Higher-order Cycles Research Articles

Using graph theory to describe and model chromosome aberrations.

A comprehensive description of chromosome aberrations is introduced that is suitable for all cytogenetic protocols (e.g. solid staining, banding, FISH, mFISH, SKY, bar coding) and for mathematical analyses. "Aberration multigraphs" systematically characterize and interrelate three basic aberration elements: (1) the initial configuration of chromosome breaks; (2) the exchange process, whose cycle structure helps to describe aberration complexity; and (3) the final configuration of rearranged chromosomes, which determines the observed pattern but may contain cryptic misrejoinings in addition. New aberration classification methods and a far-reaching generalization of mPAINT descriptors, applicable to any protocol, emerge. The difficult problem of trying to infer actual exchange processes from cytogenetically observed final patterns is analyzed using computer algorithms, adaptations of known theorems on cubic graphs, and some new graph-theoretical constructs. Results include the following: (1) For a painting protocol, unambiguously inferring the occurrence of a high-order cycle requires a corresponding number of different colors; (2) cycle structure can be computed by a simple trick directly from mPAINT descriptors if the initial configuration has no more than one break per homologue pair; and (3) higher-order cycles are more frequent than the obligate cycle structure specifies. Aberration multigraphs are a powerful new way to describe, classify and quantitatively analyze radiation-induced chromosome aberrations. They pinpoint (but do not eliminate) the problem that, with present cytogenetic techniques, one observed pattern corresponds to many possible initial configurations and exchange processes.

Lyapunov, Floquet, and singular vectors for baroclinic waves

Abstract. The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector f1 of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent l ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to f1 are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to f1. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wave-dynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors.

High-resolution strontium isotope stratigraphy across the Cambrian-Ordovician transition

We have analyzed 214 fossil apatite samples from nine stratigraphic sections worldwide that cover the lower Cambrian to lower Ordovician for their strontium isotope ratios. Of these samples, 180 from six sections that cover the Cambrian-Ordovician transition (<5 Ma) reveal how the extent of isotopic alteration can differ greatly according to lithology and sample type. From three limestone sections, 126 euconodont samples yield the most consistent 87Sr/86Sr ratios and are used in this article to constrain seawater 87Sr/86Sr across the Cambrian-Ordovician boundary. By contrast, protoconodonts, paraconodonts, and inarticulate brachiopods are far less likely to retain a primary 87Sr/86Sr signature and appear to be of only limited value for chemostratigraphy. Seawater 87Sr/86Sr fell from maximum values for the Phanerozoic of more than 0.7092 during the early Late Cambrian to 0.7090 by the earliest Ordovician. High-resolution sampling has permitted the recognition of sinusoidal variations with a wavelength on the order of <1 Ma that are superimposed on the overall fall in 87Sr/86Sr. These variations may be caused by high-order cycles in seawater 87Sr/86Sr or diagenetic alteration.

A mathematical model for the mammalian melatonin rhythm

Abstract We propose and analyse a model for the mammalian melatonin rhythm consisting of first order equations describing the underlying biochemical reactions and pathways. The model includes activation through the secondary messenger system and end product inhibition of activator molecules through cooperative binding of melatonin to the light sensitive suprachiasmatic nucleus (SCN). The model has robust free-running limit-cycle solutions for a range of parameter values. Assuming light sensitivity of parameters describing the binding of melatonin to the SCN we show that the limit-cycle solutions become entrained to the imposed day/night cycle when the period of the cycle is not too far from the free-running period. Higher order cycles and “irregular” behaviour results when the period of the day/night cycle is above or below certain thresholds relative to the free-running period.

The auriferous Late Archaean Central Rand Group of South Africa: sea-level control of sedimentation?

