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DOE OSTI2023

Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

2D analysis of tokamak divertor-plasma detachment-bifurcation with operational parameters and geometries

UEDGE simulations with density scans for various input power, transport coefficients and outer poloidal leg length are performed to study the conditions for the existence of a bifurcation-like drop of T e at the outer strike point, commonly referred to as a detachment cliff, when transitioning to a detached plasma from an attached plasma in the outer divertor as the upstream density increases (McLean et al., 2015). The simulation results show that a detachment cliff tends to occur with a higher power input regardless of diffusivities and leg length. Further analysis of change of plasma profiles at a cliff indicate that, in addition to the sharp reduction of the E x B drift fluxes in the outer divertor studied in Jaervinen et al., (2018), the substantial change of the Mach number in the outer divertor and the decrease of the outer mid-plane T e due to the radiation front moving across the separatrix into the confinement region above the X-point consistently occur for all UEDGE density scans that have a detachment cliff. UEDGE time-dependent simulation of the evolution of a detachment cliff shows that the rapid increase of radiation above the X-point occurs in a time scale of ~0.3–0.5, which could possibly be the trigger for the formation of a detachment cliff, quicker than the Mach number change in a time scale of ~1 ms and the drop of T e in a time scale of ~2–3 ms in the outer divertor.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Learning to self-fold at a bifurcation

Not Available

Arinze, Chukwunonso↗

DOE OSTI2025

Electron Bifurcating Hydrogenases

The importance of electron-bifurcating enzymes is manifest by their ability to maximize energy efficiency. Specifically, they couple a downhill oxidation-reduction (redox) reaction with an uphill redox reaction. Since the rapid increase in the discovery of bifurcating enzymes starting in 2008, there has been interest in incorporating their mechanistic principles into artificial/semiartificial systems to drive chemically challenging reactions. This has yet to be achieved, partly because the details of electron bifurcation, i.e. mechanisms, are largely elusive. Nevertheless, much progress has been made in understanding reactivities, structures, and some mechanistic aspects of these enzymes. Notable examples are electron-bifurcating hydrogenases, which are the focus of this chapter. The chapter is organized as follows. Section 11.1 provides an overview of hydrogenases and electron bifurcation. In Section 11.2, some physiological roles of electron-bifurcating hydrogenases are highlighted. Additionally, electron-bifurcating subunit compositions and biochemical reactivities are comprehensively tabulated, and some key points/considerations about these are noted. In Section 11.3, we discuss the known structures of these enzymes, which provide insight into their complex arrangements of redox cofactors, such as iron-sulfur clusters. Also provided are tabulations and discussions of some biophysical properties of the cofactors. In Section 11.4, we discuss the mechanistic proposals of these enzymes, which are primarily based on structural information. Areas of research that are much needed are outlined in Section 11.5. We conclude on the note that what is learned from electron-bifurcating hydrogenases has applicability to other bifurcating enzymes, nonbifurcating analogs, and mechanistic enzymology at large.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRSDate not supplied

A Fundamental Study of Bifurcation Acoustic Treatment Effects on Aft-Fan Engine Noise

Increasing air traffic and more stringent aircraft noise regulations continue to expand requirements on aircraft noise levels for conventional and unconventional aircraft configurations. A major component of the overall aircraft noise is the sound associated with the propulsion system mounted in the engine nacelle. Acoustic liners mounted in the aircraft engine nacelles provide a significant portion of the current fan noise reduction. However, they must be further optimized if challenging noise reduction goals are to be achieved. One location within the aft bypass duct that may be an excellent candidate for increased attention is the acoustic treatment on the engine bifurcations (i.e., engine pylon and lower bifurcation). This paper presents the continuation of a fundamental study of the effects of bifurcation treatment on simulated aft fan noise and the validation of numerical tools to predict such effects. Five bifurcation configurations (four treated and one hardwall) were fabricated and tested in the NASA Langley Curved Duct Test Rig. Results show that mode scattering may occur due to both the presence of the bifurcation, as well as variable impedance distributions on the bifurcation surface. Future work will also include optimization of bifurcation treatments for testing in the Curved Duct Test Rig. These initial results are promising and this work provides valuable information for further study and improvement of the performance of bifurcation acoustic treatments.

