Bulletin of the American Physical Society
APS March Meeting 2013
Volume 58, Number 1
Monday–Friday, March 18–22, 2013; Baltimore, Maryland
Session W30: Nonlinear Dynamics |
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Sponsoring Units: GSNP Chair: David Egolf, Georgetown University Room: 338 |
Thursday, March 21, 2013 2:30PM - 2:42PM |
W30.00001: Time Reversal Experiments in Chaotic Cavities Bo Xiao, Edwart Ott, Thomas Antonsen, Steven Anlage Wave focusing through a strongly scattering medium has been an intriguing topic in the fields of optics, acoustics and electromagnetics. By introducing the time reversal technique, prior knowledge about each transmission channel is no longer needed since the step of sending waves through the medium measures this information. Many approaches have been explored to achieve better focusing quality, which is influenced by several factors, such as the propagation loss. We present two methods to conduct time reversal experiments in ray-chaotic billiards or cavities. The first method uses a ray-tracing algorithm to calculate orbit information from knowledge of the cavity geometry. We then use this information to generate a synthetic signal, which is then sent into the cavity as if it's the time reversed signal in the traditional time-reversal scheme. This method tries to obtain channel information numerically but has limited accuracy due to the chaotic properties of the cavity. Another method is to utilize the transmission scattering parameter, obtained from the time domain response of the cavity between two ports. We amplify the time-reversed signal for each frequency channel in proportion to the loss it experiences during the transmission. The experimental results show that the amplitude of side lobes around the reconstructed signal is reduced significantly and the correlation between the reconstruction and the initial signal is improved from 0.8 to 0.98 in a low-mode density cavity. [Preview Abstract] |
Thursday, March 21, 2013 2:42PM - 2:54PM |
W30.00002: Statistical fluctuations in chains of chaotic electromagnetic enclosures Gabriele Gradoni, Thomas Antonsen, Steven Anlage, Edward Ott Today, the statistical analysis of complex electromagnetic cavities constitutes a very active field of research in applied electromagnetics and statistical physics. The Random Coupling Model (RCM) provides a framework for predicting the statistics of scattering of radiation in complicated enclosures. RCM makes use of results from random matrix theory (RMT) to model the mode spectrum of irregular cavities. Here, we show how to use the RCM to study the scenario of two (or more) three-dimensional cavities interconnected by apertures. We imagine exciting the first cavity of the so formed chain with a small antenna, and receiving a signal in the last cavity with a similar antenna. Recently, we derived the probability distribution of the power flowing through the cavity chain. A closed form solution of the trans-impedance between the two ports is derived, and its statistics discussed. Variations of cavity losses and aperture geometry are discussed within our statistical framework, for which distribution functions are generated by the Monte Carlo method. In the high-loss limit we are able to identify self- and cavity-cavity interaction terms. The extreme case of an irregular aperture connecting to an irregular cavity is also proposed and investigated. [Preview Abstract] |
Thursday, March 21, 2013 2:54PM - 3:06PM |
W30.00003: Finding equilibrium statistical mechanics in spatiotemporal chaos C. Clark Esty, Christopher C. Ballard, John A. Kerin, David A. Egolf Ruelle has argued that the extensivity of the complicated dynamics of spatiotemporal chaos is evidence that these systems can be viewed as a gas of weakly-interacting regions of a characteristic size. We have performed large-scale computational studies of spatiotemporal chaos in the 1D complex Ginzburg-Landau equation and have found that histograms of the number of maxima in the amplitude are well-described by an {\it equilibrium} Tonks gas (and variants) in the grand canonical ensemble. Furthermore, for small system sizes, the average number of particles in the Tonks gas (with particle sizes and temperatures determined from fits to the CGL histograms) exhibits oscillatory, decaying deviations from extensivity in agreement with the deviations in the fractal dimension found by Fishman and Egolf. This result not only supports Ruelle's picture but also suggests that the coarse-grained behavior of this far-from-equilibrium system might be understood using equilibrium statistical mechanics. [Preview Abstract] |
Thursday, March 21, 2013 3:06PM - 3:18PM |
