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Dec 1999

Volume 9, Issue 4, pp. 819-931

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Global analysis of periodic orbit bifurcations in coupled Morse oscillator systems: time-reversal symmetry, permutational representations and codimension-2 collisions

Masa Tsuchiya and Gregory S. Ezra

Chaos 9, 819 (1999); http://dx.doi.org/10.1063/1.166466 (22 pages) | Cited 1 time

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In this paper we study periodic orbit bifurcation sequences in a system of two coupled Morse oscillators. Time-reversal symmetry is exploited to determine periodic orbits by iteration of symmetry lines. The permutational representation of Tsuchiya and Jaffé is employed to analyze periodic orbit configurations on the symmetry lines. Local pruning rules are formulated, and a global analysis of possible bifurcation sequences of symmetric periodic orbits is made. Analysis of periodic orbit bifurcations on symmetry lines determines bifurcation sequences, together with periodic orbit periodicities and stabilities. The correlation between certain bifurcations is explained. The passage from an integrable limit to nointegrability is marked by the appearance of tangent bifurcations; our global analysis reveals the origin of these ubiquitous tangencies. For period-1 orbits, tangencies appear by a simple disconnection mechanism. For higher period orbits, a different mechanism involving 2-parameter collisions of bifurcations is found. © 1999 American Institute of Physics.
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05.45.Xt Synchronization; coupled oscillators

Dynamics of a gravitational billiard with a hyperbolic lower boundary

M. L. Ferguson, B. N. Miller, and M. A. Thompson

Chaos 9, 841 (1999); http://dx.doi.org/10.1063/1.166467 (8 pages) | Cited 4 times

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Gravitational billiards provide a simple method for the illustration of the dynamics of Hamiltonian systems. Here we examine a new billiard system with two parameters, which exhibits, in two limiting cases, the behaviors of two previously studied one-parameter systems, namely the wedge and parabolic billiard. The billiard consists of a point mass moving in two dimensions under the influence of a constant gravitational field with a hyperbolic lower boundary. An iterative mapping between successive collisions with the lower boundary is derived analytically. The behavior of the system during transformation from the wedge to the parabola is investigated for a few specific cases. It is surprising that the nature of the transformation depends strongly on the parameter values. © 1999 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.60.-x Numerical approximation and analysis

Lyapunov exponents, dual Lyapunov exponents, and multifractal analysis

Aihua Fan and Yunping Jiang

Chaos 9, 849 (1999); http://dx.doi.org/10.1063/1.166468 (5 pages)

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It is shown that the multifractal property is shared by both Lyapunov exponents and dual Lyapunov exponents related to scaling functions of one-dimensional expanding folding maps. This reveals in a quantitative way the complexity of the dynamics determined by such maps. © 1999 American Institute of Physics.
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05.45.Df Fractals

Calculations of periodic orbits: The monodromy method and application to regularized systems

N. S. Simonović

Chaos 9, 854 (1999); http://dx.doi.org/10.1063/1.166457 (11 pages) | Cited 5 times

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We describe a numerical method for calculating periodic orbits, which is a generalization of the monodromy method by Baranger et al. to the case of an arbitrary autonomous dynamical system. Two variants of the method are developed, using the midpoint and the Runge–Kutta discretization of equations of motion, respectively. Particularly, we adapt the first variant for calculating periodic orbits of Hamiltonian systems when the period or the energy is given a priori. Finally, we consider the application of the monodromy method to the case of regularized mechanical systems and demonstrate the use by two examples. © 1999 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Detecting dynamical nonstationarity in time series data

Dejin Yu, Weiping Lu, and Robert G. Harrison

Chaos 9, 865 (1999); http://dx.doi.org/10.1063/1.166458 (6 pages) | Cited 7 times

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Nonlinear time series analysis is becoming an ever more powerful tool to explore complex phenomena and uncover underlying patterns from irregular data recorded from experiments. However, the existence of dynamical nonstationarity in time series data causes many results of such analysis to be questionable and inconclusive. It is increasingly recognized that detecting dynamical nonstationarity is a crucial precursor to data analysis. In this paper, we present a test procedure to detect dynamical nonstationarity by directly inspecting the dependence of nonlinear statistical distributions on absolute time along a trajectory in phase space. We test this method using a broad range of data, chaotic, stochastic and power-law noise, both computer-generated and observed, and show that it provides a reliable test method in analyzing experimental data. © 1999 American Institute of Physics.
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05.45.Tp Time series analysis
05.40.Ca Noise
02.50.Ey Stochastic processes

