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Dec 2010

Volume 20, Issue 4, Articles (04xxxx)

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Chaos 20, 041102 (2010); http://dx.doi.org/10.1063/1.3491340 (1 page)

W. R. Matson, Bruce J. Ackerson, and Penger Tong
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Introduction: Seventh Annual Gallery of Nonlinear Images (Portland, Oregon 2010)

Greg Huber, Arshad Kudrolli, Jon Machta, and Beate Schmittmann

Chaos 20, 041101 (2010); http://dx.doi.org/10.1063/1.3530426 (1 page)

Online Publication Date: 30 December 2010

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01.10.Hx Physics organizational activities
05.45.-a Nonlinear dynamics and chaos
02.50.-r Probability theory, stochastic processes, and statistics
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Patterns in a suspension contained in a horizontally rotating cylinder

W. R. Matson, Bruce J. Ackerson, and Penger Tong

Chaos 20, 041102 (2010); http://dx.doi.org/10.1063/1.3491340 (1 page)

Online Publication Date: 30 December 2010

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47.54.De Experimental aspects
47.57.E- Suspensions
47.32.Ef Rotating and swirling flows
47.15.Rq Laminar flows in cavities, channels, ducts, and conduits
47.80.Jk Flow visualization and imaging
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Faraday instability on a network

G. Delon, D. Terwagne, N. Adami, A. Bronfort, N. Vandewalle, S. Dorbolo, and H. Caps

Chaos 20, 041103 (2010); http://dx.doi.org/10.1063/1.3518693 (1 page)

Online Publication Date: 30 December 2010

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47.20.Dr Surface-tension-driven instability
47.35.-i Hydrodynamic waves
47.54.Bd Theoretical aspects
68.03.Cd Surface tension and related phenomena
66.20.Ej Studies of viscosity and rheological properties of specific liquids
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Inviscid simulations of interacting flags

Silas Alben

Chaos 20, 041104 (2010); http://dx.doi.org/10.1063/1.3491338 (1 page)

Online Publication Date: 30 December 2010

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05.45.Pq Numerical simulations of chaotic systems
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Follow the bouncing balls! Three-dimensional imaging of flowing granular suspensions

Joshua A. Dijksman, Elie Wandersman, and Martin van Hecke

Chaos 20, 041105 (2010); http://dx.doi.org/10.1063/1.3493418 (1 page) | Cited 1 time

Online Publication Date: 30 December 2010

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47.57.Gc Granular flow
82.70.Kj Emulsions and suspensions
47.80.Jk Flow visualization and imaging
78.20.Ci Optical constants (including refractive index, complex dielectric constant, absorption, reflection and transmission coefficients, emissivity)
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Scale-local velocity fields from particle-tracking data

Douglas H. Kelley and Nicholas T. Ouellette

Chaos 20, 041106 (2010); http://dx.doi.org/10.1063/1.3489891 (1 page)

Online Publication Date: 30 December 2010

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47.80.-v Instrumentation and measurement methods in fluid dynamics
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Statistics of the general circulation from cumulant expansions

J. B. Marston

Chaos 20, 041107 (2010); http://dx.doi.org/10.1063/1.3490719 (1 page) | Cited 4 times

Online Publication Date: 30 December 2010

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92.60.Ry Climatology, climate change and variability
92.60.Bh General circulation
02.70.Rr General statistical methods
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Communities in multislice voting networks

Peter J. Mucha and Mason A. Porter

Chaos 20, 041108 (2010); http://dx.doi.org/10.1063/1.3518696 (1 page) | Cited 2 times

Online Publication Date: 30 December 2010

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89.75.Hc Networks and genealogical trees
02.60.Dc Numerical linear algebra
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The Lagrangian picture of heat transfer in convective turbulence

Maik Boltes, Herwig Zilken, and Jörg Schumacher

Chaos 20, 041109 (2010); http://dx.doi.org/10.1063/1.3497270 (1 page)

Online Publication Date: 30 December 2010

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47.27.te Turbulent convective heat transfer
47.27.nb Boundary layer turbulence
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Competition for popularity in bipartite networks

Mariano Beguerisse Díaz, Mason A. Porter, and Jukka-Pekka Onnela

Chaos 20, 043101 (2010); http://dx.doi.org/10.1063/1.3475411 (12 pages) | Cited 3 times

