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NASA NTRS1994

Topological analysis of chaotic orbits: Revisiting Hyperion

There is emerging interest in the possibility of chaotic evolution in astrophysical systems. To mention just one example, recent well-sampled ground-based observations of the Saturian satellite Hyperion strongly suggest that it is exhibiting chaotic behavior. We present a general technique, the method of close returns, for the analysis of data from astronomical objects believed to be exhibiting chaotic motion. The method is based on the extraction of pieces of the evolution that exhibit nearly periodic behavior-episodes during which the object stays near in phase space to some unstable periodic orbit. Such orbits generally act as skeletal features, tracing the topological organization of the manifold on which the chaotic dynamics takes place. This method does not require data sets as lengthy as other nonlinear analysis techniques do and is therefore well suited to many astronomical observing programs. Well sampled data covering between twenty and forty characteristic periods of the system have been found to be sufficient for the application of this technique. Additional strengths of this method are its robustness in the presence of noise and the ability for a user to clearly distinguish between periodic, random, and chaotic behavior by inspection of the resulting two-dimensional image. As an example of its power, we analyze close returns in a numerically generated data set, based on a model for Hyperion extensively studied in the literature, corresponding to nightly observations of the satellite. We show that with a small data set, embedded unstable periodic orbits can be extracted and that these orbits can be responsible for nearly periodic behavior lasting a substantial fraction of the observing run.

Boyd, Patricia T.↗

DOE OSTI2024

Complementarity for a dynamical black hole

Black hole complementarity posits that the interior of a black hole is not independent from its Hawking radiation. This leads to an apparent violation of causality: the interior can be acausally affected by operators acting solely on the radiation. We argue that this perspective is misleading and that the black hole interior must be viewed as existing in the causal past of the Hawking radiation, despite the fact that they are spacelike separated in the semiclassical description. Consequently, no operation on the Hawking radiation—no matter how complex—can affect the experience of an infalling observer. The black hole interior and the radiation only appear spacelike separated in the semiclassical description because an infalling observer’s ability to access complex information is limited; the chaotic dynamics on the horizon, as viewed from the exterior, then converts any effect caused by such an observer to information in the Hawking radiation which cannot be accessed at the semiclassical level. We arrive at the picture described above by considering a unitary exterior description in which the flow of information is strictly causal, which we extend to apply throughout the entire history of black hole evolution, including its formation. This description uses the stretched event horizon as an inner edge of spacetime, on which the information inside is holographically encoded. We argue that the global spacetime picture arises from coarse graining over black hole microstates, and discuss its relationship with the exterior description.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Understanding sextupole

In this study, we reassess the dynamics within a simple accelerator lattice featuring a single degree of freedom and incorporating a sextupole magnet. In the initial segment, we revisit the H\'enon quadratic map, a representation of a general transformation with quadratic nonlinearity. In the subsequent section, we unveil that a conventional sextupole is essentially a composite structure, comprising an integrable McMillan sextupole and octupole, along with non-integrable corrections of higher orders. This fresh perspective sheds light on the fundamental nature of the sextupole magnet, providing a more nuanced understanding of its far-from-trivial chaotic dynamics. Importantly, it enables the description of driving terms of the second and third orders and introduces associated nonlinear Courant-Snyder invariant.

43 PARTICLE ACCELERATORS↗

DOE OSTI2024

Understanding Sextupole

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

An optimization method for chaotic turbulent flow

Evidence indicates that quantities-of-interest in some turbulent flows can be controlled despite the overall chaotic dynamics. It is typically thought that this is via relatively deterministic, larger-scale components of the turbulence. However, finding such controls, if they exist, is challenging because chaos causes sensitivity gradients to explode and the search space to become intractably non-convex. This challenge is analyzed, and a penalty method is introduced to cope with it. In the new approach, the time domain is broken into segments approximately matching the chaos time scales, so that the solution within each segment is both physical and relatively deterministic. The initial condition of each segment is included in an adjoint-based gradient optimization, which temporarily introduces artificial Δq discontinuities in the overall solution. The optimization then proceeds in stages with increasing penalization of Δq. The method is developed and illustrated for a logistic map, the Lorenz Equation, and an advection augmented Kuramoto–Sivashinsky Equation. These examples show how the Δq temporarily increases the search scale prior to the strong Δq → 0 penalization that recovers a physical solution. It is then applied to turbulent Kolmogorov flow, for which it also far outperforms a standard adjoint-based gradient search. Finally, the utility of such an optimized chaotic solution is discussed.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Temporal Subsampling Diminishes Small Spatial Scales in Recurrent Neural Network Emulators of Geophysical Turbulence