Abstract Most earth scientists agree that either, or both, eustacy and tectonism play an important role in controlling sedimentation. In the case of the auriferous late Archaean Central Rand Group of South Africa, most early workers favoured a marine depositional environment for the gold placers. Transgressions and regressions, used to explain vertical alternations of coarse- and fine-grained deposits, were attributed solely to vertical movements of the basin floor, with no reference to eustatic variations. Tidal indicators are regarded as unequivocal evidence for a basin having been connected to the ocean, and their recognition is considered a preliminary requirement for investigation of sea-level fluctuations. Unambiguous recognition of sea-level fluctuations in the rock record is problematic in some cases, especially in non-fossiliferous deposits. To facilitate recognition in such deposits, the following criteria are proposed: (a) presence of thick, laterally continuous mudstones; (b) alternating coastal/marine and continental facies in the vertical profile; (c) onshore–offshore oscillating baylines; (d) marine facies unconformably overlain by fluvial facies; (e) turbidites, subaqueous debris flows and submarine canyons; (f) incised valleys. The Turffontein Subgroup of the Vredefort Structure, situated roughly at the centre of the preserved Witwatersrand Basin, is subdivided into five formations, Rt1 through Rt5. Tidal bundles in Rt3, some with mud-draped foresets, provide evidence for connection of the Witwatersrand Basin to the ocean during Central Rand times. Formation Rt4 is interpreted as being shallow marine in origin, and is unconformably overlain by a sandy fluvial succession, Rt5, thereby suggesting a regression and sea-level fall. In the Heidelberg area, in the northeastern part of the Witwatersrand Basin, the lower part of the Turffontein Subgroup consists predominantly of sandstone, with subordinate conglomerate, mudstone and diamictite. This succession also exhibits most of the characteristics, a to e, given above. Two deep scours that eliminate the rich Carbon Leader occur in parts of the Carletonville goldfield. The eastern scour has a fill comprising predominantly mudstone, and displays several additional characteristics of incised valley systems, such as a downstream increase in depth, and typical lateral and vertical facies arrangements. Texturally mature fluvial units occurring at different palaeotopographic levels indicate possible multiple episodes of incision and fill, suggesting lower-order sea-level fluctuations, superimposed upon a higher-order cycle. The results of the case studies presented provide clear evidence for sea-level control of deposition in the distal parts of the Witwatersrand Basin during Central Rand times. However, in proximal regions, sedimentation was probably governed by source area tectonism.

Climatically driven sedimentary cycles in the Late Miocene sediments of the Pannonian Basin, Hungary

Abstract By detailed cycle analysis of marginal, mostly delta plain facies successions of Pannonian sediments the frequency of water-level changes of the Late Miocene Pannonian Lake, Hungary, can be identified. Cycle analysis is presented with the object of defining the orbital record preserved in the strata. We use the bed thickness as input data for our cycle analysis. Statistical analysis revealed a slow, upward increase of both the frequency and thickness of sand beds in the boreholes resulting from infilling of the lake by a basinward migration of the marginal facies. Cycles of different order were detected in the sediments. The higher-order cycles are very close to or are equivalent with the Milankovitch climatic cycles: ∼19 ka caused by precession, ∼50 ka caused by obliquity, and ∼400 ka is the longer period of eccentricity. The longer, >1 m.y., cycle is very poorly developed here. It is probably the analogue of the 3rd-order, global climatic cycle. Cycle analysis suggests, that one of the main controlling factors of the water-level changes of the Pannonian Lake was high-frequency climatic oscillations. The long-term climatic changes only had a faint impact on the sediments. The record of the 3rd-order water-level changes is very subtle. Our results show that the water-level changes followed the rhythm of that of the eustatic changes, but being asynchronous to them, i.e., at high sea-level stage the Pannonian lake-level was low, and vice versa.

Anatomy of Extremely Thin Marine Sequences Landward of a Passive-Margin Hinge Zone: Neogene Calvert Cliffs Succession, Maryland, U.S.A.

