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Dec 2000

Volume 10, Issue 4, pp. 747-875

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Embedding dynamics for round-off errors near a periodic orbit

J. H. Lowenstein and F. Vivaldi

Chaos 10, 747 (2000); http://dx.doi.org/10.1063/1.1322027 (9 pages) | Cited 8 times

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We study the propagation of round-off errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the two-dimensional discrete phase space (a lattice) in a higher-dimensional torus, where points sharing the same round-off error are uniformly distributed within finitely many convex polyhedra. The embedding dynamics is linear and discontinuous, with algebraic integer coefficients. This representation affords efficient algorithms for classifying and computing the orbits and their exact densities, which we apply to the case of rational rotation number with denominator 7, corresponding to certain algebraic integers of degree three. We provide evidence that the hierarchical arrangement of orbits previously detected in quadratic cases [Lowenstein et al., Chaos 7, 49–66 (1997)] disappears, and that the growth of the number of orbits with the period is algebraic.© 2000 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

The Sinai billiard, square torus, and field chaos

Richard L. Liboff and Jack Liu

Chaos 10, 756 (2000); http://dx.doi.org/10.1063/1.1322028 (4 pages) | Cited 1 time

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An experiment is reported in which the Sinai quantum billiard and square-torus quantum billiard are compared for field chaos. In this mode of chaos, electromagnetic fields in a waveguide are analogous to the wave function. It is found that power loss in the square-torus guide exceeds that in the Sinai-billiard guide by approximately 3.5 dB, thereby illustrating larger field chaos for the square-torus quantum billiard than for the Sinai quantum billiard. Solutions of the Helmholtz equation are derived for the rectangular coaxial guide that illustrate that transverse electric or transverse magnetic modes exist in the guide provided the ratio of edge lengths of the outer rectangle to parallel edge lengths of the inner rectangle is rational. Eigenfunctions partition into four sets depending on even or odd reflection properties about Cartesian axis on which the concentric rectangles are oriented. These eigenfunctions are uniquely determined by four coaxial parameters and two eigen numbers. Justification of experimental findings is based on the argument that the rationals comprise a set of measure zero with respect to the irrationals. Consequently, from an observational point of view, these modes do not exist, which is in accord with the reported experiment. © 2000 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
05.45.Mt Quantum chaos; semiclassical methods

N-dimensional dynamical systems exploiting instabilities in full

J. Rius, M. Figueras, R. Herrero, J. Farjas, F. Pi, and G. Orriols

Chaos 10, 760 (2000); http://dx.doi.org/10.1063/1.1324650 (11 pages) | Cited 1 time

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We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions. © 2000 American Institute of Physics.
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05.45.Jn High-dimensional chaos

Anticontrol of chaos in continuous-time systems via time-delay feedback

Xiao Fan Wang, Guanrong Chen, and Xinghuo Yu

Chaos 10, 771 (2000); http://dx.doi.org/10.1063/1.1322358 (9 pages) | Cited 53 times

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In this paper, a systematic design approach based on time-delay feedback is developed for anticontrol of chaos in a continuous-time system. This anticontrol method can drive a finite-dimensional, continuous-time, autonomous system from nonchaotic to chaotic, and can also enhance the existing chaos of an originally chaotic system. Asymptotic analysis is used to establish an approximate relationship between a time-delay differential equation and a discrete map. Anticontrol of chaos is then accomplished based on this relationship and the differential-geometry control theory. Several examples are given to verify the effectiveness of the methodology and to illustrate the systematic design procedure. © 2000 American Institute of Physics.
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05.45.Gg Control of chaos, applications of chaos
02.40.Hw Classical differential geometry

Fractals and quantum mechanics

Nick Laskin

Chaos 10, 780 (2000); http://dx.doi.org/10.1063/1.1050284 (11 pages) | Cited 43 times