The immense computational cost of traditional numerical weather and climate models has sparked the development of machine learning (ML) based emulators. Because ML methods benefit from long records of training data, it is common to use data sets that are temporally subsampled relative to the time steps required for the numerical integration of differential equations. Here, we investigate how this often overlooked processing step affects the quality of an emulator's predictions. We implement two ML architectures from a class of methods called reservoir computing: (a) a form of Nonlinear Vector Autoregression (NVAR), and (b) an Echo State Network (ESN). Despite their simplicity, it is well documented that these architectures excel at predicting low dimensional chaotic dynamics. We are therefore motivated to test these architectures in an idealized setting of predicting high dimensional geophysical turbulence as represented by Surface Quasi-Geostrophic dynamics. In all cases, subsampling the training data consistently leads to an increased bias at small spatial scales that resembles numerical diffusion. Interestingly, the NVAR architecture becomes unstable when the temporal resolution is increased, indicating that the polynomial based interactions are insufficient at capturing the detailed nonlinearities of the turbulent flow. The ESN architecture is found to be more robust, suggesting a benefit to the more expensive but more general structure. Spectral errors are reduced by including a penalty on the kinetic energy density spectrum during training, although the subsampling related errors persist. Future work is warranted to understand how the temporal resolution of training data affects other ML architectures.

58 GEOSCIENCES↗

DOE OSTI2023

Spatio-temporal patterning of extensile active stresses in microtubule-based active fluids

Abstract Microtubule-based active fluids exhibit turbulent-like autonomous flows, which are driven by the molecular motor powered motion of filamentous constituents. Controlling active stresses in space and time is an essential prerequisite for controlling the intrinsically chaotic dynamics of extensile active fluids. We design single-headed kinesin molecular motors that exhibit optically enhanced clustering and thus enable precise and repeatable spatial and temporal control of extensile active stresses. Such motors enable rapid, reversible switching between flowing and quiescent states. In turn, spatio-temporal patterning of the active stress controls the evolution of the ubiquitous bend instability of extensile active fluids and determines its critical length dependence. Combining optically controlled clusters with conventional kinesin motors enables one-time switching from contractile to extensile active stresses. These results open a path towards real-time control of the autonomous flows generated by active fluids.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Constant Overhead Entanglement Distillation via Scrambling

High-fidelity quantum entanglement enables key quantum networking capabilities such as secure communication and distributed quantum computing, but long-distance entanglement distribution is limited by noise and loss. Entanglement distillation protocols address this problem by extracting high-fidelity Bell pairs from multiple noisy ones. The primary objective is minimizing the resource overhead: the number of noisy input pairs needed to distill each high-fidelity output pair. While protocols achieving optimal overhead are known in theory, they often require complex decoding operations that make practical implementation challenging. We circumvent this challenge by introducing protocols that use quantum scrambling—the spreading of quantum information under chaotic dynamics—through random Clifford operations. Based on this scrambling mechanism, our protocol maintains asymptotically constant overhead, independent of the desired output error rate $\bar{𝜖}$ , and can be implemented with shallow quantum circuits of depth 𝑂⁡(poly log log⁡ $\bar{𝜖}$ −1 ) and memory 𝑂⁡(poly log⁡ $\bar{𝜖}$ −1 ). Our protocol remains effective even with noisy quantum gates. By incorporating error correction, our protocol achieves state-of-the-art performance: starting with pairs of 10% initial infidelity, we require only seven noisy inputs per output pair to distill a single Bell pair with infidelity $\bar{𝜖}$ =10 −12 , substantially outperforming existing schemes. We demonstrate the utility of our protocols for quantum repeater networks.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Slow thermalization and subdiffusion in $U$(1) conserving Floquet random circuits

Random quantum circuits are paradigmatic models of minimally structured and analytically tractable chaotic dynamics. We study a family of Floquet unitary circuits with Haar random U(1) charge conserving dynamics; the minimal such model has nearest-neighbor gates acting on spin-1/2 qubits, and a single layer of even/odd gates repeated periodically in time. We find that this minimal model is not robustly thermalizing at numerically accessible system sizes, and displays slow subdiffusive dynamics for long times. We map out the thermalization dynamics in a broader parameter space of charge conserving circuits, and understand the origin of the slow dynamics in terms of proximate localized and integrable regimes in parameter space. In contrast, we find that small extensions to the minimal model are sufficient to achieve robust thermalization; these include (i) increasing the interaction range to three-site gates, (ii) increasing the local Hilbert space dimension by appending an additional unconstrained qubit to the conserved charge on each site, or (iii) using a larger Floquet period comprised of two independent layers of gates. Furthermore, our results should inform future studies of charge conserving circuits, which are relevant for a wide range of topical theoretical questions.

1-dimensional spin chains↗

NASA NTRS1983

Sea Height Information from Complex SAR Data

Synthetic aperture radar data was used to construct an estimation algorithm for development of information on long waves. The evolution of chaotic dynamic systems was also explored.