W30.00004: Dynamic Scaling of Synchronization in Kuramoto-type Globally Coupled Oscillators Meesoon Ha, Chulho Choi, Byungnam Kahng We investigate the dynamic scaling behavior of the phase synchronization order parameter in the framework of the original Kuramoto model with Gaussian natural frequecies near and at the critical value of the coupling strength. The temporal behavior has been never paid attention to in the earlier studies of synhronization and its transition nature including finite-size scaling (FSS), whereas the stationary critical behavior has been widely studied. We focus on the scaling behavior of the order parameter until the system reaches its steady state from various initial conditions in the context of the dynamic scaling form at criticality. It is found that dynamic scaling of synchronization can indicate the critical valule of the coupling strength and also estimate all critical exponents of the continuous synchronization transition, based on the scaling relation of the earlier suggested FSS theory. Moreover, we figure out that the dynamic scaling analysis is quite useful even though the system does not reach its steady state, provided that the system size is not too small. Finally, we argue how the generating method of natural frequecies and the thermal effect of phases affect dynamic scaling with the change of the dynamic exponent, which are numerically confirmed. [Preview Abstract] |
Thursday, March 21, 2013 3:18PM - 3:30PM |
W30.00005: Observation of Asymmetric Transport in Structures with Active Nonlinearities Nicholas Bender, Samuel Factor, Josh Bodyfelt, Hamidreza Ramezani, Fred Ellis, Tsampikos Kottos A mechanism for asymmetric transport based on the interplay between the fundamental symmetries of parity (P) and time (T ) with nonlinearity is presented. We experimentally demonstrate and theoretically analyze the phenomenon using, as a reference system, a pair of coupled van der Pol oscillators, one with anharmonic gain and the other with the complementary time reversed anharmonic loss, connected to two transmission lines. An increase of the degree of the gain/loss strength or of the number of PT -symmetric nonlinear dimers in a chain, increases the non-reciprocality effect. [Preview Abstract] |
Thursday, March 21, 2013 3:30PM - 3:42PM |
W30.00006: Wave scattering from cavities with both regular and chaotic ray trajectories Ming-Jer Lee, Thomas Antonsen, Edward Ott The random plane wave hypothesis has been used to characterize fields inside chaotic cavities where all ray trajectories are chaotic and visit the available phase space uniformly. We consider incident and reflected waves in channels connecting to a chaotic cavity. From Random Matrix Theory, the impedance, obtained from the scattering matrix, for pure chaotic cavities can be described as a Lorentzian random variable with predictable mean and width. For some shapes of cavities, called mixed systems, some rays are chaotic and visit subregions of phase space ergodically, while some rays are regular staying on invariant troi. We generalize the previous chaotic cavity theory to mixed systems by separating the impedance into regular and chaotic parts. We test the theory by numerically solving for eigenmodes of the Helmholtz equation in a mushroom shaped cavity where there is a clear separation between regular and chaotic regions of phase space. We compare our theoretical predictions with numerical calculations for one-port and two-ports cases with different port positions. [Preview Abstract] |
Thursday, March 21, 2013 3:42PM - 3:54PM |
W30.00007: Testing the Predictions of Random Matrix Theory in Low Loss Wave Chaotic Scattering Systems Jen-Hao Yeh, Thomas Antonsen, Edward Ott, Steven Anlage Wave chaos is a field where researchers apply random matrix theory (RMT) to predict the statistics of wave properties in complicated wave scattering systems. The RMT predictions have successfully demonstrated universality of the distributions of these wave properties, which only depend on the loss parameter of the system and the physical symmetry. Examination of these predictions in very low loss systems is interesting because extreme limits for the distribution functions and other predictions are encountered. Therefore, we use a wave-chaotic superconducting cavity to establish a low loss environment and test RMT predictions, including the statistics of the scattering (S) matrix and the impedance (Z) matrix, the universality (or lack thereof) of the Z- and S-variance ratios, and the statistics of the proper delay times of the Wigner-Smith time-delay matrix. We have applied an in-situ microwave calibration method (Thru-Reflection-Line method) to calibrate the cryostat system, and we also applied the random coupling model to remove the system-specific features. Our experimental results of different properties agree with the RMT predictions. [Preview Abstract] |
Thursday, March 21, 2013 3:54PM - 4:06PM |
