Bulletin of the American Physical Society
2005 APS March Meeting
Monday–Friday, March 21–25, 2005; Los Angeles, CA
Session J23: Stochastic Processes and Nonlinear Dynamics |
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Sponsoring Units: GSNP Chair: Bob Ecke, LANL Room: LACC 410 |
Tuesday, March 22, 2005 11:15AM - 11:27AM |
J23.00001: Stochastic Resonance in Bistable Nanomechanical Oscillators Robert Badzey, Pritiraj Mohanty The bistable doubly-clamped nanomechanical beam resonator has recently been shown [1] to be a promising step in the development of mechanical memory cells. One of the major obstacles to the full implementation of this scheme lies in the ability to effectively control the two states [2] in a noisy or high-temperature environment. Here, we present the observation of stochastic resonance in these nanomechanical systems. This is a counter-intuitive effect in which the addition of noise to a noisy system results in coherent response. Over the past two decades, it has been seen in a wealth of physical systems. Aside from adding to knowledge in this area, the observation of stochastic resonance here lays the foundation for its effective use in the areas of signal processing, quantum information, and quantum control. This work is supported by NSF Nanoscale Exploratory Research (NER) Program and NSF (DMR, CCF, ECS), DOD (ARL), ACS (PRF), and the Sloan Foundation. [1] R. Badzey et al., Appl. Phys. Lett. 85, 3587 (2004). [2] R. Badzey et al., (To be Published, Appl. Phys. Lett. 1/20/05). [Preview Abstract] |
Tuesday, March 22, 2005 11:27AM - 11:39AM |
J23.00002: Precession aided magnetic stochastic resonance in ferromagnetic nanoparticles with cubic anisotropy William Coffey, Yuri Kalmykov, Yuri Raikher, Sergey Titov It is shown that the signal-to-noise ratio (SNR) in the magnetic stochastic resonance of single-domain ferromagnetic nanoparticles having cubic anisotropy exhibits a strong intrinsic dependence on the decay rate \textit{$\alpha $} of the Larmor precession. This dependence (precession aided relaxation) is due to coupling between longitudinal relaxation and transverse (precessional) modes arising from the lack of axial symmetry. It is most pronounced in the intermediate to low damping (Kramers turnover) region 0.1 $<$ \textit{$\alpha $} $<$ 1. The effect which does not exist for axially symmetric potentials may be used to determine \textit{$\alpha $}. [Preview Abstract] |
Tuesday, March 22, 2005 11:39AM - 11:51AM |
J23.00003: Delayed stochastic control Tadaaki Hosaka, Toru Ohira, Christian Lucian, Juan Luis Cabrera, John Milton Time-delayed feedback control becomes problematic in situations in which the time constant of the system is fast compared to the feedback reaction time. In particular, when perturbations are unpredictable, traditional feedback or feed-forward control schemes can be insufficient. Nonethless a human can balance a stick at their fingertip in the presence of fluctuations that occur on time scales shorter than their neural reaction times. Here we study a simple model of a repulsive delayed random walk and demonstrate that the interplay between noise and delay can transiently stabilize an unstable fixed-point. This observation leads to the concept of ``delayed stochastic control,'' i.e. stabilization of tasks, such as stick balancing at the fingertip, by optimally tuning the noise level with respect to the feedback delay time. \\ \\ References:(1)J.L.Cabrera and J.G.Milton, PRL 89 158702 (2002);(2) T. Ohira and J.G.Milton, PRE 52 3277 (1995);(3)T.Hosaka, T.Ohira, C.Lucian, J.L.Cabrera, and J.G.Milton, Prog. Theor. Phys. (to appear). [Preview Abstract] |
Tuesday, March 22, 2005 11:51AM - 12:03PM |
J23.00004: Non-exponential time-correlation function for random physical processes T.R.S. Prasanna Dielectric relaxation in gases is reconsidered phenoemenologically and it is shown that the dipole moment correlation function must have an inflection point at the mean collision time. The exponential function, used in the Debye and Van Vleck-Weisskopf models, does not possess an inflection point at finite times and must be rejected. New models that correctly represent the effects of collisions are necessary. A new time-correlation function is proposed that differs little numerically from the exponential function, exhibits an inflection point, is analytic at t = 0 and its power spectrum has finite moments to all orders. Problems related to divergence vanish. A new lineshape function is obtained that is indistinguishable from the Lorentzian lineshape. The new correlation function implies that the process is non-Markovian, which is theoretically consistent for all processes where the derivatives have physical meaning, including those described in terms of linear response theory. In addition, its mathematical superiority implies that it is advantageous to use this function over the exponential function for all such processes. [Preview Abstract] |
Tuesday, March 22, 2005 12:03PM - 12:15PM |
J23.00005: Mean-field behavior of the Burridge-Knopoff model with long-range interactions Junchao Xia, Harvey Gould, William Klein, John Rundle The one-dimensional Burridge-Knopoff model with a variable-range Kac-like interaction is simulated using molecular dynamics, and the number of earthquake-like events is obtained. In agreement with Carlson and Langer (Phys. Rev. A 40, 6470 (1989)) the event size distribution is found to exhibit power law scaling for nearest-neighbor interactions over a limited range of event sizes. We find that long-range interactions yield mean-field exponents only if the parameter characterizing the ratio of the largest characteristic slipping speed to the speed at which the dynamical friction is appreciably reduced is sufficiently small. In this limit the dynamical behavior of the long-range Burridge-Knopoff model becomes similar to the cellular automaton model of Rundle, Jackson, and Brown and Olami, Feder, and Christensen. [Preview Abstract] |