Harger, R. O.↗

NASA NTRS1990

Visions of visualization aids: Design philosophy and experimental results

Aids for the visualization of high-dimensional scientific or other data must be designed. Simply casting multidimensional data into a two- or three-dimensional spatial metaphor does not guarantee that the presentation will provide insight or parsimonious description of the phenomena underlying the data. Indeed, the communication of the essential meaning of some multidimensional data may be obscured by presentation in a spatially distributed format. Useful visualization is generally based on pre-existing theoretical beliefs concerning the underlying phenomena which guide selection and formatting of the plotted variables. Two examples from chaotic dynamics are used to illustrate how a visulaization may be an aid to insight. Two examples of displays to aid spatial maneuvering are described. The first, a perspective format for a commercial air traffic display, illustrates how geometric distortion may be introduced to insure that an operator can understand a depicted three-dimensional situation. The second, a display for planning small spacecraft maneuvers, illustrates how the complex counterintuitive character of orbital maneuvering may be made more tractable by removing higher-order nonlinear control dynamics, and allowing independent satisfaction of velocity and plume impingement constraints on orbital changes.

Ellis, Stephen R.↗

NASA NTRS1990

Discretization chaos - Feedback control and transition to chaos

Problems in the design of feedback controllers for chaotic dynamical systems are considered theoretically, focusing on two cases where chaos arises only when a nonchaotic continuous-time system is discretized into a simpler discrete-time systems (exponential discretization and pseudo-Euler integration applied to Lotka-Volterra competition and prey-predator systems). Numerical simulation results are presented in extensive graphs and discussed in detail. It is concluded that care must be taken in applying standard dynamical-systems methods to control systems that may be discontinuous or nondifferentiable.

Grantham, Walter J.↗

NASA NTRS2000

On The Computational Capabilities of Physical Systems: Relationship With Conventional Computer Science - Part 2

In the first of this pair of papers, it was proven that there cannot be a physical computer to which one can properly pose any and all computational tasks concerning the physical universe. It was then further proven that no physical computer C can correctly carry out all computational tasks that can be posed to C. As a particular example, this result means that no physical computer that can, for any physical system external to that computer, take the specification of that external system's state as input and then correctly predict its future state before that future state actually occurs; one cannot build a physical computer that can be assured of correctly "processing information faster than the universe does". These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine. This generality is a direct consequence of the fact that a novel definition of computation - "physical computation" - is needed to address the issues considered in these papers, which concern real physical computers. While this novel definition does not fit into the traditional Chomsky hierarchy, the mathematical structure and impossibility results associated with it have parallels in the mathematics of the Chomsky hierarchy. This second paper of the pair presents a preliminary exploration of some of this mathematical structure. Analogues of Chomskian results concerning universal Turing Machines and the Halting theorem are derived, as are results concerning the (im)possibility of certain kinds of error-correcting codes. In addition, an analogue of algorithmic information complexity, "prediction complexity", is elaborated. A task-independent bound is derived on how much the prediction complexity of a computational task can differ for two different reference universal physical computers used to solve that task, a bound similar to the "encoding" bound governing how much the algorithm information complexity of a Turing machine calculation can differ for two reference universal Turing machines. Finally, it is proven that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity), in that there is one and only version of it that can be applicable throughout our universe.

Wolpert, David H.↗

NASA NTRS2000

On the Computational Capabilities of Physical Systems: The Impossibility of Infallible Computation - Part 1

In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical universe. Next it is proven that no physical computer C can correctly carry out any computational task in the subset of such tasks that can be posed to C. This result holds whether the computational tasks concern a system that is physically isolated from C, or instead concern a system that is coupled to C. As a particular example, this result means that there cannot be a physical computer that can, for any physical system external to that computer, take the specification of that external system's state as input and then correctly predict its future state before that future state actually occurs; one cannot build a physical computer that can be assured of correctly 'processing information faster than the universe does'. The results also mean that there cannot exist an infallible, general-purpose observation apparatus, and that there cannot be an infallible, general-purpose control apparatus. These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine. This generality is a direct consequence of the fact that a novel definition of computation - a definition of 'physical computation' - is needed to address the issues considered in these papers. While this definition does not fit into the traditional Chomsky hierarchy, the mathematical structure and impossibility results associated with it have parallels in the mathematics of the Chomsky hierarchy. The second in this pair of papers presents a preliminary exploration of some of this mathematical structure, including in particular that of prediction complexity, which is a 'physical computation analogue' of algorithmic information complexity. It is proven in that second paper that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity), in that there is one and only version of it that can be applicable throughout our universe.