bifurcation effects↗

NASA NTRSDate not supplied

A Fundamental Study of Bifurcation Acoustic Treatment Effects on Aft-Fan Engine Noise

bifurcation effects↗

NASA NTRSDate not supplied

A Fundamental Study of Bifurcation Acoustic Treatment Effects on Aft-Fan Engine Noise

bifurcation effects↗

NASA NTRS1985

Bifurcation theory applied to aircraft motions

Bifurcation theory is used to analyze the nonlinear dynamic stability characteristics of single-degree-of-freedom motions of an aircraft or a flap about a trim position. The bifurcation theory analysis reveals that when the bifurcation parameter, e.g., the angle of attack, is increased beyond a critical value at which the aerodynamic damping vanishes, a new solution representing finite-amplitude periodic motion bifurcates from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solution is stable (supercritical) or unstable (subcritical). For the pitching motion of a flap-plate airfoil flying at supersonic/hypersonic speed, and for oscillation of a flap at transonic speed, the bifurcation is subcritical, implying either that exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop. On the other hand, for the rolling oscillation of a slender delta wing in subsonic flight (wing rock), the bifurcation is found to be supercritical. This and the predicted amplitude of the bifurcation periodic motion are in good agreement with experiments.

Hui, W. H.↗

NASA NTRS1985

Bifurcation theory applied to aircraft motions

The bifurcation theory is used to analyze the nonlinear dynamic stability characteristics of single-degree-of-freedom motions of an aircraft or a flap about a trim position. The bifurcation theory analysis reveals that when the bifurcation parameter, e.g., the angle of attack, is increased beyond a critical value at which the aerodynamic damping vanishes, a new solution representing finite-amplitude periodic motion bifurcates from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solution is stable (supercritical) or unstable (critical). For the pitching motion of a flap-plate airfoil flying at supersonic/hypersonic speed, and for oscillation of a flap at transonic speed, the bifurcation is subcritical, implying either that exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop. On the other hand, for the rolling oscillation of a slender delta wing in subsonic flight (wing rock), the bifurcation is found to be supercritical. This and the predicted amplitude of the bifurcation periodic motion are in good agreement with the experiments.

Hui, W. H.↗

NASA NTRS2000

Experiment Evaluation of Bifurcation in Sands

The basic principles of bifurcation analysis have been established by several investigators, however several issues remain unresolved, specifically how do stress level, grain size distribution, and boundary conditions affect general bifurcation phenomena in pressure sensitive and dilatant materials. General geometrical and kinematics conditions for moving surfaces of discontinuity was derived and applied to problems of instability of solids. In 1962, the theoretical framework of bifurcation by studying the acceleration waves in elasto-plastic (J2) solids were presented. Bifurcation analysis for more specific forms of constitutive behavior was examined by studying localization in pressure-sensitive, dilatant materials, however, analyses were restricted to plane deformation states only. Bifurcation analyses were presented and applied to predict shear band formations in sand under plane strain condition. The properties of discontinuous bifurcation solutions for elastic-plastic solids under axisymmetric and plane strain loading conditions were studied. The study focused on theory, but also references and comparisons to experiments were made. The current paper includes a presentation of a summary of bifurcation analyses for biaxial and triaxial (axisymmetric) loading conditions. The Coulomb model is implemented using incremental piecewise scheme to predict the constitutive relations and shear band inclination angles. Then, a comprehensive evaluation of bifurcation phenomena is presented based on data from triaxial experiments performed under microgravity conditions aboard the Space Shuttle under very low effective confining pressure (0.05 to 1.30 kPa), in which very high peak friction angles (47 to 75 degrees) and dilatancy angles (30 to 31 degrees) were measured. The evaluation will be extended to include biaxial experiments performed on the same material under low (10 kPa) and moderate (100 kPa) confining pressures. A comparison between the behavior under biaxial and triaxial loading conditions will be presented, and related issues concerning influence of confining pressure will be discussed.

