Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session Z9: Nonlinear Dynamics and Chaotic Systems |
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Sponsoring Units: GSNP Chair: Robert Behringer, Duke University, Room: 303 |
Friday, March 20, 2009 11:15AM - 11:27AM |
Z9.00001: Rotating Space Elevator: Classical and Statistical Mechanics of cosmic scale spinning strings Steven Knudsen, Leonardo Golubovic We introduce a novel and unique nonlinear dynamical system, the Rotating Space Elevator (RSE). The RSE is a multiply rotating system of cables (strings) reaching beyond the Earth geo-synchronous satellite orbit. Strikingly, objects sliding along the RSE cable do not require internal engines or propulsion to be transported far away from the Earth's surface. The RSE action employs, in a very fundamental way, basic natural phenomena -- gravitation and inertial forces. The RSE exhibits interesting nonlinear dynamics and statistical physics phenomena. Its kinetic phase diagram involves both chaotic and quasi-periodic states of motion separated by a morphological phase transition that occurs with changing the RSE angular frequency. [Preview Abstract] |
Friday, March 20, 2009 11:27AM - 11:39AM |
Z9.00002: Numerical investigation on the chaos assisted tunneling for a coupled microwave cavity Hoshik Lee, Louis Pecora, Dong Ho Wu It is known that chaos-assisted dynamical tunneling may occur in nonintegrable (chaotic) systems. Recently we investigate wave chaotic systems to see if the system may promote the chaos-assisted spatial tunneling in addition to the dynamic tunneling. Our previous experiments suggest some enhancement of the spatial tunneling rate in a coupled, wave-chaotic 2D microwave double cavity, indicating that the presence of chaotic modes changes not only the dynamical tunneling rate but also the spatial tunneling rate. To understand the underlying physics we carry out numerical simulations on nonintegrable 2D cavities as well as on integrable 2D cavities. We will present details about the experiments and numerical simulation results. [Preview Abstract] |
Friday, March 20, 2009 11:39AM - 11:51AM |
Z9.00003: Scaling properties of delay times in one-dimensional random media Joshua Bodyfelt, Antonio Mendez-Bermudez, Andrey Chabanov, Tsampikos Kottos The scaling properties of the inverse moments of Wigner delay times are investigated in finite one-dimensional (1D) random media with one channel attached to the boundary of the sample. We find that they follow a simple scaling law which is independent of the microscopic details of the random potential. Our theoretical considerations are confirmed numerically for systems as diverse as 1D disordered wires and optical lattices to microwave waveguides with correlated scatterers. [Preview Abstract] |
Friday, March 20, 2009 11:51AM - 12:03PM |
Z9.00004: Lyapunov exponent calculation via reconstruction of the invariant density function from iterative Chebyshev maximum entropy approach Nagendra Dhakal, Hiro Shimoyama, Parthapratim Biswas We apply a maximum entropy approach (MaxEnt) to compute invariant density functions to obtain the Lyapunov exponents. The method gives the solution by iteratively calculating the Lagrange multipliers within the maximum entropy method from moment constraints. We illustrate our method by reproducing known invariant densities for several cases of discrete maps (in both chaotic and non-chaotic regime). The global convergence of invariant density function is studied with particular emphasis on Lynapunov exponent of the maps for varying number of moments. We demonstrate that Lyapunov exponent of a chaotic map can be computed with a high degree of precision from this approach. [Preview Abstract] |
Friday, March 20, 2009 12:03PM - 12:15PM |
Z9.00005: Time-Shifted Synchronization of Chaotic Oscillator Chains without Explicit Coupling Delays Jonathan Blakely, Mark Stahl, Ned Corron It has recently been reported that time-shifted synchronization (i.e., lag or anticipation) of chaotic oscillators can result from forms of coupling that do not contain explicit delay terms. Identical time-shifted synchronization is not a solution in these systems so the dynamics are a form of generalized synchronization where trajectories are similar but not exactly identical. Here we examine chains of uni-directionally coupled oscillators in which time-shifted synchronization occurs without delays in the coupling. We observe the distortion of the waveforms of the response oscillators located far from the drive oscillator. Under weak coupling, we see much less distortion occurs over chains with significant total time shift than predicted by a recently introduced theoretical estimate. Under stronger coupling, we find better agreement with the theoretical prediction and, despite sometimes severe attenuation, generalized synchronization is maintained over long chain lengths. We report results from numerical models as well as from an experimental system of electronic circuits. Such oscillator chains may prove useful in applications requiring a variable delay such as chaotic radar or beam forming. [Preview Abstract] |
Friday, March 20, 2009 12:15PM - 12:27PM |