Wolpert, David H.↗

NASA NTRS2018

Mission analysis for the EM-1 CubeSats EQUULEUS and OMOTENASHI

EQUULEUS is a Lunar L2 orbiter and a 6-Unit CubeSat by JAXA and the University of Tokyo. OMOTENASHI is a 6-Unit CubeSat by JAXA, the world’s smallest Lunar lander. EQUULEUS and OMOTENASHI are among the 13 secondary payloads selected by NASA to be launched with Exploration Mission-1 in 2019. Despite their limited size and cost, EQUULEUS and OMOTENASHI are challenging missions, especially in terms of trajectory design and control. EQUULEUS exploits the Earth-Sun-Moon chaotic dynamics and enter a libration point orbit around the L2 of the Earth-Moon system, using a new water propulsion system with low thrust and little propellant. This “Orbit Control Experiment” is one of the main objectives of the mission. OMOTENASHI executes a semi-hard landing that requires breaking the spacecraft to a stop just a few-hundred meters above the Moon’s surface. Both missions present new and unique challenges, where the design of the nominal trajectory is mainly driven by the constrains on orbital control capabilities, and operational and robustness considerations. This paper presents the current baselines, and give an overview of the new techniques developed for their design.

Oshima, Kenta↗

DOE OSTI2026

Chaos as the cause of randomness in tearing mode onset times in DIII-D ITER baseline scenario plasmas

The first evidence of widespread chaotic plasma dynamics relevant to the onset of disruptive tearing modes is found in DIII-D tokamak discharges. This evidence is obtained by calculating positive maximal Lyapunov exponents from Mirnov signals using the Rosenstein algorithm, implying the chaotic growth and/or rotation of magnetohydrodynamic modes in DIII-D [Luxon, Nucl. Fusion 42, 614 (2002)] plasmas. Such chaotic dynamics offer an explanation for the random tearing mode onset times previously observed in ITER baseline scenario discharges. Further, the Lyapunov timescales associated with this chaos could inhibit reliable prediction of tearing mode onset beyond the angular momentum confinement timescale; this constraint should inform the design of plasma control systems on ITER.

Chaotic dynamics↗

NASA NTRS1981

Chaotic behavior and fluid dynamics

It is shown that while fluids are very complicated systems having many degrees of freedom, the basic mechanism underlying the observed chaotic behavior of fluid turbulence may not require the consideration of many degrees of freedom. Simple dynamical systems display erratic and chaotic behavior reminiscent of turbulence, the key to which is the 'sensitivity to initial conditions' introduced by Ruelle (1978). Simple models which are sensitive to initial data and have permitted workers to study chaotic behavior are described, with attention to the Lorenz equations, Landau's idea of a continuous transition to turbulence via an infinite cascade of bifurcations, and one-dimensional maps.

Yorke, J. A.↗

NASA NTRS1994

Nonlinear dynamics of drops and bubbles and chaotic phenomena

Nonlinear phenomena associated with the dynamics of free drops and bubbles are investigated analytically, numerically and experimentally. Although newly developed levitation and measurement techniques have been implemented, the full experimental validation of theoretical predictions has been hindered by interfering artifacts associated with levitation in the Earth gravitational field. The low gravity environment of orbital space flight has been shown to provide a more quiescent environment which can be utilized to better match the idealized theoretical conditions. The research effort described in this paper is a closely coupled collaboration between predictive and guiding theoretical activities and a unique experimental program involving the ultrasonic and electrostatic levitation of single droplets and bubbles. The goal is to develop and to validate methods based on nonlinear dynamics for the understanding of the large amplitude oscillatory response of single drops and bubbles to both isotropic and asymmetric pressure stimuli. The first specific area on interest has been the resonant coupling between volume and shape oscillatory modes isolated gas or vapor bubbles in a liquid host. The result of multiple time-scale asymptotic treatment, combined with domain perturbation and bifurcation methods, has been the prediction of resonant and near-resonant coupling between volume and shape modes leading to stable as well as chaotic oscillations. Experimental investigations of the large amplitude shape oscillation modes of centimeter-size single bubbles trapped in water at 1 G and under reduced hydrostatic pressure, have suggested the possibility of a low gravity experiment to study the direct coupling between these low frequency shape modes and the volume pulsation, sound-radiating mode. The second subject of interest has involved numerical modeling, using the boundary integral method, of the large amplitude shape oscillations of charged and uncharged drops in the presence of a static or time-varying electric field. Theoretically predicted non linearity in the resonant frequency of the fundamental quadrupole mode has been verified by the accompanying experimental studies. Additional phenomena such as hysteresis in the frequency response of ultrasoncially levitated droplets in the presence of a time varying electric field, and mode coupling in the oscillations of ultrasonically modulated droplets, have also been uncovered. One of the results of this ground-based research has been the identification and characterization of phenomena strictly associated with the influence of the gravitational field. This has also allowed us to identify the specific requirements for potential microgravity investigations yielding new information not obtainable on Earth.

Trinh, Eugene H.↗