Regions Of Space Research Articles

Oral candidiasis in liver transplant patients: species identification and antifungal susceptibility profile.

This study aimed to verify oral candidiasis, identify the causative species, and investigate the antifungal susceptibility of yeasts isolated from liver transplant patients. A descriptive analysis of 97 patients who underwent liver transplantation was conducted at a hospital. Two clinical examinations (Collections A and B) of the oral cavity were performed. Oral material was collected from all patients, inoculated in Sabouraud Dextrose Agar, and incubated at 35℃ for 48 hours. Samples were identified by molecular sequencing of the internal trascribed space region of rDNA. An antifungal susceptibility test with fluconazole, amphotericin B, and micafungin was performed using the Clinical and Laboratory Standards Institute yeast broth microdilution method. Among the patients, 15 presented with oral candidiasis: eight in Collection A and seven in Collection B. The primary type of candidiasis was atrophic, followed by pseudomembranous candidiasis. The most prevalent species was Candida albicans (nine), followed by Candida glabrata (three), Candida tropicalis (two), and Candida dubliniensis (one). Regarding susceptibility to fluconazole, of the 15 samples, 11 were susceptible, three were susceptible in a dose-dependent manner, and one was resistant. The most commonly identified type of candidiasis was atrophic, with C. albicans and C. glabrata being the most prevalent causative species. One fluconazole-resistant isolate each of C. tropicalis and C. albicans were identified.

Stability, Chemical Bonding, and Electron Lone Pair Localization in AsN at High Pressure by Density Functional Theory Calculations.

The covalent bonding framework of crystalline single-bonded cubic AsN, recently synthesized under high pressure and high temperature conditions in a laser-heated diamond anvil cell, is here studied by means of density functional theory calculations and compared to single crystal X-ray diffraction data. The precise localization of the nonbonding electron lone pairs and the determination of their distances and orientations are related to the presence of characteristic structural motifs and space regions of the unit cell dominated by repulsive electronic interactions, with the relative orientation of the electron lone pairs playing a key role in minimizing the energy of the structure. We find that the vibrational modes associated with the expression of the lone pairs are strongly localized, an observation that may have implications for the thermal conductivity of the compound. The results indicate the thermodynamic stability of the experimentally observed structure of AsN above ∼17 GPa, provide a detailed insight into the nature of the chemical bonding network underlying the formation of this compound, and open new perspectives to the design and high pressure synthesis of new pnictogen-based advanced materials for potential applications of energetic and technological relevance.

A novel metaheuristic based on object-oriented programming concepts for engineering optimization

This paper presents a novel, robust, efficient, and simple optimization algorithm called the Object-Oriented Programming Optimization Algorithm (OOPOA) for tackling constrained and unconstrained optimization problems. The algorithm is inspired by the inheritance concept of Object-Oriented programming languages, where the features of a class are classified into three types according to inheritance probability: public, private, and protected. The object-oriented programming inheritance concept is implemented in the algorithm to update the population for the next generations. The proposed algorithm ensures exploitation by selecting the solution with the highest fitness to be inherited in each iteration. It ensures exploration by applying a mutation process that helps explore wide regions in the search space. The performance of this technique is demonstrated by solving 34 different optimization tasks, including 20 standard benchmark problems, ten IEEE Congress of Evolutionary Computation benchmark test functions, and four constrained real-world engineering design problems.

MSE superiority of the unrestricted Stein-rule estimator in a regression model with a possible structural break

This paper investigates the estimation of a linear regression model with a possible structural break at a known point. We analytically derive the exact formulae of the MSE for the restricted SR and PSR estimators, which shrinks the OLS estimator toward the restriction of no structural break. We compare the MSE performance of restricted/unrestricted SR, PSR, and least squared estimators. We analytically show that the unrestricted SR estimator can have a smaller MSE than the restricted SR estimator even when the restriction is correct. Further, our numerical results show that the unrestricted PSR estimator has the best MSE performance over a wide region of parameter space. These results indicate that the use of the unrestricted PSR estimator is recommended even when a structural break may not exist, although the unrestricted PSR estimator does not take the possibility of no structural break into consideration.