Tuesday, March 22, 2005 12:15PM - 12:27PM |
J23.00006: Repulsive Synchronization in an Array of Phase Oscillators Lev Tsimring, Nikolai Rulkov, Michael Larsen, Michael Gabbay We study the dynamics of an array of phase oscillators with repulsive coupling. Globally-coupled network of identical oscillators settles on one of a family of synchronized regimes characterized by zero mean field. However, variations of oscillator natural frequencies destroy synchronization for sufficiently large number of coupled oscillators independently of the coupling strength. In locally coupled networks (with a finite range of coupling less than the system size), the synchronization occurs even for non-identical oscillators when coupling is sufficiently strong. In the synchronized regime, a ring of repulsively coupled oscillators approaches linear phase distribution. [Preview Abstract] |
Tuesday, March 22, 2005 12:27PM - 12:39PM |
J23.00007: Periodic orbit theory of two coupled Tchebyscheff maps Domenico Lippolis, Carl Philip Dettmann Coupled map lattices have been widely used as models in several fields of physics, such as chaotic strings, turbulence, and phase transitions, as well as in other disciplines, such as biology (ecology, evolution) and information processing. This work investigates properties of periodic orbits in two coupled Tchebyscheff maps. Then zeta function cycle expansions are used to compute dynamical averages appearing in Beck's theory of chaotic strings. The results show close agreement with direct simulation for most values of the coupling parameter, and yield information about the system complementary to that of direct simulation. [Preview Abstract] |
Tuesday, March 22, 2005 12:39PM - 12:51PM |
J23.00008: Flow Equation Approach to the Statistics of Nonlinear Dynamical Systems J.B. Marston, Seungwook Ma, M.B. Hastings The probability distribution function of non-linear dynamical systems is governed by a linear framework that resembles quantum many-body theory, in which stochastic forcing and/or averaging over initial conditions play the role of non-zero $\hbar$. Besides the well-known Fokker-Planck approach, there is a related Hopf functional method\footnote{Uriel Frisch, {\it Turbulence: The Legacy of A. N. Kolmogorov} (Cambridge University Press, 1995) chapter 9.5.}; in both formalisms, zero modes of linear operators describe the stationary non-equilibrium statistics. To access the statistics, we investigate the method of continuous unitary transformations\footnote{S. D. Glazek and K. G. Wilson, Phys. Rev. D {\bf 48}, 5863 (1993); Phys. Rev. D {\bf 49}, 4214 (1994).} (also known as the flow equation approach\footnote{F. Wegner, Ann. Phys. {\bf 3}, 77 (1994).}), suitably generalized to the diagonalization of non-Hermitian matrices. Comparison to the more traditional cumulant expansion method is illustrated with low-dimensional attractors. The treatment of high-dimensional dynamical systems is also discussed. [Preview Abstract] |
Tuesday, March 22, 2005 12:51PM - 1:03PM |
J23.00009: Quantized Interest Rate At The-money for American options Lamine Dieng, Samir Lipovaca In this work, we expand the idea of Shepp for stock optimization using the Bachelier model as our model for the stock price at the money ($X=K)$ for the American call and put options. At the money ($X=K)$ for American options, the expected payoff of both the call and put options is zero. Shepp investigated several stochastic optimization problems using martingale and stopping time theories, one of the problems he investigated was how to optimize the stock price using both the Black-Scholes and the Bachelier (additive) models for the American option above the strike price $K$ (exercise price) to a stopping point. In order to explorer the non-relativistic quantum effect on the expected payoff for both the call and put options at the money ($X=K)$, we assumed the stock price to undergo a stochastic process governed by the Bachelier model given above. Further, using Ito calculus and martingale theory, we obtained a differential equation for the expected payoff for both the call and put options in terms delta and gamma. We investigated the solution of the differential equation in the limit when delta is zero, this sometimes is called hedging or delta neutral in finance. By comparison, the delta-hedged differential equation corresponded to the non-relativistic time-independent Shroedinger equation in quantum mechanics with a constant diffusion constant. We solved exactly the non-relativistic Schroedinger equation at the money and obtained a quantized interest rate in terms of volatility and stock price. [Preview Abstract] |
Tuesday, March 22, 2005 1:03PM - 1:15PM |