Z9.00006: Combining Wave Chaos and the Loschmidt Echo to Extend the Concept of Fidelity to Classical Waves Biniyam Taddese, James Hart, Thomas Antonsen, Edward Ott, Steven Anlage We propose and demonstrate a new remote sensor scheme by applying the wave mechanical concept of fidelity loss to classical waves. The sensor makes explicit use of time- reversal invariance and spatial reciprocity in a wave chaotic system to sensitively and remotely measure the presence of small perturbations to the system. The loss of fidelity is measured through a classical wave-analog of the Loschmidt echo by employing a single-channel time-reversal mirror to rebroadcast a probe signal into the perturbed system. We also introduce the use of exponential amplification of the probe signal to partially overcome the effects of propagation losses. It is demonstrated that exponential amplification can be used to vary the spatial range of sensitivity to perturbations, thereby actively modifying the range of operation of the sensor. Experimental results are presented for both electromagnetic and acoustic versions of the Loschmidt echo based sensor. [Preview Abstract] |
Friday, March 20, 2009 12:27PM - 12:39PM |
Z9.00007: Synchronization and Competition in a Double-Bump-on-Tail Instability Dmitry V. Dylov, Jason W. Fleischer We experimentally and theoretically consider a double-bump-on-tail instability by mapping the general wave-kinetic problem to a multiple beam propagation problem using statistical light. More specifically, we consider the nonlinear interaction of three spatially-incoherent beams in a self-focusing photorefractive crystal. For weak nonlinearity, we observe instability competition and sequential flattening of the bumps in momentum space, with no observable variations in position-space intensity. This joint dynamics resembles the phase synchronization of a ``classical'' laser system (relaxation from a ``non- equilibrium'' state to a lower-energy one), with the corresponding gain rates following from the optical equivalent of inverse Landau damping. For strong nonlinearity, intensity modulations appear and the triple-hump spectrum merges into a single-peaked profile with an algebraic $k^{-2}$ inertial range. This spectrum, with its associated modulations, is a definitive observation of soliton, or Langmuir, turbulence. [Preview Abstract] |
Friday, March 20, 2009 12:39PM - 12:51PM |
Z9.00008: Stratospheric Ozone and Dynamical Systems Francisco J. Uribe, Rosa Maria Velasco, Ernesto Perez-Chavela We consider the Chapman mechanism for stratospheric ozone dynamics. The resulting nonlinear differential equations are studied from the point of view of the theory of dynamical systems. In particular we calculate and analyze the nature of the critical points and show that the region in which the concentrations are non-negative is a positively invariant set, meaning that initial conditions with non-negative concentrations always give non-negative concentrations. Poincar\'e compactification is used to ellucidate the global flow. Comments about the inclusion of nitrogen oxides are also given. [Preview Abstract] |
Friday, March 20, 2009 12:51PM - 1:03PM |
Z9.00009: Chaotic Escape of Rays from a Vase-shaped Billiard: Simulations and Experiment Jaison Novick, Matthew Len Keeler, John Delos We study the escape of rays from a two dimensional, specularly-reflecting open vase-shaped cavity. The narrowest point of the vase's neck defines a dividing surface between rays that escape without return and those turned back into the cavity. Our simulations show that a point burst of rays emitted in all directions can contain both regular and chaotic scattering trajectories. The chaotic trajectories leak out in an infinitely long pulse train organized by a fractal. For escaping trajectories, we record the propagation time to escape and find that the fractal manifests itself in the escape time versus the launch angle. We have experimentally verified the early fractal structure. A two dimensional aluminum vase with reflective Teflon walls was constructed with an ultrasound transmitter as the point source. A microphone was placed at points along the vase's mouth. We find good agreement between measurements and classical simulations. [Preview Abstract] |
Friday, March 20, 2009 1:03PM - 1:15PM |
Z9.00010: Scattering a pulse from a chaotic cavity: Transitioning from algebraic to exponential decay James Hart, Thomas Antonsen, Edward Ott The ensemble averaged power scattered in and out of lossless chaotic cavities (such as microwave resonators, acoustic cavities and quantum dots) decays as a power law in time for large times. In the case of a pulse with a finite duration, the power scattered from a single realization of a cavity closely tracks the power law ensemble decay initially, but eventually transitions to an exponential decay. In this presentation, we explore the nature of this transition in the case of coupling to a single port. We find that for a given pulse shape, the properties of the transition are universal if time is properly normalized. We define the crossover time to be the time at which the deviations from the mean of the reflected power in individual realizations become comparable to the mean reflected power. We demonstrate numerically that, for randomly chosen cavity realizations and given pulse shapes, the probability distribution function of reflected power depends only on time, normalized to this crossover time. Paper: http://arxiv.org/abs/0810.1664 [Preview Abstract] |
