastic disks of diameter σ, using a constant driving field g and a constant‐viscosity linear drag force −p/τ, where p is the point–mass momentum. This combination leads to a nonequilibrium steady state which depends only upon the dimensionless ratio gτ2/σ. The long‐time‐averaged trajectory leads to multifractal phase‐space structures closely resembling those we found earlier using isokinetic equations of motion derived from Gauss’ Principle of Least Constraint. A highly damped [small τ] creeping‐flow limit describes our results for gτ2/σ less than about 0.2. The lightly damped Green–Kubo linear‐response limit for the model provides an accurate description of the dissipative dynamics for gτ2/σ greater than about 2.0.
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05.45.-a Nonlinear dynamics and chaos
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
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