Alshibi, Khalid A.↗

NASA NTRS2019

An Investigation of Bifurcation Acoustic Treatment Effects on Aft-Fan Engine Nacelle Noise

Increasing air traffic and more stringent aircraft noise regulations continue to expand requirements on aircraft noise reduction capabilities for conventional and unconventional aircraft configurations. A major component of the overall aircraft noise is the sound associated with the propulsion system mounted in the engine nacelle. Acoustic liners mounted in the aircraft engine nacelles provide a significant portion of the current fan noise reduction. However, they must be further optimized if challenging noise reduction goals are to be achieved. One area within the aft bypass duct that may be an excellent candidate for increased attention is the acoustic treatment on the engine bifurcations (i.e., engine pylon and lower bifurcation). This paper describes a fundamental study of the effects of bifurcation treatment on simulated aft fan noise, as well as the validation of numerical tools to predict such effects. Five bifurcation configurations (four treated and one hardwall) were fabricated and tested in the NASA Langley Curved Duct Test Rig. Results show that mode scattering may occur due to both the presence of the bifurcation, as well as variable impedance distributions on the bifurcation surface. Future work will also include optimization of bifurcation treatments for testing in the Curved Duct Test Rig. These initial results are promising and this work provides valuable information for further study and improvement of the performance of bifurcation acoustic treatments.

Nark, Douglas M.↗

DOE OSTI2026

The anaerobic fungus Caecomyces churrovis produces H 2 via a non-bifurcating NADH-dependent enzyme complex

ABSTRACT Hydrogenosomes are mitochondria-derived organelles that produce ATP and H 2 to support energy metabolism in anaerobic eukaryotes. H 2 production allows reoxidation of reduced cofactors generated during fermentative metabolism; however, the metabolic mechanisms for H 2 production in anaerobic eukaryotes remains incompletely understood. In particular, it remains unclear whether anaerobic fungi (AF) hydrogenosomes use a ferredoxin-dependent pathway or a distinct mechanism to regenerate NAD(P) + and link electron transfer to H 2 formation. Here, by combining genomic search, proteomic analysis, and enzymology, we reveal the molecular mechanism for H 2 production in the AF strain Caecomyces churrovis . Our enzyme assays on the organelle fraction of C. churrovis revealed the activity of H 2 :NAD + oxidoreductase but not pyruvate:ferredoxin oxidoreductase, which is usually linked to H 2 formation. We identified genes encoding [FeFe] hydrogenase (Hyd) and NADH dehydrogenase subunits E and F (NuoE, NuoF) in C. churrovis , and confirmed their expression in the isolated hydrogenosomal fractions by proteomic analysis. Combining the individually purified enzymes, we found Hyd and NuoEF proteins formed H 2 directly from NADH independently of ferredoxin, functioning as a non-bifurcating NADH-dependent enzyme rather than an electron-bifurcating enzyme. We identified homologs of hydrogenosomal NuoE, NuoF, and Hyd in many other AF, indicating this pathway is commonly shared among the AF. This work demonstrates the existence of a non-bifurcating NADH-dependent enzyme complex in eukaryotes. Moreover, this complex could potentially be exploited as a target for controlling AF H 2 production and altering fungal metabolism. IMPORTANCE H 2 production is a prominent feature of anaerobic energy metabolism, yet our understanding of eukaryotic mechanisms remains limited. Anaerobic fungi (AF) are key decomposers of lignocellulose and contribute to hydrogen flux in anaerobic environments. Although it has been more than 40 years since the H 2 production from AF was first reported, the molecular mechanism for hydrogenosomal H 2 production and redox balance remains unclear. We demonstrate that AF produce H 2 from NADH utilizing a non-bifurcating NADH-dependent enzyme complex rather than an electron-bifurcating, ferredoxin-dependent variant. We show that this enzyme complex is conserved across multiple AF lineages and thus demonstrate the occurrence of a non-bifurcating NADH-dependent enzyme in eukaryotes. This discovery expands our understanding of eukaryotic hydrogenosomal metabolism, reveals a previously unknown strategy for redox balancing, and highlights potential targets for manipulating H 2 production. These insights have broad implications for microbial energy metabolism, anaerobic ecosystems, and bioengineering of H 2 -producing systems.

Zhang, Bo [Department of Chemical Engineering, Uni↗

NASA NTRS1981

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigenmode of the panel. In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths. The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.

