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DOE OSTI2021

Data-driven Chaos Indicator for Nonlinear Dynamics and Applications on Storage Ring Lattice Design

A data-driven chaos indicator concept is introduced to characterize the degree of chaos for nonlinear dynamical systems. The indicator is represented by the prediction accuracy of surrogate models established purely from data. It provides a metric for the predictability of nonlinear motions in a given system. When using the indicator to implement a tune-scan for a quadratic Hénon map, the main resonances and their asymmetric stop-band widths can be identified. When applied to particle transportation in a storage ring, as particle motion becomes more chaotic, its surrogate model prediction accuracy decreases correspondingly. Therefore, the prediction accuracy, acting as a chaos indicator, can be used directly as the objective for nonlinear beam dynamics optimization. This method provides a different perspective on nonlinear beam dynamics and an efficient method for nonlinear lattice optimization. Applications in dynamic aperture optimization are demonstrated as real world examples.

43 PARTICLE ACCELERATORS↗

DOE OSTI2025

Fuzzy spheres in stringy matrix models: quantifying chaos in a mixed phase space

We consider a truncation of the BMN matrix model to a configuration of two fuzzy spheres, described by two coupled non-linear oscillators dependent on the mass parameter μ. The classical phase diagram of the system generically (μ ≠ 0) contains three equilibrium points: two centers and a center-saddle; as μ → 0 the system exhibits a pitchfork bifurcation. We demonstrate that the system is exactly integrable in quadratures for μ = 0, while for very large values of μ, it approaches another integrable point characterized by two harmonic oscillators. The classical phase space is mixed, containing both integrable islands and chaotic regions, as evidenced by the classical Lyapunov spectrum. At the quantum level, we explore indicators of early and late time chaos. The eigenvalue spacing is best described by a Brody distribution, which interpolates between Poisson and Wigner distributions; it dovetails, at the quantum level, the classical results and reemphasizes the notion that the quantum system is mixed. We also study the spectral form factor and the quantum Lyapunov exponent, as defined by out-of-time-ordered correlators. These two indicators of quantum chaos exhibit weak correlations with the Brody distribution. We speculate that the behavior of the system as μ → 0 dominates the spectral form factor and the quantum Lyapunov exponent, making these indicators of quantum chaos less effective in the context of a mixed phase space.

AdS-CFT correspondence↗

DOE OSTI2024

Chaos in inhomogeneous neutrino fast flavor instability

In dense neutrino gases, the neutrino-neutrino coherent forward scattering gives rise to a complex flavor oscillation phenomenon not fully incorporated in simulations of neutron star mergers (NSM) and core collapse supernovae (CCSNe). Moreover, it has been proposed to be chaotic, potentially limiting our ability to predict neutrino flavor transformations in simulations. To address this issue, we explore how small flavor perturbations evolve in the nonlinear regime of the neutrino quantum kinetic equation within a narrow centimeter-scale region inside a NSM and a toy neutrino distribution. Our findings reveal that paths in the flavor state space of solutions with similar initial conditions diverge exponentially, exhibiting chaos. This inherent chaos makes the microscopic scales of neutrino flavor transformations unpredictable. However, the domain-averaged neutrino density matrix remains relatively stable, with chaos minimally affecting it. This particular property suggests that domain-averaged quantities remain reliable despite the exponential amplification of errors. Published by the American Physical Society 2024

Astronomy & Astrophysics↗

DOE OSTI2024

Lyapunov exponent as a signature of dissipative many-body quantum chaos

A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large- q limit of a q -body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q ≥ 4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024

García-García, Antonio M.↗

DOE OSTI2024

Quantifying Quantum Chaos through Microcanonical Distributions of Entanglement

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space. Published by the American Physical Society 2024

Physics↗

DOE OSTI2025

BPS chaos

Black holes are chaotic quantum systems that are expected to exhibit random matrix statistics in their finite energy spectrum. Lin, Maldacena, Rozenberg and Shan (LMRS) have proposed a related characterization of chaos for the ground states of BPS black holes with finite area horizons. On a separate front, the “fuzzball program” has uncovered large families of horizon-free geometries that account for the entropy of holographic BPS systems, but only in situations with sufficient supersymmetry to exclude finite area horizons. The highly structured, non-random nature of these solutions seems in tension with strong chaos. We verify this intuition by performing analytic and numerical calculations of the LMRS diagnostic in the corresponding boundary quantum system. In particular we examine the 1/2 and 1/4-BPS sectors of \mathcal{N}=4 𝒩 = 4 SYM, and the two charge sector of the D1-D5 CFT. We find evidence that these systems are only weakly chaotic, with a Thouless time determining the onset of chaos that grows as a power of N N . In contrast, finite horizon area BPS black holes should be strongly chaotic, with a Thouless time of order one. In this case, finite energy chaotic states become BPS as N N is decreased through the recently discovered “fortuity” mechanism. Hence they can plausibly retain their strongly chaotic character.

