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Jun 2013

Volume 23, Issue 2 (partial)

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Attracting and repelling Lagrangian coherent structures from a single computation

Mohammad Farazmand and George Haller

Chaos 23, 023101 (2013); http://dx.doi.org/10.1063/1.4800210 (11 pages)

Online Publication Date: 12 April 2013

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Hyperbolic Lagrangian Coherent Structures (LCSs) are locally most repelling or most attracting material surfaces in a finite-time dynamical system. To identify both types of hyperbolic LCSs at the same time instance, the standard practice has been to compute repelling LCSs from future data and attracting LCSs from past data. This approach tacitly assumes that coherent structures in the flow are fundamentally recurrent, and hence gives inconsistent results for temporally aperiodic systems. Here, we resolve this inconsistency by showing how both repelling and attracting LCSs are computable at the same time instance from a single forward or a single backward run. These LCSs are obtained as surfaces normal to the weakest and strongest eigenvectors of the Cauchy-Green strain tensor.
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02.30.-f Function theory, analysis
02.10.Ud Linear algebra

Intermittency in relation with 1/f noise and stochastic differential equations

J. Ruseckas and B. Kaulakys

Chaos 23, 023102 (2013); http://dx.doi.org/10.1063/1.4802429 (8 pages)

Online Publication Date: 17 April 2013

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One of the models of intermittency is on-off intermittency, arising due to time-dependent forcing of a bifurcation parameter through a bifurcation point. For on-off intermittency, the power spectral density (PSD) of the time-dependent deviation from the invariant subspace in a low frequency region exhibits 1/math power-law noise. Here, we investigate a mechanism of intermittency, similar to the on-off intermittency, occurring in nonlinear dynamical systems with invariant subspace. In contrast to the on-off intermittency, we consider the case where the transverse Lyapunov exponent is zero. We show that for such nonlinear dynamical systems, the power spectral density of the deviation from the invariant subspace can have 1/fβ form in a wide range of frequencies. That is, such nonlinear systems exhibit 1/f noise. The connection with the stochastic differential equations generating 1/fβ noise is established and analyzed, as well.
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05.45.-a Nonlinear dynamics and chaos
02.30.Jr Partial differential equations
02.50.Ey Stochastic processes
05.40.Ca Noise

Beyond long memory in heart rate variability: An approach based on fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity

Argentina Leite, Ana Paula Rocha, and Maria Eduarda Silva

Chaos 23, 023103 (2013); http://dx.doi.org/10.1063/1.4802035 (10 pages)

Online Publication Date: 19 April 2013

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Heart Rate Variability (HRV) series exhibit long memory and time-varying conditional variance. This work considers the Fractionally Integrated AutoRegressive Moving Average (ARFIMA) models with Generalized AutoRegressive Conditional Heteroscedastic (GARCH) errors. ARFIMA-GARCH models may be used to capture and remove long memory and estimate the conditional volatility in 24 h HRV recordings. The ARFIMA-GARCH approach is applied to fifteen long term HRV series available at Physionet, leading to the discrimination among normal individuals, heart failure patients, and patients with atrial fibrillation.
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87.85.Ng Biological signal processing
02.50.Ey Stochastic processes
87.19.Hh Cardiac dynamics
87.19.X- Diseases

Bifurcations in a low-order nonlinear model of tropical Pacific sea surface temperatures derived from observational data

Mei Hong, Ren Zhang, Hui-Zan Wang, Jing-jing Ge, and Ao-Da Pan

Chaos 23, 023104 (2013); http://dx.doi.org/10.1063/1.4802036 (10 pages)

Online Publication Date: 19 April 2013

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Aiming at tackling the difficulty in exactly constituting the sea surface temperature (SST) dynamical model, the paper introduces the dynamical system reconstruction idea and establishes the nonlinear dynamical model of SST field based on 1963-2010 monthly average Hadley SST data. Time coefficients series after empirical orthogonal functions decomposition are taken as the dynamical model variables and Genetic Algorithms is used to optimize and retrieve the model parameters. The stability of the equilibrium in the reconstructed model is analyzed and dynamical actions such as bifurcation and mutation are discussed. Also the activity configuration and aberrance mechanism of the SST field are developed upon the actual activity characteristics of the SST field in the Tropical Pacific Ocean in that year. Results reveal that the bifurcation action of the SST field system from one stable high-value equilibrium to another stable low-value equilibrium accords with the La Niña process while the mutation action of the SST field system from two stable equilibriums to another stable equilibrium accords with the El Niño process.
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92.10.af Thermohaline convection
92.10.am El Nino Southern Oscillation
93.30.Pm Pacific Ocean
02.30.Oz Bifurcation theory