Noor, A. K.↗

NASA NTRS2004

Effects of Bifurcations on Aft-Fan Engine Nacelle Noise

Aft-fan engine nacelle noise is a significant factor in the increasingly important issue of aircraft community noise. The ability to predict such noise within complex duct geometries is a valuable tool in studying possible noise attenuation methods. A recent example of code development for such predictions is the ducted fan noise propagation and radiation code CDUCT-LaRC. This work focuses on predicting the effects of geometry changes (i.e. bifurcations, pylons) on aft fan noise propagation. Beginning with simplified geometries, calculations show that bifurcations lead to scattering of acoustic energy into higher order modes. In addition, when circumferential mode number and the number of bifurcations are properly commensurate, bifurcations increase the relative importance of the plane wave mode near the exhaust plane of the bypass duct. This is particularly evident when the bypass duct surfaces include acoustic treatment. Calculations involving more complex geometries further illustrate that bifurcations and pylons clearly affect modal content, in both propagation and radiation calculations. Additionally, results show that consideration of acoustic radiation results may provide further insight into acoustic treatment effectiveness for situations in which modal decomposition may not be straightforward. The ability of CDUCT-LaRC to handle complex (non-axisymmetric) multi-block geometries, as well as axially and circumferentially segmented liners, allows investigation into the effects of geometric elements (bifurcations, pylons).

Nark, Douglas M.↗

NASA NTRS2014

Bifurcation Analysis Using Rigorous Branch and Bound Methods

For the study of nonlinear dynamic systems, it is important to locate the equilibria and bifurcations occurring within a specified computational domain. This paper proposes a new approach for solving these problems and compares it to the numerical continuation method. The new approach is based upon branch and bound and utilizes rigorous enclosure techniques to yield outer bounding sets of both the equilibrium and local bifurcation manifolds. These sets, which comprise the union of hyper-rectangles, can be made to be as tight as desired. Sufficient conditions for the existence of equilibrium and bifurcation points taking the form of algebraic inequality constraints in the state-parameter space are used to calculate their enclosures directly. The enclosures for the bifurcation sets can be computed independently of the equilibrium manifold, and are guaranteed to contain all solutions within the computational domain. A further advantage of this method is the ability to compute a near-maximally sized hyper-rectangle of high dimension centered at a fixed parameter-state point whose elements are guaranteed to exclude all bifurcation points. This hyper-rectangle, which requires a global description of the bifurcation manifold within the computational domain, cannot be obtained otherwise. A test case, based on the dynamics of a UAV subject to uncertain center of gravity location, is used to illustrate the efficacy of the method by comparing it with numerical continuation and to evaluate its computational complexity.

Smith, Andrew P.↗

DOE OSTI2025

Electrochemical Observation and pH Dependence of All Three Expected Redox Couples in an Extremophilic Bifurcating Electron Transfer Flavoprotein with Fused Subunits

Bifurcating enzymes employ energy from a favorable electron transfer to drive unfavorable transfer of a second electron, thereby generating a more reactive product. They are therefore highly desirable in catalytic systems, for example, to drive challenging reactions such as nitrogen fixation. While most bifurcating enzymes contain air-sensitive metal centers, bifurcating electron transfer flavoproteins (bETFs) employ flavins. However, they have not been successfully deployed on electrodes. Herein, we demonstrate immobilization and expected thermodynamic reactivity of a bETF from a hyperthermophilic archaeon, Sulfolobus acidocaldarius (SaETF). SaETF differs from previously biochemically characterized bETFs in being a single protein, representing a concatenation of the two subunits of known ETFs. However, SaETF retains the chemical properties of heterodimeric bETFs, including possession of two FADs: one that undergoes sequential 1-electron (1e) reductions at high E° and forms an anionic semiquinone, and another that is amenable to lower-E° 2e reduction, including by NADH. We found homologous monomeric ETF genes in archaeal and bacterial genomes, accompanied by genes that also commonly flank heterodimeric ETFs, and SaETF’s sequence conservation is 50% higher with bETFs than with canonical ETFs. Thus, SaETF is best described as a bETF. Our direct electrochemical trials capture reversible redox couples for all three thermodynamically expected redox events. We document electrochemical activity over a range of pH values and reveal a conformational change coupled to proton acquisition that affects the electrochemical activity of the higher-E° FAD. Thus, this well-behaved monomeric bETF opens the door to bioinspired bifurcating devices or bifurcation on a chip.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Insight Into the High-Potential Branch of the Alternate Nad+-Dependent NADPH:Ferredoxin Oxidoreductase II (NfnII) from Pyrococcus Furiosus. Evidence for the Gating Step in Electron Bifurcation