J23.00010: A Unified Approach to Attractor Reconstruction Louis Pecora, Linda Moniz, Jon Nichols, Thomas Carroll Reconstruction of attractors for dynamical systems has typically focused on solving seemingly separate problems of finding a proper time delay and then finding a proper embedding dimension. Techniques for solving these problems are somewhat heuristic. We show that the two problems of time delay and embedding dimension are actually the same problem. Using Taken's theorem we derive a mathemetical criterion for adding new components to reconstruction vectors. We also show how several statistics that gauge functional dependence between multivariate data sets can fullfill a practical application of the theory and solve at once the problems of determining time delays, getting embedding dimension, and optimally choosing time series to use from a multivariate data set. This unified approach is compared to ``standard'' approaches and is shown to be superior in requiring fewer embedding dimensions. [Preview Abstract] |
Tuesday, March 22, 2005 1:15PM - 1:27PM |
J23.00011: Perturbations of L\'evy Processes Using the Feynman Functional Arjuna Flenner, Brian DeFacio The Wiener process is an example of a stable L\'{e}vy process, and the stable L\'{e}vy processes have been used as a model for many Physical systems as well as Financial Mathematics. Using the heat semigroup, the Feynman-Kac formula provides a connection between the Wiener process and Feynman's path integral. DeFacio, Johnson, and Lapidus were able to expand upon this method to formulate a rigorous Feynman Functional for certain semigroup operators. It is possible to add a potential term to these operators in order to investigate the effect perturbations have on stable L\'{e}vy processes. With restrictions on the potential, these perturbations can be calculated using the methods developed for the Feynman path integral, and formal manipulation of the perturbations yields previous results in the literature. An overview of the Feynman Functional formulation for L\'{e}vy processes will be given, a few examples of perturbed L\'{e}vy processes will be shown, and some Mathematical and computational difficulties will be discussed. [Preview Abstract] |
Tuesday, March 22, 2005 1:27PM - 1:39PM |
J23.00012: Non-linear Logit Models for Binarized High Frequency Financial Data Naoya Sazuka We propose a non-linear logit model for binarized high frequency data of yen-dollar exchange rate indicating up or down price movement. We show a non-trivial probability structure from the binarized data, which is invisible from the price change itself. The model successfully captures the structure, which is not possible by the conventional analysis such as an AR model and a logit model. In addition, a similar and a stronger bias can be observed from other binarized high frequency active stock data on NYSE, for example GE, INTL, MSFT, WMT and so on. Our model could be useful for a wide range of binary time series with non-trivial dynamical structures. \\ References [1]N. Sazuka and T. Ohira, in {\it Computational Finance and its application}, pp.275-305, WIT press 2004. [2]N. Sazuka, et. al., Physica A 324 pp.366-371, 2003. [3]T. Ohira, et. al., Physica A 308 N1-4, pp.368-374, 2002. [Preview Abstract] |
Tuesday, March 22, 2005 1:39PM - 1:51PM |
J23.00013: The Opinion Dynamics of Majority Rule Sidney Redner, Pu Chen We investigate the long-time behavior of a majority rule opinion dynamics model in finite spatial dimensions. Each site of the system is endowed with a finite-state spin variable that evolves by majority rule. In a single update event, a group of spins with a fixed (odd) size is specified and all members of the group adopt the local majority state. For the case of two states, repeated application of this update step leads to a coarsening mosaic of spin domains and ultimate consensus in a finite system. The approach to consensus is governed by two disparate time scales, with the longer time scale arising from realizations in which spins organize into coherent single- opinion bands. The extension to more than two states leads to a surprising faster evolution as soon as one state establishes itself as a local majority somewhere in the system. [Preview Abstract] |
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J23.00014: Theoretical evidence for the link between geomagnetic reversal and glacial events: stochastic resonance in the geodynamo model Chih-Yuan Tseng, Chien-Chih Chen Not yet a theoretical analysis can explain the coincident temporal correlation between the geomagnetic reversal and glacial events, which both have a quasi-period of about 100 kyr, although there exists dozens of observational evidences for such correlation. The geodynamo has widely been thought to be an intuitive and self-sustained model of the Earth's magnetic field$^{10}$. In this letter we report how possible a signal with 100 kyr quasi-period can be embedded in the geomagnetic filed \textit{via} the mechanism of stochastic resonance in a forced Rikitake dynamo. We thus suggest one common triggering for the geomagnetic reversal and glacial events, neither the glaciation controls the geomagnetic reversal nor vice versa. Instead, both kinds of catastrophes may result from the cyclic variation of the Earth's orbital eccentricity. [Preview Abstract] |
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J23.00015: Multi-scale continuum mechanics: From global bifurcations to noise induced high dimensional chaos Ira Schwartz, Lora Billings, Ying-Cheng Lai, David Morgan Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural/mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater that the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high dimensional chaos. Constrained invariant sets are computed to reveal a process from low dimensional to high dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. [Preview Abstract] |
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