Nonreciprocity and odd viscosity in chiral active fluids

Odd viscosity couples stress to strain rate in a dissipationless way. It has been studied in plasmas under magnetic fields, superfluid [Formula: see text], quantum-Hall fluids, and recently in the context of chiral active matter. In most of these studies, odd terms in the viscosity obey Onsager reciprocal relations. Although this is expected in equilibrium systems, it is not obvious that Onsager relations hold in active materials. By directly coarse-graining the kinetic energy and independently using both the Poisson-bracket formalism and a kinetic theory derivation, we find that the appearance of a nonvanishing angular momentum density, which is a hallmark of chiral active materials, necessarily breaks Onsager reciprocal relations. This leads to a non-Hermitian dynamical matrix for the total hydrodynamic momentum and to the appearance of odd viscosity and other nondissipative contributions to the viscosity. Furthermore, by accounting for both the angular momentum density and interactions that lead to odd viscosity, we find regions in the parameter space in which 3D odd mechanical waves propagate and regions in which they are mechanically unstable. The lines separating these regions are continuous lines of exceptional points, suggesting a possible nonreciprocal phase transition.

Laser induced spark ignition of a gaseous methane–oxygen model rocket combustor

The spatio-temporal evolution of laser induced spark ignition of non-premixed gaseous methane and oxygen is investigated. The reactants are injected into an optically accessible combustion chamber from an oxidizer centered shear-coaxial injector. High-speed schlieren imaging and measurements of laser energy deposited are used to characterize ignition behavior at various spark locations throughout the chamber. A spatial map of ignition probability is generated from tests at multiple axial and radial locations from the point of propellant injection. The rate of pressure rise from successful tests reveals two modes of ignition dependent on the location of the laser-induced breakdown, direct and indirect, based on laser-induced breakdown occurring within reactant jet or in the recirculation region, respectively. Physical processes occurring over multiple timescales throughout the direct ignition method are first discussed. Ignition outcomes associated with the indirect method are determined by the hydrodynamic ejection protruding from the laser spark, the behavior of which is explored in detail using flow statistics extracted from images taken at a single spark location. The spatial–temporal progression of this jet is found to be dependent on deposited laser energy, leading to a relationship between the amount of energy added to the flow and ignition outcome. General bounds for spark location, deposited laser energy, and ejection behavior are identified to predict ignition outcomes. Cases that are an exception from these bounds are investigated in detail to understand the cause of unique behavior. These cases lie within regions of the variable space that are susceptible to stochastic elements of the flow.Novelty and Significance Statement: This work investigates laser induced spark ignition of O2 and CH4 issuing from a shear-coaxial injector in a model rocket combustor. Probability of ignition as a function of deposited energy and spatial location is studied with high-speed schlieren imaging and pressure measurements. Two distinct ignition modes that have previously not been identified, direct and indirect, are dependent on the laser deposition location with respect to the reactant jet. For indirect cases, where laser deposition is at a distance from the reactants, the hydrodynamic jet ejection generated from the breakdown of the plasma plays a critical role on ignition probability. The behavior of the ejection jet is characterized to provide a phenomenological description of the ignition process. The results of this research show the important interaction between chemical, fluid dynamic, and thermodynamic processes and their impact on laser ignition.

EXPRESS: Admission Control in Multi-server Systems under Binary Reward Structure

We study a multi-server queueing system where a customer is satisfied (and generates a unit revenue) if their queueing time is at most a given constant. If the queueing time of the admitted customer exceeds this constant, the customer gets served, but is unsatisfied and generates no revenue. Such queueing systems arise in the context of modeling service systems where excessive delays are of concern. A key challenge is how to design an admission control policy to maximize the number of satisfied customers per unit time in the long run, assuming that we can observe the number of customers in the system at any time. We call this the binary reward structure system and show that a threshold-type admission policy is optimal. The optimal threshold policy has to be computed numerically. Hence we propose a square-root admission policy to approximate the optimal admission control policy, and compare the performance of these two policies. We derive an analytical upper bound on the performance of optimal admission control policy by deriving an optimal admission policy assuming we have full information over the queueing time of the admitted customers. This is equivalent to a queueing system where customers abandon the queue (i.e., leave without service) if their queueing time exceeds the given constant. We demonstrate that the optimal policy that includes customer abandonment, or alternatively, the optimal policy under full information, the optimal threshold policy, and the square-root admission policy, all exhibit identical performance in the asymptotic regions of the parameter space. Our numerical results indicate that the worst optimality gap of the square-root admission policy is within 3.9% of the optimal revenue, and implementing the square-root admission policy in the observable queueing system leads to a revenue loss that is at most 5.6% of the maximum possible revenue rate in the full information system. We also compare the binary reward structure with the more common linear reward structure where the system incurs holding cost per unit queueing time per customer. In addition, we also show that the analysis based on queueing time is applicable to the system time as well.