Transport in finite size systems: An exit time approach

P. Castiglione, M. Cencini, A. Vulpiani, and E. Zambianchi

Chaos 9, 871 (1999); http://dx.doi.org/10.1063/1.166459 (9 pages) | Cited 4 times

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In the framework of chaotic scattering we analyze passive tracer transport in finite systems. In particular, we study models with open streamlines and a finite number of recirculation zones. In the nontrivial case with a small number of recirculation zones a description by means of asymptotic quantities (such as the eddy diffusivity) is not appropriate. The nonasymptotic properties of dispersion are characterized by means of the exit time statistics, which shows strong sensitivity on initial conditions. This yields a probability distribution function with long tails, making impossible a characterization in terms of a unique typical exit time. © 1999 American Institute of Physics.
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05.60.Cd Classical transport
05.45.-a Nonlinear dynamics and chaos
02.50.Cw Probability theory

Threshold, excitability and isochrones in the Bonhoeffer–van der Pol system

A. Rabinovitch and I. Rogachevskii

Chaos 9, 880 (1999); http://dx.doi.org/10.1063/1.166460 (7 pages) | Cited 10 times

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Some new insight is obtained for the structure of the Bonhoeffer–van der Pol system. The problems of excitability and threshold are discussed for all three types of the system classified according to the existing attractors: a focus only, a limit cycle only and a limit cycle together with a focus. These problems can be treated by the T-repellers and the T-attractors of the system which are mutually reciprocal under time inversion. The threshold depends on the structure of the T-repeller (unstable part of integral manifold). This structure is then used to understand the behavior and the properties of the two different types of isochrones: Winfree isochrones (W-isochrones) and regular isochrones. Winfree’s description of a W-isochrone is extended to excitable systems. Both W-isochrones and regular isochrones are calculated for the Bonhoeffer–van der Pol system in its limit cycle and excitable regimes. The important role of the T-repeller as an asymptotic limit for both types of isochrones is manifested. © 1999 American Institute of Physics.
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05.45.Pq Numerical simulations of chaotic systems
05.45.Tp Time series analysis

An expansion of system with time delayed feedback control into spatio-temporal state space

Takashi Hikihara and Yoshisuke Ueda

Chaos 9, 887 (1999); http://dx.doi.org/10.1063/1.166461 (6 pages) | Cited 5 times

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Time delayed feedback control is well known as an effective continuous control method for stabilizing the unstable periodic orbit embedded in chaotic attractors. As for the system with time delay, the solution is in a function space and shows characteristics governed by an infinite dimension. Therefore it is difficult to understand the system behavior analytically. In this paper, it is shown that, when the state space for the system with time delay is expanded into the spatio-temporal state space, the solution propagates in the space as a wave theoretically and numerically. The dynamic behavior experimentally obtained in the sinusoidally excited magneto-elastic beam system under time delayed feedback control is also discussed by the transformation into the expanded spatio-temporal state space. © 1999 American Institute of Physics.
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05.45.Gg Control of chaos, applications of chaos
75.80.+q Magnetomechanical effects, magnetostriction

Traveling waves and chaotic properties in cellular automata

M. Courbage, D. Mercier, and S. Yasmineh

Chaos 9, 893 (1999); http://dx.doi.org/10.1063/1.166462 (9 pages) | Cited 2 times

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Traveling wave solutions of cellular automata (CA) with two states and nearest neighbors interaction on one-dimensional (1-D) infinite lattice are computed. Space and time periods and the number of distinct waves have been computed for all representative rules, and each velocity ranging from 2 to 22. This computation shows a difference between spatially extended systems, generating only temporal chaos and those producing as well spatial complexity. In the first case wavelengths are simply related to the velocity of propagation and the dispersivity is an affine function, while in the second case (which coincides with Wolfram class 3), the dispersivity is multiform and its dependence on the velocities is highly random and discontinuous. This property is typical of space–time chaos in CA. © 1999 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Anomalous spatio-temporal chaos in a two-dimensional system of nonlocally coupled oscillators

Hiroya Nakao

Chaos 9, 902 (1999); http://dx.doi.org/10.1063/1.166463 (8 pages) | Cited 6 times

