Bulletin of the American Physical Society
APS March Meeting 2017
Volume 62, Number 4
Monday–Friday, March 13–17, 2017; New Orleans, Louisiana
Session R15: Chaos and Nonlinear Dynamics |
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Sponsoring Units: GSNP Chair: Cichel Pleimling, Virginia Tech University Room: 274 |
Thursday, March 16, 2017 8:00AM - 8:12AM |
R15.00001: A universal dichotomy for dynamical systems with variable delay G\"{u}nter Radons, David M\"{u}ller, Andreas Otto We show that the dynamics of systems with time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating to the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with constant delay. The new, second class is characterized by modelocking behavior of their access maps and by an asymptotically linear, instead of a logarithmic scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay. [Preview Abstract] |
Thursday, March 16, 2017 8:12AM - 8:24AM |
R15.00002: Response to an external field of a system of coupled Kuramoto oscillators Shadi Esmaeili, Darka Labavic, Hildegard Meyer-Ortmanns, Michel Pleimling Many structures in nature reach spontaneous order and synchrony. The Kuramoto model has successfully explained a vast variety of synchronization patterns observed in nature. We study the response to an external field of a system of Kuramoto oscillators coupled through frustrated bonds. The emerging physical aging in this system is investigated through autocorrelation and response functions. The observed aging phenomena are very robust and show qualitative features that are only weakly dependent on system parameters and on the type of perturbation. [Preview Abstract] |
Thursday, March 16, 2017 8:24AM - 8:36AM |
R15.00003: Direct observation of coherent energy transfer in nonlinear micro-mechanical oscillators Changyao Chen, Damian Zanette, David Czaplewski, Jeffrey Guest, Steve Shaw, Mark Dykman, Daniel Lopez Energy dissipation is an unavoidable phenomenon of physical systems that are directly coupled to an external environmental bath. The ability to engineer the processes responsible for dissipation and coupling is fundamental to manipulate the state of the systems. This is particularly important in oscillatory states whose dynamic response is crucial for many applications, e.g., micro and nano mechanical resonators for sensing and timing, qubits for quantum engineering, and vibrational modes for optomechanical devices. In situations where stable oscillations are required, the energy dissipated by the vibrational modes is usually compensated by replenishment from external energy sources. Consequently, if the external energy supply is removed, the amplitude of oscillations start to decay immediately since there is no means to counteract the energy dissipated. Here, we experimentally demonstrate a novel strategy to maintain stable oscillations, i.e. constant amplitude and frequency, without supplying external energy to compensate losses. The fundamental intrinsic mechanism of mode coupling is used to redistribute and store mechanical energy among vibrational modes and coherently transfer it back to the principal mode when the external excitation is off. [Preview Abstract] |
Thursday, March 16, 2017 8:36AM - 8:48AM |
R15.00004: First-principles calculation of phase and amplitude dynamics for coupled Wien-bridge oscillators David Mertens, Lars English, Panos Kevrekidis We present the nonlinear phase and amplitude dynamics of coupled Wien-bridge oscillators. A Wien-bridge oscillator is a classic electronic realization of a tunable autonomous oscillator in which positive feedback drives a clipped frequency filter. Such oscillators have been recently utilized in synchronization experiments; simplicity and low cost makes this design a prime candidate for experiments on large populations. However, there has been no established link between the underlying electronic components and the amplitude and phase dynamics. Using the method of multiple time scales, we derived the equations governing the slow evolution of the oscillators. In this talk we will present the result of our analysis as well as a comparison with measurements for two oscillators. We will finish by generalizing the theory to the case of many interacting oscillators. [Preview Abstract] |
Thursday, March 16, 2017 8:48AM - 9:00AM |
R15.00005: Correlations in Coupled Chaotic Diode Lasers Jose Rios Leite, Wendson Barbosa, Edison Rosero Correlations in different time scales were studied experimentally in the chaotic time behavior of pairs of electrically coupled diode lasers. As the lasers are induced to chaotic oscillation by optical feedback, we find features with in-phase and anti-phase fluctuations that appear when the lasers are coupled in parallel from a high impedance current source. The chaotic synchronization of low frequency power drop fluctuations is observed simultaneously with fast anti-phase power fluctuations associated to pump current competition from the common source. [Preview Abstract] |