W30.00008: Phase dynamics of coupled oscillators reconstructed from data Michael Rosenblum, Bjoern Kralemann, Arkady Pikovsky We present a technique for invariant reconstruction of the phase dynamics equations for coupled oscillators from data. The invariant description is achieved by means of a transformation of phase estimates (protophases) obtained from general scalar observables to genuine phases. Staring from the bivariate data, we obtain the coupling functions in terms of these phases. We discuss the importance of the protophase-to-phase transformation for characterization of strength and directionality of interaction. To illustrate the technique we analyse the cardio-respiratory interaction on healthy humans. Our invariant approach is confirmed by high similarity of the coupling functions obtained from different observables of the cardiac system. Next, we generalize the technique to cover the case of small networks of coupled periodic units. We use the partial norms of the reconstructed coupling functions to quantify directed coupling between the oscillators. We illustrate the method by different network motifs for three coupled oscillators. We also discuss nonlinear effects in coupling. [Preview Abstract] |
Thursday, March 21, 2013 4:06PM - 4:18PM |
W30.00009: The Universal $\alpha$-Family of Maps Mark Edelman We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal $\alpha$-Family of Maps depending on a single parameter $\alpha > 0$ which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We show that many well-known regular maps, like integer n- dimensional (area/volume preserving for $n>1$) quadratic maps (including for $n=1$ the Logistic Map which is not measure preserving) and n-dimensional (volume preserving for $n>2$) standard maps (including the non-measure preserving Circle Map and the area preserving Standard Map), can be considered as particular forms of the Universal $\alpha$-Family of Maps. In the case of the fractional $\alpha$ corresponding maps, which are maps with memory, demonstrate various types of attractors including cascade of bifurcation types trajectories. Maps with memory can be applied for modeling biological systems and circuit elements with memory. [Preview Abstract] |
Thursday, March 21, 2013 4:18PM - 4:30PM |
W30.00010: Synchronization Dynamics of Coupled Anharmonic Plasma Oscillators John Laoye, Uchechukwu Vincent, Taiwo Roy-Layinde The synchronization of two identical mutually driven coupled plasma oscillators modeled by anharmonic oscillators was investigated. Each plasma oscillator was described by a nonlinear differential equation of the form: The model employed the spring-type coupling. Numerical simulations, including Poincare sections, time series analysis, and bifurcation diagram were performed using the fourth-order Runge-Kutta scheme. The numerical value of the threshold coupling Kth was determined to be approximately 0.15. [Preview Abstract] |
Thursday, March 21, 2013 4:30PM - 4:42PM |
W30.00011: Nonlinear Time-Reversal in a Wave Chaotic System Matthew Frazier, Steven Anlage, Biniyam Taddese, Edward Ott, Thomas Antonsen Time reversal mirrors are particularly simple to implement in wave chaotic systems and form the basis for a new class of sensors [1-3]. The sensors make explicit use of time-reversal invariance and spatial reciprocity in a wave chaotic system to sensitively measure the presence of small perturbations to the system. The underlying ray chaos increases the sensitivity to small perturbations throughout the volume explored by the waves. We extend our time-reversal mirror to include a discrete element with a nonlinear dynamical response [4]. The initially injected pulse interacts with the nonlinear element, generating new frequency components originating at the element. By selectively filtering for and applying the time-reversal mirror to the new frequency components, we focus a brief-in-time excitation only onto the nonlinear element, without knowledge of its location. Furthermore, we demonstrate a model which captures the essential features of our time-reversal mirror, modeling the wave-chaotic system as a network of transmission lines arranged as a star graph, with the discrete nonlinearity modeled as a diode terminating a particular line. [1] Appl. Phys. Lett. 95, 114103 (2009) [2] J. Appl. Phys. 108, 114911 (2010) [3] Acta Physica Polonica A 112, 569 (2007) [4] arXiv:1207.1667 [Preview Abstract] |
Thursday, March 21, 2013 4:42PM - 4:54PM |