We have investigated the rapid-reaction kinetics of the NAD+-dependent NADPH:ferredoxin oxidoreductase II (NfnII) from Pyrococcus furiosus, permitting a comparison with recent work done with the paralog NfnI from the same organism. The half-potentials of the electron-bifurcating L-FAD are highly crossed in both NfnI, meaning the potential for the quinone/semiquinone couple is significantly lower than that for the semiquinone/hydroquinone couple so that the semiquinone oxidation state is thermodynamically unstable. The same appears to be the case with NfnII on the basis of its similar behavior in transient absorption spectroscopy experiments and the absence of any evidence for FAD*- accumulation in the course of reductive titrations (which would be manifested as a transient increase in absorbance at ~380 nm)1. Reductive titrations with both the one-electron donor sodium dithionite and the obligate two-electron donor NADPH demonstrate that, in contrast to NfnI, little FADH* accumulates in the high-potential pathway of NfnII in the course of reduction, as reflected in the absence of a transient increase in absorbance in the 500-600 nm region. This indicates that the half-potentials of the S-FAD are crossed in NfnII by a minimum of 120 mV. Rapid-reaction experiments mixing oxidized NfnII with NADPH also show no evidence of S-FADH* accumulation. Furthermore, the enzyme is only partially reduced at the end of the reaction with NADPH, indicating that there is little electron transfer into the high-potential pathway of NfnII. When the reaction is carried out in the presence of the one-electron carrier ferredoxin, only approximately one equivalent of ferredoxin becomes reduced, in contrast to the maximum of three equivalents seen with NfnI. This is consistent with only a single electron bifurcation event taking place under these conditions with NfnII, with only a single high-potential electron passing into the [2Fe-2S] cluster of the high-potential pathway of NfnII but not progressing further to the S-FAD. This accounts for the observation that NfnII has greatly reduced bifurcating activity compared to NfnI. The lack of electron transfer into the S-FAD of NfnII due to its crossed half-potentials prevents successive bifurcation activity and indicates that electron transfer within the high-potential branch gates bifurcation activity in NfnI.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Generative learning for slow manifolds and bifurcation diagrams

In dynamical systems characterized by separation of time scales, the approximation of so called “slow manifolds”, on which the long term dynamics lie, is a useful step for model reduction. Initializing on such slow manifolds is a useful step in modeling, since it circumvents fast transients, and is crucial in multiscale algorithms (like the equation-free approach) alternating between fine scale (fast) and coarser scale (slow) simulations. In a similar spirit, when one studies the infinite time dynamics of systems depending on parameters, the system attractors (e.g., its steady states) lie on bifurcation diagrams (curves for one-parameter continuation, and more generally, on manifolds in state parameter space. Sampling these manifolds gives us representative attractors (here, steady states of ODEs or PDEs) at different parameter values. Algorithms for the systematic construction of these manifolds (slow manifolds, bifurcation diagrams) are required parts of the “traditional” numerical nonlinear dynamics toolkit. In more recent years, as the field of Machine Learning develops, conditional score-based generative models (cSGMs) have been demonstrated to exhibit remarkable capabilities in generating plausible data from target distributions that are conditioned on some given label. It is tempting to exploit such generative models to produce samples of data distributions (points on a slow manifold, steady states on a bifurcation surface) conditioned on (consistent with) some quantity of interest (QoI, observable). In this work, we present a framework for using cSGMs to quickly (a) initialize on a low-dimensional (reduced-order) slow manifold of a multi-time-scale system consistent with desired value(s) of a QoI (a “label”) on the manifold, and (b) approximate steady states in a bifurcation diagram consistent with a (new, out-of-sample) parameter value. This conditional sampling can help uncover the geometry of the reduced slow-manifold and/or approximately “fill in” missing segments of steady states in a bifurcation diagram. Finally, the quantity of interest, which determines how the sampling is conditioned, is either known a priori or identified using manifold learning-based dimensionality reduction techniques applied to the training data.

Dynamical systems↗