HDL and plaque regression in a multiphase model of early atherosclerosis

Atherosclerosis is a chronic disease of the arteries characterised by the accumulation of lipids and lipid-engorged cells in the artery wall. Early plaque growth is aggravated by the deposition of low density lipoproteins (LDL) in the wall and the subsequent immune response. High density lipoproteins (HDL) counterbalance the effects of LDL by accepting cholesterol from macrophages and removing it from the plaque. In this paper, we develop a free boundary multiphase model to investigate the effects of LDL and HDL on early plaque development. We examine how the rates of LDL and HDL deposition affect cholesterol accumulation in macrophages, and how this impacts cell death rates and emigration. We identify a region of LDL–HDL parameter space where plaque growth stabilises for low LDL and high HDL influxes, due to macrophage emigration and HDL clearance that counterbalances the influx of new cells and cholesterol. We explore how the efferocytic uptake of dead cells and the recruitment of new macrophages affect plaque development for a range of LDL and HDL influxes. Finally, we consider how changes in the LDL–HDL profile can change the course of plaque development. We show that changes towards lower LDL and higher HDL can slow plaque growth and even induce regression. We find that these changes have less effect on larger, more established plaques, and that temporary changes will only slow plaque growth in the short term.

Dissipative Dark Substructure: The Consequences of Atomic Dark Matter on Milky Way Analog Subhalos

Dissipative Dark Substructure: The Consequences of Atomic Dark Matter on Milky Way Analog Subhalos

Earth’s geomagnetic environment—progress and gaps in understanding, prediction, and impacts

Understanding of Earth’s geomagnetic environment is critical to mitigating the space weather impacts caused by disruptive geoelectric fields in power lines and other conductors on Earth’s surface. These impacts are the result of a chain of processes driven by the solar wind and linking Earth’s magnetosphere, ionosphere, thermosphere and Earth’s surface. Tremendous progress has been made over the last two decades in understanding the solar wind driving mechanisms, the coupling mechanisms connecting the magnetically controlled regions of near-Earth space, and the impacts of these collective processes on human technologies on Earth’s surface. Studies of solar wind drivers have been focused on understanding the responses of the geomagnetic environment to spatial and temporal variations in the solar wind associated with Coronal Mass Ejections, Corotating Interaction Regions, Interplanetary Shocks, High-Speed Streams, and other interplanetary magnetic field structures. Increasingly sophisticated numerical models are able to simulate the magnetospheric response to the solar wind forcing associated with these structures. Magnetosphere-ionosphere-thermosphere coupling remains a great challenge, although new observations and sophisticated models that can assimilate disparate data sets have improved the ability to specify the electrodynamic properties of the high latitude ionosphere. The temporal and spatial resolution needed to predict the electric fields, conductivities, and currents in the ionosphere is driving the need for further advances. These parameters are intricately tied to auroral phenomena—energy deposition due to Joule heating and precipitating particles, motions of the auroral boundary, and ion outflow. A new view of these auroral processes is emerging that focuses on small-scale structures in the magnetosphere and their ionospheric effects, which may include the rapid variations in current associated with geomagnetically induced currents and the resulting perturbations to geoelectric fields on Earth’s surface. Improvements in model development have paralleled the advancements in understanding, yielding coupled models that better replicate the spatial and temporal scales needed to simulate the interconnected domains. Many realizations of such multi-component systems are under development, each with its own limitations and advantages. Challenges remain in the ability of models to quantify uncertainties introduced by propagation of solar wind parameters, to account for numerical effects in model codes, and to handle the special conditions occurring during extreme events. The impacts to technical systems on the ground are highly sensitive to the local electric properties of Earth’s surface, as well as to the specific technology at risk. Current research is focused on understanding the characteristics of geomagnetic disturbances that are important for geomagnetically induced currents, the development of earth conductivity models, the calculation of geoelectric fields, and the modeling of induced currents in the different affected systems. Assessing and mitigating the risks to technical systems requires quantitative knowledge of the range of values to be expected under all possible geomagnetic and technical conditions. Considering the progress that has been made in studying the chain of events leading to hazardous geomagnetic disturbances, the path forward will require concerted efforts to reveal missing physics, improve modeling capabilities, and deploy new observational assets. New understanding should be targeted to accurately quantify solar wind driving, magnetosphere-ionosphere-thermosphere coupling, and the impacts on specific technologies. The research, modeling, and observations highlighted here provide a framework for constructing a plan by which the international science community can comprehensively address the growing threat to human technologies caused by geomagnetic disturbances.