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A two-dimensional system of nonlocally coupled complex Ginzburg–Landau oscillators is investigated numerically for the first time. As previously shown for the one-dimensional case, this two-dimensional system exhibits anomalous spatio-temporal chaos characterized by power-law spatial correlations. In this chaotic regime, the amplitude difference between neighboring elements displays temporal noisy on–off intermittency. The system is also spatially intermittent in this regime, as revealed by multiscaling analysis: The amplitude field is multiaffine and the difference field is multifractal. Correspondingly, the probability distribution function of the measure defined for each field is strongly non-Gaussian, exhibiting scale-dependent deviations in the tail due to intermittency. © 1999 American Institute of Physics.
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05.45.Df Fractals
02.50.Cw Probability theory

Chaotic mixing of granular material in slowly rotating containers as a discrete mapping

T. Elperin and A. Vikhansky

Chaos 9, 910 (1999); http://dx.doi.org/10.1063/1.166464 (6 pages) | Cited 8 times

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Chaotic mixing of granular material in a two-dimensional slowly rotating noncircular container in the absence of granular diffusivity is studied analytically and numerically as a discrete mapping. The noncircularity of a drum produces a time periodic disturbance and chaotization of the flow field. The location of the fixed points of the mapping and the separatrices of the hyperbolic points are determined in a closed analytical form.© 1999 American Institute of Physics.
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45.70.Mg Granular flow: mixing, segregation and stratification
05.45.-a Nonlinear dynamics and chaos
47.55.Kf Particle-laden flows
47.60.-i Flow phenomena in quasi-one-dimensional systems
47.32.-y Vortex dynamics; rotating fluids

Hysteresis and bistability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells

Ali R. Yehia, Dominique Jeandupeux, Francisco Alonso, and Michael R. Guevara

Chaos 9, 916 (1999); http://dx.doi.org/10.1063/1.166465 (16 pages) | Cited 29 times

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The transmembrane potential of a single quiescent cell isolated from rabbit ventricular muscle was recorded using a suction electrode in whole-cell recording mode. The cell was then driven with a periodic train of current pulses injected into the cell through the same recording electrode. When the interpulse interval or basic cycle length (BCL) was sufficiently long, 1:1 rhythm resulted, with each stimulus pulse producing an action potential. Gradual decrease in BCL invariably resulted in loss of 1:1 synchronization at some point. When the pulse amplitude was set to a fixed low level and BCL gradually decreased, N+1:N rhythms (N ≥ 2) reminiscent of clinically observed Wenckebach rhythms were seen. Further decrease in BCL then yielded a 2:1 rhythm. In contrast, when the pulse amplitude was set to a fixed high level, a period-doubled 2:2 rhythm resembling alternans rhythm was seen before a 2:1 rhythm occurred. With the pulse amplitude set to an intermediate level (i.e., to a level between those at which Wenckebach and alternans rhythms were seen), there was a direct transition from 1:1 to 2:1 rhythm as the BCL was decreased: Wenckebach and alternans rhythms were not seen. When at that point the BCL was increased, the transition back to 1:1 rhythm occurred at a longer BCL than that at which the {1:1→2:1} transition had initially occurred, demonstrating hysteresis. With the BCL set to a value within the hysteresis range, injection of a single well-timed extrastimulus converted 1:1 rhythm into 2:1 rhythm or vice versa, providing incontrovertible evidence of bistability (the coexistence of two different periodic rhythms at a fixed set of stimulation parameters). Hysteresis between 1:1 and 2:1 rhythms was also seen when the stimulus amplitude, rather than the BCL, was changed. Simulations using numerical integration of an ionic model of a single ventricular cell formulated as a nonlinear system of differential equations provided results that were very similar to those found in the experiments. The steady-state action potential duration restitution curve, which is a plot of the duration of the action potential during 1:1 rhythm as a function of the recovery time or diastolic interval immediately preceding that action potential, was determined. Iteration of a finite-difference equation derived using the restitution curve predicted the direct {1:1↔2:1} transition, as well as bistability, in both the experimental and modeling work. However, prediction of the action potential duration during 2:1 rhythm was not as accurate in the experiments as in the model. Finally, we point out a few implications of our findings for cardiac arrhythmias (e.g., Mobitz type II block, ischemic alternans). © 1999 American Institute of Physics.
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87.17.-d Cell processes
87.19.R- Mechanical and electrical properties of tissues and organs
87.19.Hh Cardiac dynamics
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