Characterization of multiscroll attractors using Lyapunov exponents and Lagrangian coherent structures

Filipe I. Fazanaro, Diogo C. Soriano, Ricardo Suyama, Romis Attux, Marconi K. Madrid, and José Raimundo de Oliveira

Chaos 23, 023105 (2013); http://dx.doi.org/10.1063/1.4802428 (10 pages)

Online Publication Date: 19 April 2013

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multimedia

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The present work aims to apply a recently proposed method for estimating Lyapunov exponents to characterize—with the aid of the metric entropy and the fractal dimension—the degree of information and the topological structure associated with multiscroll attractors. In particular, the employed methodology offers the possibility of obtaining the whole Lyapunov spectrum directly from the state equations without employing any linearization procedure or time series-based analysis. As a main result, the predictability and the complexity associated with the phase trajectory were quantified as the number of scrolls are progressively increased for a particular piecewise linear model. In general, it is shown here that the trajectory tends to increase its complexity and unpredictability following an exponential behaviour with the addition of scrolls towards to an upper bound limit, except for some degenerated situations where a non-uniform grid of scrolls is attained. Moreover, the approach employed here also provides an easy way for estimating the finite time Lyapunov exponents of the dynamics and, consequently, the Lagrangian coherent structures for the vector field. These structures are particularly important to understand the stretching/folding behaviour underlying the chaotic multiscroll structure and can provide a better insight of phase space partition and exploration as new scrolls are progressively added to the attractor.
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05.45.Ac Low-dimensional chaos
05.45.Df Fractals
05.45.Tp Time series analysis
02.40.Pc General topology

Child allowances, fertility, and chaotic dynamics

Hung-Ju Chen and Ming-Chia Li

Chaos 23, 023106 (2013); http://dx.doi.org/10.1063/1.4802034 (9 pages)

Online Publication Date: 24 April 2013

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This paper analyzes the dynamics in an overlapping generations model with the provision of child allowances. Fertility is an increasing function of child allowances and there exists a threshold effect of the marginal effect of child allowances on fertility. We show that if the effectiveness of child allowances is sufficiently high, an intermediate-sized tax rate will be enough to generate chaotic dynamics. Besides, a decrease in the inter-temporal elasticity of substitution will prevent the occurrence of irregular cycles.
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87.85.-d Biomedical engineering
02.30.Oz Bifurcation theory
05.45.-a Nonlinear dynamics and chaos
87.10.Pq Elasticity theory

Effective suppressibility of chaos

Álvaro G. López, Jesús M. Seoane, and Miguel A. F. Sanjuán

Chaos 23, 023107 (2013); http://dx.doi.org/10.1063/1.4803521 (9 pages)

Online Publication Date: 8 May 2013

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Suppression of chaos is a relevant phenomenon that can take place in nonlinear dynamical systems when a parameter is varied. Here, we investigate the possibilities of effectively suppressing the chaotic motion of a dynamical system by a specific time independent variation of a parameter of our system. In realistic situations, we need to be very careful with the experimental conditions and the accuracy of the parameter measurements. We define the suppressibility, a new measure taking values in the parameter space, that allows us to detect which chaotic motions can be suppressed, what possible new choices of the parameter guarantee their suppression, and how small the parameter variations from the initial chaotic state to the final periodic one are. We apply this measure to a Duffing oscillator and a system consisting on ten globally coupled Hénon maps. We offer as our main result tool sets that can be used as guides to suppress chaotic dynamics.
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05.45.-a Nonlinear dynamics and chaos
02.30.Uu Integral transforms

Robust global synchronization of two complex dynamical networks

Mohammad Mostafa Asheghan and Joaquín Míguez

Chaos 23, 023108 (2013); http://dx.doi.org/10.1063/1.4803522 (11 pages)