Black Hole–Disk Interactions in Magnetically Arrested Active Galactic Nuclei: General Relativistic Magnetohydrodynamic Simulations Using a Time-dependent, Binary Metric

Perturber objects interacting with supermassive black hole accretion disks are often invoked to explain observed quasiperiodic behavior in active galactic nuclei (AGN). We present global, 3D general relativistic magnetohydrodynamic (GRMHD) simulations of black holes on inclined orbits colliding with magnetically arrested thick AGN disks using a binary black hole spacetime with mass ratio 0.1. We do this by implementing an approximate time-dependent binary black hole metric into the GRMHD Athena++ code. The secondary enhances the unbound mass outflow rate 2–4 times above that provided by the disk in quasiperiodic outbursts, eventually merging into a more continuous outflow at larger distances. We present a simple analytic model that qualitatively agrees well with this result and can be used to extrapolate to unexplored regions of parameter space. We show self-consistently for the first time that spin–orbit coupling between the primary black hole spin and the binary orbital angular momentum causes the accretion disk and jet directions to precess significantly (by 60°–80°) on long timescales (e.g., ∼20 times the binary orbital period). Because this effect may be the only way for thick AGN disks to consistently precess, it could provide strong evidence of a secondary black hole companion if observed in such a system. Besides this new phenomenology, the time-average properties of the disk and accretion rates onto the primary are only marginally altered by the presence of the secondary, consistent with our estimate for a perturbed thick disk. This situation might drastically change in cooled thin disks.

A mathematical model of the Bray–Liebhafsky reaction

It is known that the widely studied Bray–Liebhafsky reaction typically exhibits complex chemical behaviour. Numerous mathematical systems have been proposed to describe the iodine oscillations that occur during this process. Recently, a four-variable model of the Bray–Liebhafsky reaction has been proposed and analytical and numerical investigations suggested that chaotic solutions may exist. We revisit this four-variable model here and perform what appears to be the first detailed work on this system. We suggest that this model is perhaps not chaotic after all. Informed by these fresh insights, we propose a reduced two-variable model based upon the four-variable system. This model is created with the twin goals of enabling simpler mathematical analysis while retaining the underlying chemical mechanisms. We are able to demonstrate that our reduced problem performs very well when compared with the full model for realistic parameter values. In particular, key regions of parameter space are identified within which temporal oscillations can occur. Moreover, these persistent oscillations are consistent with the available qualitative experimental observations.