Chen, Yiming (ORCID:0000000218613230)↗

NASA NTRS1990

The route to chaos for the Kuramoto-Sivashinsky equation

The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609... . Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Papageorgiou, Demetrios T.↗

NASA NTRS1991

Predicting chaos for infinite dimensional dynamical systems: The Kuramoto-Sivashinsky equation, a case study

The results of extensive computations are presented in order to accurately characterize transitions to chaos for the Kuramoto-Sivashinsky equation. In particular, the oscillatory dynamics in a window that supports a complete sequence of period doubling bifurcations preceding chaos is followed. As many as thirteen period doublings are followed and used to compute the Feigenbaum number for the cascade and so enable, for the first time, an accurate numerical evaluation of the theory of universal behavior of nonlinear systems, for an infinite dimensional dynamical system. Furthermore, the dynamics at the threshold of chaos exhibit a fractal behavior which is demonstrated and used to compute a universal scaling factor that enables the self-similar continuation of the solution into a chaotic regime.

Smyrlis, Yiorgos S.↗

NASA NTRS1991

The route to chaos for the Kuramoto-Sivashinsky equation

The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609 .... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Papageorgiou, Demetrios T.↗

NASA NTRS1994

EEG and chaos: Description of underlying dynamics and its relation to dissociative states

The goal of this work is the identification of states especially as related to the process of error production and lapses of awareness as might be experienced during aviation. Given the need for further articulation of the characteristics of 'error prone state' or 'hazardous state of awareness,' this NASA grant focused on basic ground work for the study of the psychophysiology of these states. In specific, the purpose of this grant was to establish the necessary methodology for addressing three broad questions. The first is how the error prone state should be conceptualized, and whether it is similar to a dissociative state, a hypnotic state, or absent mindedness. Over 1200 subjects completed a variety of psychometric measures reflecting internal states and proneness to mental lapses and absent mindedness; the study suggests that there exists a consistency of patterns displayed by individuals who self-report dissociative experiences such that those individuals who score high on measures of dissociation also score high on measures of absent mindedness, errors, and absorption, but not on scales of hypnotizability. The second broad question is whether some individuals are more prone to enter these states than others. A study of 14 young adults who scored either high or low on the dissociation experiences scale performed a series of six tasks. This study suggests that high and low dissociative individuals arrive at the experiment in similar electrocortical states and perform cognitive tasks (e.g., mental math) in a similar manner; it is in the processing of internal emotional states that differences begin to emerge. The third question to be answered is whether recent research in nonlinear dynamics, i.e., chaos, offer an addition and/or alternative to traditional signal processing methods, i.e., fast Fourier transforms, and whether chaos procedures can be modified to offer additional information useful in identifying brain states. A preliminary review suggests that current nonlinear dynamical techniques such as dimensional analysis can be successfully applied to electrocortical activity. Using the data set developed in the study of the young adults, chaos analyses using the Farmer algorithm were performed; it is concluded that dimensionality measures reflect information not contained in traditional EEG Fourier analysis.

Ray, William J.↗

NASA NTRS2003

A Case for Hydrothermal Gray Hematite in Aram Chaos

The Thermal Emission Spectrometer (TES) on Mars Global Surveyor has detected deposits of coarsegrained, gray crystalline hematite in Sinus Meridiani, Aram Chaos, and Vallis Marineris [1]. Detailed features in the hematite spectral signature of the Sinus Meridiani region show that the spectrum is consistent with emission dominated by crystal c-faces of hematite, implying that the hematite is specular [2]. Gray specular hematite (also known as specularite ) is a particular gray crystalline form that has intergrown, hexagonal plates with a silvery metallic luster. We believe that the key to the origin of specularite is that it requires crystallization at temperatures in excess of about 100 C. In reviewing the occurrence of gray hematite on Earth, we find no exceptions to this warm temperature requirement [3]. Thermal crystallization on Mars could occur (1) as diagenesis at a depth of a few kilometers of sediments originally formed in lowtemperature waters, or (2) as direct precipitation from hydrothermal solution. Aram Chaos has unique chaotic terrain that offers more clues to the formation of the hematite than the relatively featureless flat terrain (as seen from orbit) of Sinus Meridiani. Aram Chaos provides the opportunity to look at a combination of TES data, Mars Orbiter Camera images, and Mars Orbiter Laser Altimeter (MOLA) topography. This combination of data suggests that high concentrations of hematite were formed in planar strata and have since been exposed by erosion of an overlying light-toned, caprock. Lesser concentrations of hematite are found adjacent to these strata at lower elevations, which we interpret as perhaps a lag deposit. The topography and the collapsed nature of the chaotic terrain favor a hydrothermally charged aquifer as the original setting where the hematite formed. An alternative sedimentary origin requires post-depositional burial to a depth of ~3-5 km to induce thermally driven recrystallization of fine-grained iron oxides to coarse-grained hematite.