W30.00012: Quantifying Transport in Chaotic Rayleigh-Benard Convection Christopher Mehrvarzi, Mark Paul The transport of a scalar species in a complex flow field is important in many areas of current interest such as the combustion of premixed gases, the dynamics of particles in the atmosphere and oceans, and the reaction of chemicals in a mixture. There has been significant progress in understanding transport in steady periodic flows such as a ring of vortices. In addition, transport in turbulent flow has an extensive literature. Here we focus on the transport of a scalar species in a three-dimensional time-dependent flow field given by the spiral defect chaos state of Rayleigh-Benard convection. We use a highly efficient and parallel spectral element approach to simultaneously evolve the Boussinesq equations and the reaction-advection-diffusion equation in large cylindrical domains with experimentally relevant boundary conditions. We explore the active and passive transport of a scalar species in a chaotic flow field to quantify the transport enhancement for a range of Lewis and Damkholer numbers. [Preview Abstract] |
Thursday, March 21, 2013 4:54PM - 5:06PM |
W30.00013: Quantifying Spatiotemporal Chaos in Rayleigh-Benard Convection: Using Numerics to Connect Theory and Experiment Mu Xu, Alireza Karimi, Jeffrey Tithof, Miro Kramar, Vidit Nanda, Michael Schatz, Konstantin Mischaikow, Mark Paul Spatiotemporal chaos is a common and important feature of spatially-extended systems that are driven far-from-equilibrium. Many open questions remain regarding the high-dimensional chaotic dynamics that describe fluid systems for laboratory conditions. In this talk we explore the spiral defect chaos state of Rayleigh-Benard convection. Recent advances in computing algorithms and available supercomputing resources have made possible the computation of fundamentally important quantities of theoretical importance that are currently inaccessible to experiment. For example, the temporal variation of the spectrum of Lyapunov exponents, the spatial and temporal variation of the Lyapunov vectors, and the variation of the fractal dimension with system parameters. We use large-scale parallel numerical simulations to compute theoretically important diagnostics of spatiotemporal chaos, such as these, with particular interest in connecting these numerical results with experimentally accessible quantities that describe the pattern dynamics. [Preview Abstract] |
Thursday, March 21, 2013 5:06PM - 5:18PM |
W30.00014: Effect of Size Polydispersity on Diffusion Behaviors of Traces in Random Obstacle Matrices Hyun Woo Cho, Bong June Sung, Arun Yethiraj Diffusion behavior on random obstacle matrices has been studied extensively for several decades to explain dynamic behaviors in disordered systems, such as dynamic arrest in colloidal glass phase and anomalous diffusion in crowded biological systems. We present the effect of size polydispersity of the obstacles on diffusion behavior in two-dimensional random obstacle matrices. We generate the random matrices by randomly locating non-overlapping hard disks in two-dimensional space, and consider the diffusion behavior of the tracers. We show that the diffusion behavior is sensitive to the size polydispersity of the obstacles even though their average sizes are the same. In addition, we locate the percolation threshold of void space, and find that diffusion constant D follows scaling relation $D\sim \left( {\varphi_{c} -\varphi } \right)^{\mu -\beta }$ regardless of the size polydispersity, where $\varphi $ and $\varphi_{c} $ is the area fraction of the obstacles and its value at percolation threshold, respectively. The value of the dynamic scaling constant $\mu $ is, however, not universal. We will also discuss briefly non-universal dynamic scaling exponents of two-dimensional random obstacle matrices. [Preview Abstract] |
Thursday, March 21, 2013 5:18PM - 5:30PM |
W30.00015: Colloidal Bandpass and Bandgap Filters Benjamin Yellen, Mukarram Tahir, Yuyu Ouyang, Franco Nori Thermally or deterministically-driven transport of objects through asymmetric potential energy landscapes (ratchet-based motion) is of considerable interest as models for biological transport and as methods for controlling the flow of information, material, and energy. Here, we provide a general framework for implementing a colloidal bandpass filter, in which particles of a specific size range can be selectively transported through a periodic lattice, whereas larger or smaller particles are dynamically trapped in closed-orbits. Our approach is based on quasi-static (adiabatic) transition in a tunable potential energy landscape composed of a multi-frequency magnetic field input signal with the static field of a spatially-periodic magnetization. By tuning the phase shifts between the input signal and the relative forcing coefficients, large-sized particles may experience no local energy barriers, medium-sized particles experience only one local energy barrier, and small-sized particles experience two local energy barriers. The odd symmetry present in this system can be used to nudge the medium-sized particles along an open pathway, whereas the large or small beads remain trapped in a closed-orbit, leading to a bandpass filter, and vice versa for a bandgap filter. [Preview Abstract] |
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