A wide-angle formulation of foreground filters for HI intensity mapping

Neutral hydrogen intensity mapping can in principle deliver rapid and large-volume cosmological surveys with exquisitely accurate redshifts that are determined directly from imaging. However, intensity maps suffer from very strong foreground contamination. Future surveys will require efficient data pipelines to remove the foregrounds and reveal the cosmological signal. It is expected that this cleaning will not remove the signal in substantial parts of the available Fourier space and that significant loss of signal due to imperfect cleaning will be confined to specific regions of Fourier space. This suggests a strategy which is useful for simplified estimates and rapid computations — i.e., to apply foreground filters that avoid the regions where loss of signal is significant. The standard Fourier-space power spectrum and foreground filters use a flat-sky approximation and thus exclude wide-angle correlations. We provide a new geometrical formulation of foreground filters in harmonic space, which naturally includes all wide-angle effects in the power spectrum. Foreground filtering leads to a loss of isotropy in Fourier space. In harmonic space this produces off-diagonal correlations. We derive analytical expressions for the generalised HI power spectrum and its cross-power with CMB lensing, for both single-dish and interferometer mode surveys. We show numerically that the off-diagonal contributions are negligible for the auto power. In the cross power, there is a non-negligible off-diagonal contribution, but only for a small interval of the largest available scales. For auto and cross power, the signal loss due to foreground avoidance decreases with increasing multipole (i.e. smaller scales), and the loss in interferometer mode is equal to, or slightly greater than, in single-dish mode. We find that the cross power in single-dish mode vanishes below a critical multipole, ℓ < ℓ 0. For an SKA-like survey, ℓ 0 ∼ 20 – 40 over redshifts z = 1 – 3. This feature is not seen in interferometer mode as the pertinent angular scales are larger than those allowed by the minimum baseline.

On the optimality of DIA-estimators: theory and applications

In this contribution, we introduce, in analogy to penalized ambiguity resolution, the concept of penalized misclosure space partitioning, with the goal of directing the performance of the DIA-estimator towards its application-dependent tolerable risk objectives. We assign penalty functions to each of the decision regions in misclosure space and use the distribution of the misclosure vector to determine the optimal partitioning by minimizing the mean penalty. As each minimum mean penalty partitioning depends on the given penalty functions, different choices can be made, in dependence of the application. For the DIA-estimator, we introduce a special set of penalty functions that penalize its unwanted outcomes. It is shown how this set allows one to construct the optimal DIA-estimator, being the estimator that within its class has the largest probability of lying inside a user specified tolerance region. Further elaboration shows how these penalty functions are driven by the influential biases of the different hypotheses and how they can be used operationally. Hereby the option is included of extending the misclosure partitioning with an additional undecided region to accommodate situations when it will be hard to discriminate between some of the hypotheses or when identification is unconvincing. By extending the analogy with integer ambiguity resolution to that of integer-equivariant ambiguity resolution, we also introduce the maximum probability estimator within the similar larger class.

The Sun as a target for axion dark matter detection

The exploration of the parameter space of axion and axion-like particle dark matter is a major aim of the future program of astroparticle physics investigations. In this context, we present a possible strategy that focuses on detecting radio emissions arising from the conversion of dark matter axions in the Sun's magnetic field, including conversion in sunspots. We demonstrate that near-future low-frequency radio telescopes, such as the SKA Low, may access regions of unexplored parameter space for masses ma≲10−6 eV.

DyMMM-LEAPS: An ML-based framework for modulating evenness and stability in synthetic microbial communities

There have been a growing number of computational strategies to aid in the design of synthetic microbial consortia. A framework to identify regions in parametric space to maximize two essential properties, evenness and stability, is critical. In this study, we introduce DyMMM-LEAPS (dynamic multispecies metabolic modeling—locating evenness and stability in large parametric space), an extension of the DyMMM framework. Our method explores the large parametric space of genetic circuits in synthetic microbial communities to identify regions of evenness and stability. Due to the high computational costs of exhaustive sampling, we utilize adaptive sampling and surrogate modeling to reduce the number of simulations required to map the vast space. Our framework predicts engineering targets and computes their operating ranges to maximize the probability of the engineered community to have high evenness and stability. We demonstrate our approach by simulating five cocultures and one three-strain culture with different social interactions (cooperation, competition, and predation) employing quorum-sensing-based genetic circuits. In addition to guiding circuit tuning, our pipeline gives an opportunity for a detailed analysis of pockets of evenness and stability for the circuit under investigation, which can further help dissect the relationship between the two properties. DyMMM-LEAPS is easily customizable and can be expanded to a larger community with more complex interactions.