D C Catling↗

DOE OSTI2023

Reminiscence of Classical Chaos in Driven Transmons

Transmon qubits are ubiquitously used in superconducting quantum information processor architectures. Strong drives are required to realize fast, high-fidelity, gates and measurements, including parametrically activated processes. Here, we show that even off-resonant drives, in regimes routinely used in experiments, can cause strong modifications to the structure of the transmon spectrum rendering a large part of it chaotic. Accounting for the full nonlinear dynamics of the transmon in a Floquet-Markov formalism, we find that these chaotic states, often neglected through the hypothesis that the anharmonicity is weak, strongly impact the lifetime of the transmon’s computational states. In particular, we observe that chaos-assisted quantum phase slips greatly enhance band dispersions. In the presence of a measurement resonator, we find that approaching chaotic behavior correlates with strong transmon-resonator hybridization, and an average resonator response centered on the bare resonator frequency. These results lead to a photon-number threshold characterizing the appearance of chaos-induced quantum demolition effects during strong-drive operations, such as dispersive qubit readout. The phenomena described here are expected to be present in all circuits based on low-impedance Josephson junctions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2005

Stochastic Representation of Chaos using Terminal Attractors

A nonlinear version of the Liouville equation based upon terminal attractors is proposed for describing post-instability motions of dynamical systems with exponential divergence of trajectories such as those leading to chaos and turbulence. As a result, the post-instability motions are represented by expectations, variances, and higher moments of the state variables as functions of time. The proposed approach can be applied to conservative chaos, and in particular, to n-bodies problem, as well as to dissipative systems, and in particular, to chaotic attractors and turbulence.

stochastic processes↗

DOE OSTI2023

Effective description of sub-maximal chaos: stringy effects for SYK scrambling

It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective ‘scramblon’ modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

Sextupole reduction via chaos suppression at the National Synchrotron Light Source II

We revisit the nonlinear lattice design approach for the National Synchrotron Light Source II (NSLS-II) storage ring. By suppressing chaos, we identify alternative sextupole configurations to the original design, which relied on the conventional strategy of simultaneously minimizing resonance driving terms (RDTs) and amplitude-dependent detuning (ADD). These alternatives achieve comparable performance while requiring fewer sextupoles. A detailed comparison of two representative solutions is presented and supported by experimental validation. Our results show that the dynamic aperture correlates more strongly with global chaos than with individual RDTs, and that the importance of minimizing ADD may have been overstated in earlier design strategies.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Towards the web of quantum chaos diagnostics

We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also investigated the relationship between a specific circuit complexity and LE by using the inverted oscillator model and made a conjecture about their relationship. These relationships signal a deeper connection between these three probes of quantum chaos.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Sextupole reduction via chaos suppression at NSLS-II

We revisit the nonlinear lattice design approach for the National Synchrotron Light Source II (NSLS-II) storage ring. By suppressing chaos, we identify alternative sextupole configurations to the original design, which relied on the conventional approach of simultaneously minimizing Resonance Driving Terms (RDTs) and Amplitude-Dependent Detuning (ADD). These alternatives achieve comparable performance while requiring fewer sextupoles. A detailed comparison of two representative solutions is presented and supported by experimental validation. Our results indicate that dynamic aperture correlates more strongly with global chaos than with individual RDTs, and that the importance of minimizing ADD may have been overstated in earlier design approach.

43 PARTICLE ACCELERATORS↗

NASA NTRS1989

Low-dimensional chaos in turbulence

Direct numerical simulations are being performed on two different fluid flows in an attempt to discover the mechanism underlying the transition to turbulence in each. The first system is Taylor-Couette flow; the second, two-dimensional flow over an airfoil. Both flows exhibit a gradual transition to high-dimensional turbulence through low-dimensional chaos. The hope is that the instabilities leading to chaos will be easier to relate to physical processes in this case, and that the understanding of these mechanisms can then be applied to a wider array of turbulent systems.

Vastano, John A.↗