Online Publication Date: 8 May 2013

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We investigate the synchronization of two coupled complex dynamical networks, a problem that has been termed outer synchronization in the literature. Our approach relies on (a) a basic lemma on the eigendecomposition of matrices resulting from Kronecker products and (b) a suitable choice of Lyapunov function related to the synchronization error dynamics. Starting from these two ingredients, a theorem that provides a sufficient condition for outer synchronization of the networks is proved. The condition in the theorem is expressed as a linear matrix inequality. When satisfied, synchronization is guaranteed to occur globally, i.e., independently of the initial conditions of the networks. The argument of the proof includes the design of the gain of the synchronizer, which is a constant square matrix with dimension dependent on the number of dynamic variables in a single network node, but independent of the size of the overall network, which can be much larger. This basic result is subsequently elaborated to simplify the design of the synchronizer, to avoid unnecessarily restrictive assumptions (e.g., diffusivity) on the coupling matrix that defines the topology of the networks and, finally, to obtain synchronizers that are robust to model errors in the parameters of the coupled networks. An illustrative numerical example for the outer synchronization of two networks of classical Lorenz nodes with perturbed parameters is presented.
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89.75.Hc Networks and genealogical trees
02.10.Yn Matrix theory
02.40.Pc General topology
05.45.Xt Synchronization; coupled oscillators

Eigenstates and instabilities of chains with embedded defects

J. D'Ambroise, P. G. Kevrekidis, and S. Lepri

Chaos 23, 023109 (2013); http://dx.doi.org/10.1063/1.4803523 (10 pages)

Online Publication Date: 8 May 2013

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We consider the eigenvalue problem for one-dimensional linear Schrödinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples, we discuss both linear and nonlinear, Hamiltonian and PT-symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra

Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency

Seiji Uenohara, Takahito Mitsui, Yoshito Hirata, Takashi Morie, Yoshihiko Horio, and Kazuyuki Aihara

Chaos 23, 023110 (2013); http://dx.doi.org/10.1063/1.4804181 (9 pages)

Online Publication Date: 8 May 2013

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We experimentally study strange nonchaotic attractors (SNAs) and chaotic attractors by using a nonlinear integrated circuit driven by a quasiperiodic input signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. It is a difficult problem to distinguish between SNAs and chaotic attractors experimentally. If a system has an SNA as a unique attractor, the system produces an identical response to a repeated quasiperiodic signal, regardless of the initial conditions, after a certain transient time. Such reproducibility of response outputs is called consistency. On the other hand, if the attractor is chaotic, the consistency is low owing to the sensitive dependence on initial conditions. In this paper, we analyze the experimental data for distinguishing between SNAs and chaotic attractors on the basis of the consistency.
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05.45.-a Nonlinear dynamics and chaos

Four dimensional chaos and intermittency in a mesoscopic model of the electroencephalogram

Mathew P. Dafilis, Federico Frascoli, Peter J. Cadusch, and David T. J. Liley

Chaos 23, 023111 (2013); http://dx.doi.org/10.1063/1.4804176 (7 pages)

Online Publication Date: 9 May 2013

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The occurrence of so-called four dimensional chaos in dynamical systems represented by coupled, nonlinear, ordinary differential equations is rarely reported in the literature. In this paper, we present evidence that Liley's mesoscopic theory of the electroencephalogram (EEG), which has been used to describe brain activity in a variety of clinically relevant contexts, possesses a chaotic attractor with a Kaplan-Yorke dimension significantly larger than three. This accounts for simple, high order chaos for a physiologically admissible parameter set. Whilst the Lyapunov spectrum of the attractor has only one positive exponent, the contracting dimensions are such that the integer part of the Kaplan-Yorke dimension is three, thus giving rise to four dimensional chaos. A one-parameter bifurcation analysis with respect to the parameter corresponding to extracortical input is conducted, with results indicating that the origin of chaos is due to an inverse period doubling cascade. Hence, in the vicinity of the high order, strange attractor, the model is shown to display intermittent behavior, with random alternations between oscillatory and chaotic regimes. This phenomenon represents a possible dynamical justification of some of the typical features of clinically established EEG traces, which can arise in the case of burst suppression in anesthesia and epileptic encephalopathies in early infancy.
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87.19.le EEG and MEG
87.19.L- Neuroscience
87.85.D- Applied neuroscience
05.45.-a Nonlinear dynamics and chaos

Shunting inhibitory cellular neural networks with chaotic external inputs

M. U. Akhmet and M. O. Fen

Chaos 23, 023112 (2013); http://dx.doi.org/10.1063/1.4805022 (9 pages)

Online Publication Date: 17 May 2013

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Taking advantage of external inputs, it is shown that shunting inhibitory cellular neural networks behave chaotically. The analysis is based on the Li-Yorke definition of chaos. Appropriate illustrations which support the theoretical results are depicted.
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07.05.Mh Neural networks, fuzzy logic, artificial intelligence
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