Optimal scheduling of multi-regional energy system considering demand response union and shared energy storage

In the current context of the scarcity of fossil energy and the large-scale development and utilization of new energy sources, the power system is developing in the direction of multi-source synergy and interconnection. Demand response technology and energy storage technology have become important adjustment means of integrated energy system because of their efficient coordination ability and flexible adjustment ability. However, in the system where multiple energy systems operate cooperatively, there are still some limitations in time and capacity scales. Therefore, in order to enhance the demand-side response capability in multi-energy systems and give full play to the function of energy storage power stations, this paper proposes an optimal scheduling model for multi-area energy systems that considers joint demand response and shared energy storage. First, the system energy coupling matrix is constructed based on energy hubs, and the modeling framework for joint demand-side response and shared energy storage is established. Second, the multi-objective optimization problem is solved using NSGA-II algorithm solution with the optimization objectives of minimizing the total operating cost of the system and maximizing the net environmental impact. Finally, the simulation analysis is carried out. The simulation results show that the addition of joint demand response and shared energy storage can guide the scheduling optimization of multiple energy sources in each region in time and space, and realize the energy complementarity and mutual assistance of multi-regional energy systems. Compared with the traditional multi-regional energy system optimization scheduling scheme, the operation cost of the scheme is reduced by 4.2 %, and the environmental protection is improved by 41.9 %, which proves the feasibility of the optimal scheduling model proposed in this paper.

Impact of Ejecta Temperature and Mass on the Strength of Heavy Element Signatures in Kilonovae

A kilonova, the electromagnetic emission produced by compact binary mergers, is formed through a delicate interplay of physical processes, involving r-process nucleosynthesis and interactions between heavy elements and photons through radiative transfer. This complexity makes it difficult to achieve a comprehensive understanding of kilonova spectra. In this study, we aim to enhance our understanding and establish connections between physical parameters and observables through radiative-transfer simulations. Specifically, we investigate how ejecta temperature and element mass influence the resulting kilonova spectrum. For each species, the strength of its line features depends on these parameters, leading to the formation of a distinct region in the parameter space, dubbed the resonance island, where the line signature of that species is notably evident in the kilonova spectrum. We explore its origin and applications. Among explored r-process elements (31 ≤ Z ≤ 92), we find that four species—SrII, YII, BaII, and CeII—exhibit large and strong resonance islands, suggesting their significant contributions to kilonova spectra at specific wavelengths. In addition, we discuss potential challenges and future perspectives in observable heavy elements and their masses in the context of the resonance island.

Cosmological implications of gauged U(1) B-L on ΔN eff in the CMB and BBN

We calculate the effects of a light, very weakly-coupled boson X arising from a spontaneously broken U(1) B-L symmetry on ΔN eff as measured by the CMB and Yp from BBN. Our focus is the mass range 1 eV ≲ mX ≲ 100 MeV; masses lighter than about an eV have strong constraints from fifth-force law constraints, while masses heavier than about 100 MeV are constrained by other probes, including terrestrial experiments. We do not assume N eff began in thermal equilibrium with the SM; instead, we allow N eff to freeze-in from its very weak interactions with the SM. We find U(1) B-L is more strongly constrained by ΔN eff than previously considered. The bounds arise from the energy density in electrons and neutrinos slowly siphoned off into N eff bosons, which become nonrelativistic, redshift as matter, and then decay, dumping their slightly larger energy density back into the SM bath causing ΔN eff > 0. While some of the parameter space has complementary constraints from stellar cooling, supernova emission, and terrestrial experiments, we find future CMB observatories including Simons Observatory and CMB-S4 can access regions of mass and coupling space not probed by any other method. In gauging U(1) B-L , we assume the [U(1) B-L ]3 anomaly is canceled by right-handed neutrinos, and so our ΔN eff calculations have been carried out in two scenarios: neutrinos have Dirac masses, or, right-handed neutrinos acquire Majorana masses. In the latter scenario, we comment on the additional implications of thermalized right-handed neutrinos decaying during BBN. We also briefly consider the possibility that X decays into dark sector states. If these states behave as radiation, we find weaker constraints, whereas if they are massive, there are stronger constraints, though now from ΔN eff < 0.

Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres.

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.