Online Publication Date: 8 October 2010

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We present a dynamical model for rewiring and attachment in bipartite networks. Edges are placed between nodes that belong to catalogs that can either be fixed in size or growing in size. The model is motivated by an empirical study of data from the video rental service Netflix, which invites its users to give ratings to the videos available in its catalog. We find that the distribution of the number of ratings given by users and that of the number of ratings received by videos both follow a power law with an exponential cutoff. We also examine the activity patterns of Netflix users and find bursts of intense video-rating activity followed by long periods of inactivity. We derive ordinary differential equations to model the acquisition of edges by the nodes over time and obtain the corresponding time-dependent degree distributions. We then compare our results with the Netflix data and find good agreement. We conclude with a discussion of how catalog models can be used to study systems in which agents are forced to choose, rate, or prioritize their interactions from a large set of options.
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89.75.Hc Networks and genealogical trees
89.65.Ef Social organizations; anthropology
02.30.Hq Ordinary differential equations

Perturbation series for calculation of invariant surface splitting in volume-preserving maps

N. Korneev

Chaos 20, 043102 (2010); http://dx.doi.org/10.1063/1.3496401 (8 pages)

Online Publication Date: 2 November 2010

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The invariant surface splittings for small perturbation are described for two-dimensional and three-dimensional sample volume-preserving maps by explicit analytic expressions obtained from perturbation series for the self-adjoint operator related to the Frobenius–Perron operator.
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05.45.-a Nonlinear dynamics and chaos

Motion of vortices outside a cylinder

Serdar Tulu and Oguz Yilmaz

Chaos 20, 043103 (2010); http://dx.doi.org/10.1063/1.3497915 (11 pages) | Cited 1 time

Online Publication Date: 2 November 2010

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The problem of motion of the vortices around an oscillating cylinder in the presence of a uniform flow is considered. The Hamiltonian for vortex motion for the case with no uniform flow and stationary cylinder is constructed, reduced, and constant Hamiltonian (energy) curves are plotted when the system is shown to be integrable according to Liouville. By adding uniform flow to the system and by allowing the cylinder to vibrate, we model the natural vibration of the cylinder in the flow field, which has applications in ocean engineering involving tethers or pipelines in a flow field. We conclude that in the chaotic case forces on the cylinder may be considerably larger than those on the integrable case depending on the initial positions of vortices and that complex phenomena such as chaotic capture and escape occur when the initial positions lie in a certain region.
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47.32.-y Vortex dynamics; rotating fluids
47.60.Dx Flows in ducts and channels
47.52.+j Chaos in fluid dynamics
02.30.Rz Integral equations
47.85.Np Fluidics

Hyperbolic chaos in the klystron-type microwave vacuum tube oscillator

V. V. Emel’yanov, S. P. Kuznetsov, and N. M. Ryskin

Chaos 20, 043104 (2010); http://dx.doi.org/10.1063/1.3494156 (8 pages)

Online Publication Date: 2 November 2010

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The ring-loop oscillator consisting of two coupled klystrons which is capable of generating hyperbolic chaotic signal in the microwave band is considered. The system of delayed-differential equations describing the dynamics of the oscillator is derived. This system is further reduced to the two-dimensional return map under the assumption of the instantaneous build-up of oscillations in the cavities. The results of detailed numerical simulation for both models are presented showing that there exists large enough range of control parameters where the sustained regime corresponds to the structurally stable hyperbolic chaos.
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05.45.Vx Communication using chaos
84.40.Fe Microwave tubes (e.g., klystrons, magnetrons, traveling-wave, backward-wave tubes, etc.)
02.30.Hq Ordinary differential equations

Extensive chaos in the Lorenz-96 model

A. Karimi and M. R. Paul

Chaos 20, 043105 (2010); http://dx.doi.org/10.1063/1.3496397 (11 pages) | Cited 5 times

Online Publication Date: 2 November 2010

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We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.
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05.45.Df Fractals
05.45.Jn High-dimensional chaos

Interactions between oscillatory modes near a 2:3 resonant Hopf-Hopf bifurcation

G. Revel, D. M. Alonso, and J. L. Moiola

Chaos 20, 043106 (2010); http://dx.doi.org/10.1063/1.3509771 (8 pages) | Cited 2 times

Online Publication Date: 2 November 2010

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In this paper, the dynamics near a 2:3 resonant Hopf-Hopf bifurcation is studied. The main result is the identification of a distinctive structure connecting 1:2 and 1:3 strong resonances of limit cycles. This structure is found near the Hopf-Hopf point revealing that it may be associated to the resonant case, and may provide useful information about the dynamics generated by this codimension 3 bifurcation.
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05.45.Xt Synchronization; coupled oscillators

Stochastic solutions of Navier–Stokes equations: An experimental evidence

Ivan Djurek, Danijel Djurek, and Antonio Petošić

Chaos 20, 043107 (2010); http://dx.doi.org/10.1063/1.3495962 (7 pages)