Thursday, March 16, 2017 9:00AM - 9:12AM |
R15.00006: Geometry in a dynamical system without space: Hyperbolic Geometry in Kuramoto Oscillator Systems Jan Engelbrecht, Bolun Chen, Renato Mirollo Kuramoto oscillator networks have the special property that their time evolution is constrained to lie on 3D orbits of the M\"obius group acting on the $N$-fold torus $T^N$ which explains the $N-3$ constants of motion discovered by Watanabe and Strogatz. The dynamics for phase models can be further reduced to 2D invariant sets in $T^{N-1}$ which have a natural geometry equivalent to the unit disk $\Delta$ with hyperbolic metric. We show that the classic Kuramoto model with order parameter $Z_1$ (the first moment of the oscillator configuration) is a gradient flow in this metric with a unique fixed point on each generic 2D invariant set, corresponding to the hyperbolic barycenter of an oscillator configuration. This gradient property makes the dynamics especially easy to analyze. We exhibit several new families of Kuramoto oscillator models which reduce to gradient flows in this metric; some of these have a richer fixed point structure including non-hyperbolic fixed points associated with fixed point bifurcations. [Preview Abstract] |
Thursday, March 16, 2017 9:12AM - 9:24AM |
R15.00007: Testing Wave Chaos Statistical Predictions in Scaled Electromagnetic Cavities Bo Xiao, Thomas Antonsen, Edward Ott, Zachary Drikas, Jesus Gil Gil, Steven Anlage Predicting the induced voltage at locations inside a complex enclosure subject to an incident electromagnetic wave is a focus in many fields such as electromagnetic compatibility and telecommunication. Real life enclosures are usually ray-chaotic and the exact solution of the fields heavily depends on the geometry details and is very sensitive to small changes. Thus a statistical approach is more appropriate. Random Coupling Model (RCM) predicts the statistical properties of the waves inside a ray-chaotic enclosure, which has been widely discussed and accepted. Testing RCM in a network of cavities coupled through apertures is a new frontier of RCM but is difficult to demonstrate in experiment due to the size of the large structures. Here we present a novel scaled cavity experimental setup to study the statistical properties of waves in a network of cavities connected by apertures. We scale down the structure to a manageable size, then scale up the frequency and the conductivity of the walls so that the normalized loss is the same compared to the full-size structure. Our current setup can host a scaled-down cubic volume with 450 wavelengths on each dimension, equivalent to a house in full scale. We present the experimental setup and some studies on a single scaled cavity, which is the first step towards more complicated networks. [Preview Abstract] |
Thursday, March 16, 2017 9:24AM - 9:36AM |
R15.00008: Nonlinear Wave Chaos and the Random Coupling Model Min Zhou, Edward Ott, Thomas M. Antonsen, Steven Anlage The Random Coupling Model (RCM) has been shown to successfully predict the statistical properties of linear wave chaotic cavities in the highly over-moded regime. It is of interest to extend the RCM to strongly nonlinear systems. To introduce nonlinearity, an active nonlinear circuit is connected to two ports of the wave chaotic \textonequarter -bowtie cavity. The active nonlinear circuit consists of a frequency multiplier, an amplifier and several passive filters. It acts to double the input frequency in the range from 3.5 GHz to 5 GHz, and operates for microwaves going in only one direction. Measurements are taken between two additional ports of the cavity and we measure the statistics of the second harmonic voltage over an ensemble of realizations of the scattering system. We developed an RCM-based model of this system as two chaotic cavities coupled by means of a nonlinear transfer function. The harmonics received at the output are predicted to be the product of three statistical quantities that describe the three elements correspondingly. Statistical results from simulation, RCM-based modeling, and direct experimental measurements will be compared. [Preview Abstract] |
Thursday, March 16, 2017 9:36AM - 9:48AM |