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A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Lévy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Lévy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrödinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Lévy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox’s H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional “Bohr atom” and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. © 2000 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
05.40.Jc Brownian motion
03.65.Ca Formalism
05.45.Df Fractals
12.38.-t Quantum chromodynamics

Dynamical properties of chemical systems near Hopf bifurcation points

M. Ipsen and I. Schreiber

Chaos 10, 791 (2000); http://dx.doi.org/10.1063/1.1311980 (12 pages) | Cited 1 time

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In this paper, we numerically investigate local properties of dynamical systems close to a Hopf bifurcation instability. We focus on chemical systems and present an approach based on the theory of normal forms for determining numerical estimates of the limit cycle that branches off at the Hopf bifurcation point. For several numerically ill-conditioned examples taken from chemical kinetics, we compare our results with those obtained by using traditional approaches where an approximation of the limit cycle is restricted to the center subspace spanned by critical eigenvectors, and show that inclusion of higher-order terms in the normal form expansion of the limit cycle provides a significant improvement of the limit cycle estimates. This result also provides an accurate initial estimate for subsequent numerical continuation of the limit cycle. © 2000 American Institute of Physics.
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82.40.Bj Oscillations, chaos, and bifurcations
05.45.-a Nonlinear dynamics and chaos
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

A phase transition in water coupled to a local external perturbation

D. Volchenkov and R. Lima

Chaos 10, 803 (2000); http://dx.doi.org/10.1063/1.1288710 (9 pages)

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A flux of ideal fluid coupled to perturbation is investigated by nonperturbative methods of the quantum field theory. Asymptotic behavior of the flux coupled to perturbation turns out to be similar to that of superfluids. © 2000 American Institute of Physics.
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64.70.-p Specific phase transitions

Asynchronous algorithm for integration of reaction–diffusion equations for inhomogeneous excitable media

Guillaume Rousseau and Raymond Kapral

Chaos 10, 812 (2000); http://dx.doi.org/10.1063/1.1311979 (14 pages) | Cited 7 times

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An asynchronous algorithm for the integration of reaction–diffusion equations for inhomogeneous excitable media is described. Since many physical systems are inhomogeneous where either the local kinetics or the diffusion or conduction properties vary significantly in space, integration schemes must be able to account for wide variations in the temporal and spatial scales of the solutions. The asynchronous algorithm utilizes a fixed spatial grid and automatically adjusts the time step locally to achieve an efficient simulation where the errors in the solution are controlled. The scheme does not depend on the specific form of the local kinetics and is easily applied to systems with complex geometries. © 2000 American Institute of Physics.
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82.20.Db Transition state theory and statistical theories of rate constants
02.30.-f Function theory, analysis
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.60.Cd Classical transport

Bistable reaction-diffusion systems can have robust zero-velocity fronts

Jacques-Alexandre Sepulchre and Valentin I. Krinsky

Chaos 10, 826 (2000); http://dx.doi.org/10.1063/1.1328037 (8 pages) | Cited 7 times

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We show that for a class of bistable reaction-diffusion systems, zero-velocity fronts can be robust in the singular limit where one of the diffusion coefficients vanishes. In this case, stationary fronts can persist along variations of the system parameters. This property contrasts with the standard result that the front velocity v(μ), expressed as a function of a control parameter μ, is zero only at some isolated values μ0, and thus not giving robustness to zero-velocity fronts when μ is varied. © 2000 American Institute of Physics.
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05.60.Cd Classical transport
05.45.-a Nonlinear dynamics and chaos

Nonlinear interactions in a rotating disk flow: From a Volterra model to the Ginzburg–Landau equation

E. Floriani, T. Dudok de Wit, and P. Le Gal

Chaos 10, 834 (2000); http://dx.doi.org/10.1063/1.1285863 (14 pages) | Cited 3 times