Online Publication Date: 5 November 2010

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An electrodynamic loudspeaker has been operated in anharmonic regime indicated by the nonlinear ordinary differential equation when spring constant γ in restoring term, as well as, viscoelasticity of the membrane material, increases with displacement. For driving currents in the range of 2.8–3.3 A, doubling of the vibration period appears, while for currents in the range of 3.3–3.6 A, multiple sequences of subharmonic vibrations begin with f/4 and 3f/4. An application of currents higher than 3.6 A results in a spectrum, characteristic for the chaotic state. The loudspeaker was then operated in a closed chamber, and subharmonic vibrations disappeared by an evacuation. Subsequent injection of air revoked them again at ∼ 120 mbar (Re′ = 476) when air viscous forces dominate the Morse convection. At 430 mbar (Re = 538) single vibration state was restored, and the phenomenon is in an agreement with prediction of the five mode truncation procedure applied to the Navier–Stokes equations describing a two-dimensional incompressible fluid.
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43.25.Ts Nonlinear acoustical and dynamical systems
43.38.Ja Loudspeakers and horns, practical sound sources
43.25.Dc Nonlinear acoustics of solids

Synchronization of an ensemble of oscillators regulated by their spatial movement

Sumantra Sarkar and P. Parmananda

Chaos 20, 043108 (2010); http://dx.doi.org/10.1063/1.3496399 (8 pages) | Cited 3 times

Online Publication Date: 5 November 2010

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Synchronization for a collection of oscillators residing in a finite two dimensional plane is explored. The coupling between any two oscillators in this array is unidirectional, viz., master-slave configuration. Initially the oscillators are distributed randomly in space and their autonomous time-periods follow a Gaussian distribution. The duty cycles of these oscillators, which work under an on-off scenario, are normally distributed as well. It is realized that random hopping of oscillators is a necessary condition for observing global synchronization in this ensemble of oscillators. Global synchronization in the context of the present work is defined as the state in which all the oscillators are rendered identical. Furthermore, there exists an optimal amplitude of random hopping for which the attainment of this global synchronization is the fastest. The present work is deemed to be of relevance to the synchronization phenomena exhibited by pulse coupled oscillators such as a collection of fireflies.
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05.45.Xt Synchronization; coupled oscillators
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.Ng Distribution theory and Monte Carlo studies

Phase synchronization between collective rhythms of globally coupled oscillator groups: Noisy identical case

Yoji Kawamura, Hiroya Nakao, Kensuke Arai, Hiroshi Kori, and Yoshiki Kuramoto

Chaos 20, 043109 (2010); http://dx.doi.org/10.1063/1.3491344 (10 pages) | Cited 5 times

Online Publication Date: 10 November 2010

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We theoretically investigate the collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker–Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker–Planck equations.
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05.45.Xt Synchronization; coupled oscillators
05.45.Ac Low-dimensional chaos
05.40.Ca Noise
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)

Phase synchronization between collective rhythms of globally coupled oscillator groups: Noiseless nonidentical case

Yoji Kawamura, Hiroya Nakao, Kensuke Arai, Hiroshi Kori, and Yoshiki Kuramoto

Chaos 20, 043110 (2010); http://dx.doi.org/10.1063/1.3491346 (8 pages) | Cited 13 times

Online Publication Date: 10 November 2010

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We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott–Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.
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05.45.-a Nonlinear dynamics and chaos

On the role of frustration in excitable systems

Pablo Kaluza and Hildegard Meyer-Ortmanns

Chaos 20, 043111 (2010); http://dx.doi.org/10.1063/1.3491342 (11 pages) | Cited 3 times

Online Publication Date: 10 November 2010

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We study the role of frustration in excitable systems that allow for oscillations either by construction or in an induced way. We first generalize the notion of frustration to systems whose dynamical equations do not derive from a Hamiltonian. Their couplings can be directed or undirected; they should come in pairs of opposing effects like attractive and repulsive, or activating and repressive, ferromagnetic and antiferromagnetic. As examples we then consider bistable frustrated units as elementary building blocks of our motifs of coupled units. Frustration can be implemented in these systems in various ways: on the level of a single unit via the coupling of a self-loop of positive feedback to a negative feedback loop, on the level of coupled units via the topology or via the type of coupling which may be repressive or activating. In comparison to systems without frustration, we analyze the impact of frustration on the type and number of attractors and observe a considerable enrichment of phase space, ranging from stable fixed-point behavior over different patterns of coexisting options for phase-locked motion to chaotic behavior. In particular we find multistable behavior even for the smallest motifs as long as they are frustrated. Therefore we confirm an enrichment of phase space here for excitable systems with their many applications in biological systems, a phenomenon that is familiar from frustrated spin systems and less known from frustrated phase oscillators. So the enrichment of phase space seems to be a generic effect of frustration in dynamical systems. For a certain range of parameters our systems may be realized in cell tissues. Our results point therefore on a possible generic origin for dynamical behavior that is flexible and functionally stable at the same time, since frustrated systems provide alternative paths for the same set of parameters and at the same “energy costs.”
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05.45.Xt Synchronization; coupled oscillators