R15.00009: Chaos computing in hybrid digital-analog systems Vivek Kohar, Behnam Kia, John F. Lindner, William L. Ditto Nonlinear dynamical systems, especially when operating in chaotic regime, are very sensitive to noise and the deviations due to noise restrict the exploitation of the large number of dynamical behaviors contained in these systems. We discuss the super-stability of some initial conditions of nonlinear dynamical systems \footnote{V. Kohar, B. Kia, J. Lindner, W. Ditto Phys. Rev. E {\bf 93}, 032213 (2016).} and how such initial conditions can be utilized in chaos computing to implement all Boolean functions in hybrid digital-analog systems \footnote{V. Kohar, B. Kia, J. Lindner, W. Ditto, (submitted).} consisting of digital AND gates and a $3-$transistor analog circuit. We further discuss the super-linear scaling of noise robustness of these super-stable initial conditions when a number of identical nonlinear dynamical systems are coupled together in various network topologies \footnote{V. Kohar, S. Kia, B. Kia, J. Lindner, W. Ditto Nonlinear Dynamics, {\bf 84}, 1805-1812 (2016).}. [Preview Abstract] |
Thursday, March 16, 2017 9:48AM - 10:00AM |
R15.00010: Why do Reservoir Computing Networks Predict Chaotic Systems so Well? Zhixin Lu, Jaideep Pathak, Michelle Girvan, Brian Hunt, Edward Ott Recently a new type of artificial neural network, which is called a reservoir computing network (RCN), has been employed to predict the evolution of chaotic dynamical systems from measured data and without \textit{a priori} knowledge of the governing equations of the system. The quality of these predictions has been found to be spectacularly good. Here, we present a dynamical-system-based theory for how RCN works. Basically a RCN is thought of as consisting of three parts, a randomly chosen input layer, a randomly chosen recurrent network (the reservoir), and an output layer. The advantage of the RCN framework is that training is done only on the linear output layer, making it computationally feasible for the reservoir dimensionality to be large. In this presentation, we address the underlying dynamical mechanisms of RCN function by employing the concepts of generalized synchronization and conditional Lyapunov exponents. Using this framework, we propose conditions on reservoir dynamics necessary for good prediction performance. By looking at the RCN from this dynamical systems point of view, we gain a deeper understanding of its surprising computational power, as well as insights on how to design a RCN. [Preview Abstract] |
Thursday, March 16, 2017 10:00AM - 10:12AM |
R15.00011: Rotational diffusion of a molecular cat Ori Katz-Saporta, Efi Efrati We show that a simple isolated system can perform rotational random walk on account of internal excitations alone. We consider the classical dynamics of a "molecular cat": a triatomic “molecule” connected by three harmonic springs with non-zero rest lengths, suspended in free space. In this system, much like for falling cats, the angular momentum constraint is non-holonomic allowing for rotations with zero overall angular momentum. The geometric nonlinearities arising from the non-zero rest lengths of the springs suffice to break integrability and lead to chaotic dynamics. The coupling of the non-integrability of the system and its non-holonomic nature results in an angular random walk of the molecule. We study the properties and dynamics of this angular motion analytically and numerically. For low energy excitations the system displays normal-mode-like motion, while for high enough excitation energy we observe regular random-walk. In between, at intermediate energies we observe an angular Lévy-walk type motion associated with a fractional diffusion coefficient interpolating between the two regimes. [Preview Abstract] |
Thursday, March 16, 2017 10:12AM - 10:24AM |
R15.00012: Random Matrix Theory Approach to Chaotic Coherent Perfect Absorbers Huanan Li, Suwun Suwunnarat, Ragnar Fleischmann, Holger Schanz, Tsampikos Kottos |
Thursday, March 16, 2017 10:24AM - 10:36AM |
R15.00013: Abstract Withdrawn Networks containing locally active circuit elements have the potential to be `poised at the edge of chaos' and/or to express chaotic behavior, which could be a powerful non-Boolean computing primitive. Such an element has not been realized within a compact electronic device. Here we built sub-100 nm NbO$_{\mathrm{2}}$ memristors that exhibited two distinct regions of negative differential resistance in their quasi-DC i-v characteristics: current controlled and temperature controlled. We identify the latter as a `Chua Corsage' generated by the Joule-heating-driven Mott transition. We constructed a Pearson-Anson-like oscillator using these devices and observed a range of distinctly accessible sinusoidal and highly chaotic oscillations generated by applying a variable DC voltage. We show that the nanoscale device dimensions, coupled with the non-linear transport mechanism, generated chaotic oscillations that were driven by ambient thermal noise. The voltage-tunable locally active behavior of this device provides a new route to chaotic and emergent behavior in transistorless electronic circuits. |
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