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The physical system under consideration is the flow above a rotating disk and its cross-flow instability, which is a typical route to turbulence in three-dimensional boundary layers. Our aim is to study the nonlinear properties of the wavefield through a Volterra series equation. The kernels of the Volterra expansion, which contain relevant physical information about the system, are estimated by fitting two-point measurements via a nonlinear parametric model. We then consider describing the wavefield with the complex Ginzburg–Landau equation, and derive analytical relations which express the coefficients of the Ginzburg–Landau equation in terms of the kernels of the Volterra expansion. These relations must hold for a large class of weakly nonlinear systems, in fluid as well as in plasma physics. © 2000 American Institute of Physics.
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47.27.nb Boundary layer turbulence
47.27.Cn Transition to turbulence
47.32.-y Vortex dynamics; rotating fluids
47.20.-k Flow instabilities
47.15.Fe Stability of laminar flows

Nonlinear time series analysis of normal and pathological human walking

Jonathan B. Dingwell and Joseph P. Cusumano

Chaos 10, 848 (2000); http://dx.doi.org/10.1063/1.1324008 (16 pages) | Cited 117 times

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Characterizing locomotor dynamics is essential for understanding the neuromuscular control of locomotion. In particular, quantifying dynamic stability during walking is important for assessing people who have a greater risk of falling. However, traditional biomechanical methods of defining stability have not quantified the resistance of the neuromuscular system to perturbations, suggesting that more precise definitions are required. For the present study, average maximum finite-time Lyapunov exponents were estimated to quantify the local dynamic stability of human walking kinematics. Local scaling exponents, defined as the local slopes of the correlation sum curves, were also calculated to quantify the local scaling structure of each embedded time series. Comparisons were made between overground and motorized treadmill walking in young healthy subjects and between diabetic neuropathic (NP) patients and healthy controls (CO) during overground walking. A modification of the method of surrogate data was developed to examine the stochastic nature of the fluctuations overlying the nominally periodic patterns in these data sets. Results demonstrated that having subjects walk on a motorized treadmill artificially stabilized their natural locomotor kinematics by small but statistically significant amounts. Furthermore, a paradox previously present in the biomechanical literature that resulted from mistakenly equating variability with dynamic stability was resolved. By slowing their self-selected walking speeds, NP patients adopted more locally stable gait patterns, even though they simultaneously exhibited greater kinematic variability than CO subjects. Additionally, the loss of peripheral sensation in NP patients was associated with statistically significant differences in the local scaling structure of their walking kinematics at those length scales where it was anticipated that sensory feedback would play the greatest role. Lastly, stride-to-stride fluctuations in the walking patterns of all three subject groups were clearly distinguishable from linearly autocorrelated Gaussian noise. As a collateral benefit of the methodological approach taken in this study, some of the first steps at characterizing the underlying structure of human locomotor dynamics have been taken. Implications for understanding the neuromuscular control of locomotion are discussed. © 2000 American Institute of Physics.
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87.19.rs Movement
87.19.ru Locomotion
05.45.Tp Time series analysis
05.40.Ca Noise

Timely detection of dynamical change in scalp EEG signals

L. M. Hively, V. A. Protopopescu, and P. C. Gailey

Chaos 10, 864 (2000); http://dx.doi.org/10.1063/1.1312369 (12 pages) | Cited 26 times

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We present a robust, model-independent technique for quantifying changes in the dynamics underlying nonlinear time-serial data. After constructing discrete density distributions of phase-space points on the attractor for time-windowed data sets, we measure the dissimilarity between density distributions via L1-distance and χ2 statistics. The discriminating power of the new measures is first tested on data generated by the Bondarenko “synthetic brain” model. We also compare traditional nonlinear measures and the new dissimilarity measures to detect dynamical change in scalp EEG data. The results demonstrate a clear superiority of the new measures in comparison to traditional nonlinear measures as robust and timely discriminators of changing dynamics. © 2000 American Institute of Physics.
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87.19.R- Mechanical and electrical properties of tissues and organs
05.45.Tp Time series analysis
02.50.-r Probability theory, stochastic processes, and statistics
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