Impact of degree heterogeneity on the behavior of trapping in Koch networks

Zhongzhi Zhang, Shuyang Gao, and Wenlei Xie

Chaos 20, 043112 (2010); http://dx.doi.org/10.1063/1.3493406 (6 pages) | Cited 7 times

Online Publication Date: 10 November 2010

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Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent γ of power-law degree distribution P(k) ∼ k−γ, which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent γ in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number N, which is obviously independent of γ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where γ influences qualitatively the MFPT of trapping problem.
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89.75.Hc Networks and genealogical trees
02.50.-r Probability theory, stochastic processes, and statistics
05.40.Fb Random walks and Levy flights

Fundamental solitons in discrete lattices with a delayed nonlinear response

A. Maluckov, L. Hadžievski, and B. A. Malomed

Chaos 20, 043113 (2010); http://dx.doi.org/10.1063/1.3493407 (6 pages) | Cited 1 time

Online Publication Date: 10 November 2010

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The formation of unstaggered localized modes in dynamical lattices can be supported by the interplay of discreteness and nonlinearity with a finite relaxation time. In rapidly responding nonlinear media, on-site discrete solitons are stable, and their broad intersite counterparts are marginally stable, featuring a virtually vanishing real instability eigenvalue. The solitons become unstable in the case of the slowly relaxing nonlinearity. The character of the instability alters with the increase of the delay time, which leads to a change in the dynamics of unstable discrete solitons. They form robust localized breathers in rapidly relaxing media, and decay into oscillatory diffractive pattern in the lattices with a slow nonlinear response. Marginally stable solitons can freely move across the lattice.
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05.45.Yv Solitons
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Resonance tongues in a system of globally coupled FitzHugh–Nagumo oscillators with time-periodic coupling strength

Adrian Bîrzu and Katharina Krischer

Chaos 20, 043114 (2010); http://dx.doi.org/10.1063/1.3504999 (6 pages) | Cited 8 times

Online Publication Date: 10 November 2010

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We investigate the dynamics of a population of globally coupled FitzHugh–Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.
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05.45.-a Nonlinear dynamics and chaos

Recurrence-based detection of the hyperchaos-chaos transition in an electronic circuit

E. J. Ngamga, A. Buscarino, M. Frasca, G. Sciuto, J. Kurths, and L. Fortuna

Chaos 20, 043115 (2010); http://dx.doi.org/10.1063/1.3498731 (11 pages)

Online Publication Date: 11 November 2010

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Some complex measures based on recurrence plots give evidence about hyperchaos-chaos transitions in coupled nonlinear systems [ E. G. Souza et al., “Using recurrences to characterize the hyperchaos-chaos transition,” Phys. Rev. E 78, 066206 (2008) ]. In this paper, these measures are combined with a significance test based on twin surrogates to identify such a transition in a fourth-order Lorenz-like system, which is able to pass from a hyperchaotic to a chaotic behavior for increasing values of a single parameter. A circuit analog of the mathematical model has been designed and implemented and the robustness of the recurrence-based method on experimental data has been tested. In both the numerical and experimental cases, the combination of the recurrence measures and the significance test allows to clearly identify the hyperchaos-chaos transition.
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05.45.Ra Coupled map lattices
05.45.Gg Control of chaos, applications of chaos

Transport in time-dependent dynamical systems: Finite-time coherent sets

Gary Froyland, Naratip Santitissadeekorn, and Adam Monahan

Chaos 20, 043116 (2010); http://dx.doi.org/10.1063/1.3502450 (10 pages) | Cited 14 times

Online Publication Date: 11 November 2010

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We study the transport properties of nonautonomous chaotic dynamical systems over a finite-time duration. We are particularly interested in those regions that remain coherent and relatively nondispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detect maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three-dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.
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05.45.Tp Time series analysis
05.60.-k Transport processes
92.60.Wc Weather analysis and prediction
05.10.-a Computational methods in statistical physics and nonlinear dynamics
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