Friday, March 20, 2009 1:15PM - 1:27PM |
Z9.00011: Stability of large complex systems Harold Hastings We use a random matrix model to study the stability of large, complex systems. Our approach was motivated by a long-standing dilemma concerning stability of large systems. MacArthur (1955) and Hutchinson (1959) argued that more ``complex'' natural systems tended to be more stable than less complex systems based upon energy flow. May (1972) argued the opposite, using random matrix models. In prior work we showed that in some sense both are right: under reasonable scaling assumptions on interaction strength, Lyapunov stability increases but structural stability decreases as complexity is increased (c.f. Harrison, 1979; Hastings, 1984). We now apply these methods to a variety of complex systems. [Preview Abstract] |
Friday, March 20, 2009 1:27PM - 1:39PM |
Z9.00012: The Initiation of Optical Breakdown in Simple Liquids Kevin Cissner The probability of breakdown in 40 simple HPLC hydrocarbons and water from a Q-switched laser at 1064 (6ns) and 532 nm (5 ns) was measured using a variety of lenses and cell path length. In each instance a plot of the cumulative distribution function vs. the input laser fluence fits an error-function well, except at low probabilities. Care was taken to measure the light distribution \textit{in sit}u across the entire focal plane. The transmission within the HOMO{\_}LUMO gap was also measure using the long-path-length cells. Trends in the breakdown data with the optical/electronic properties of the target liquids are confounded by spherical aberration. However, the data suggest a connection to the chemical group of the liquid and especially to the C-X bond. In all cases the threshold at 1064 nm is actually less than that at 532 nm. No evidence was found for a mechanism involving dissolved air. A comparison is made to the behavior for static breakdown in gases. [Preview Abstract] |
Friday, March 20, 2009 1:39PM - 1:51PM |
Z9.00013: Analysis of Laser Breakdown Data Roger Becker Experiments on laser breakdown for ns pulses of 532 nm or 1064 nm light in water and dozens of simple hydrocarbon liquids are analyzed and compared to widely-used models and other laser breakdown experiments reported in the literature. Particular attention is given to the curve for the probability of breakdown as a function of the laser fluence at the beam focus. Criticism is made of the na\"{\i}ve forms of both ``avalanche'' breakdown and multi-photon breakdown. It appears that the process is complex and is intimately tied to the chemical group of the material. Difficulties with developing an accurate model of laser breakdown in liquids are outlined. [Preview Abstract] |
Friday, March 20, 2009 1:51PM - 2:03PM |
Z9.00014: Fidelity Gap in Dynamical Systems with Critical Chaos Carl T. West, Tomaz Prosen, Tsampikos Kottos We analyze the fidelity decay for a class of dynamical systems showing {\it critical chaos}, using a Kicked Rotor with singular kicking potential as a prototype model. We found that the classical fidelity shows a gap $F_g$ (initial drop of fidelity) which scales as $F_g(\alpha, \epsilon, \eta ) = f(\chi\equiv\frac{\eta^{3-\alpha}}{\epsilon})$ where $\alpha$ is the order of singularity of the non-analytical potential, $\eta$ is the characteristic spread of the initial phase space density and $\epsilon$ is the perturbation strength. Instead, the corresponding quantum fidelity gap is insensitive to $\alpha$ due to strong diffraction effects that dominate the quantum dynamics. [Preview Abstract] |
Friday, March 20, 2009 2:03PM - 2:15PM |
Z9.00015: Irreversibility, Poincare Recurrence and Stochasticity in Statistical Mechanics Puru Gujrati We will show that deterministic dynamics always leads to the conservation of entropy and Poincare recurrence.\footnote{P.D. Gujrati, Poincare Recurrence, Zermelo's Second Law Paradox, and Probabilistic Origin in Statistical Mechanics, http://arxiv.org/abs/0803.0983 (arXiv:0803.0983)} Thus, recurrence is incompatible with entropy change. The law of increase of entropy can only occur for systems with stochastic dynamics, and the irreversibility emerges out of their indeterminate evolution,\footnote{P.D. Gujrati, Irreversibility, Molecular Chaos, and A Simple Proof of the Second Law, http://arxiv.org/abs/0803.1099 (arXiv:0803.1099)} as we will discuss. This stochasticity requires some weak but uncontrollable interaction of the system with outside or the walls of the container. Boltzmann infuses this stochasticity in his deterministic approach by invoking the assumption of molecular chaos. The molecular chaos cannot emerge out of deterministic dynamics, as shown elsewhere in this meeting.\footnote{Pradeep Fernando and P.D. Gujrati (poster)} [Preview Abstract] |
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