ABSTRACT Detailed examination of Neogene strata in cliffs 25-35 m high along the western shore of Chesapeake Bay, Maryland, reveals the complexity of the surviving record of siliciclastic sequences 150 km inland of the structural hinge zone of the Atlantic passive margin. Previous study of the lower to middle Miocene Calvert (Plum Point Member) and Choptank Formations documented a series of third-order sequences 7-10 m thick in which lowstand deposits are entirely lacking, transgressive tracts comprise a mosaic of condensed bioclastic facies, and regressive (highstand) tracts are present but partially truncated by the next sequence boundary; smaller-scale (fourth-order) cyclic units could not be resolved. Together, these sequences constitute t e transgressive and early highstand tracts of a larger (second-order Miocene) composite sequence. The present paper documents stratigraphic relations higher in the Calvert Cliffs succession, including the upper Miocene St. Marys Formation, which represents late highstand marine deposits of the Miocene second-order sequence, and younger Neogene fluvial and tidal-inlet deposits representing incised-valley deposits of the succeeding second-order cycle. The St. Marys Formation consists of a series of tabular units 2-5 m thick, each with an exclusively transgressive array of facies and bounded by stranding surfaces of abrupt shallowing. These units, which are opposite to the flooding-surface-bounded regressive facies arrays of model parasequences, are best characterized as shaved sequences in which only the transgressive tract survives, and are stacked into larger transgressive, highstand, and forced-regression sets. Biostratigraphic analyses by others indicate that this onshore record contains the same number of third-order ( 1 my duration) units as present offshore, and so thinning landward of the hinge zone was accomplished not by omission or erosion of entire cycles of deposition, but instead by omission of some subsidiary elements (e.g., lowstand tracts), by erosional shaving of sequence tops (removing the entire regressive tract in some sequences), by a reduced number of component high-order cycles surviving per larger set, and by qualitative changes in the anatomy or composition of elements (e.g., condensed transgressive tracts; shaved sequences rather than parasequences). All of these differences can be attributed to limited accommodation, ut preservation of an onshore record of each baselevel cycle was probably also favored by the large amplitude and rapidity of eustatic fluctuations during the Miocene.

Sequence stratigraphy and stable isotopes ( δ13C, δ180) of the Late Cretaceous carbonate ramp of the western margin of the Iberian Chain (Soria, Spain)

Within the Upper Cretaceous sediments of the western margin of the northern Iberian Chain, several formations of Turonian to Campanian age form a carbonate platform succession (165–270 m thick) developed during a second-order cycle of relative sea-level rise and fall. This succession consists of two depositional sequences, deposited during third-order cycles, formed by sediments deposited in a broad, epeiric, ramp-type platform open to the north. The lower depositional sequence (Moral de Homuez Formation) began during the Cenomanian-Turonian transgression and represents the transgressive systems tract and highstand systems tract; it was terminated by a late Turonian regression resulting in a type 2 sequence boundary. Two types of parasequences (deepening-upward and shallowing-upward, respectively), 1 to 3 m thick, deposited within fourth- and higher-order cycles, have been identified, with features corresponding to lagoonal deposits. The upper depositional sequence (comprising the Hortezuelos, Caballar, Hontoria del Pinar, Burgo de Osma and Santo Domingo de Silos Formations) began during the early Coniacian transgression and was terminated by a Campanian regression. This upper depositional sequence also represents the transgressive systems tract and highstand systems tract. Eight types of shallowing-upward parasequence, 1 to 4 m thick (locally up to 10 m), deposited in subtidal high-energy (shoals and channels) and low-energy settings (lagoon, locally with rudist patch-reefs) and in tidal-flat environments have been identified. Carbon-isotope data, showing sharp and correlatable positive anomalies in different sections, strongly suggest that Oceanic Anoxic Events are recorded in these shallow platform deposits. The Cenomanian-Turonian Oceanic Anoxic Event results in a positive δ 13 C excursion of up to 4‰ that coincides with the presence of small-size, low-diversity benthic foraminifers, agglutinated and planktonic foraminifers and calcispheres, despite the paucity or complete absence of black shales. The Santonian-Campanian Oceanic Anoxic Event is recorded but is not so evident. The relative δ 13 C minimum between these two maxima apparently coincides with the Turonian-Coniacian boundary. It is concluded that short-term variations in δ 13 C can be useful for correlation in shallow platform areas, where the fossil record has little biostratigraphical precision. Accordingly, the ages previously assigned to the sediments studied have been partially modified. The oxygen-isotope results are more negative than expected for carbonates precipitated in equilibrium with normal marine waters, thus indicating diagenetic alteration by meteoric waters.

An experimental study of the partial transformation cycling of shape-memory alloys

The martensitic phase transition is responsible for many of the unusual thermomechanical properties exhibited by shape-memory alloys. The partial martensitic transformation of Cu-Zn-Al single-crystal samples, induced by a temperature change and recorded by means of the electrical resistance, is studied in this work. In this work, partial transformation cycling measurements are introduced which will be used to discuss the validity of some assumptions of a set of phenomenological models. More specifically, and as a main goal, the authors try to fit the measurements using the model for partial cycling described in references which contains as limit cases several models described in the literature. The results show a very good agreement between model and experiment for first-order partial cycles and a failure of one of the main assumptions of the models which affects higher-order cycles.

Factors controlling the sedimentary evolution of the kimmeridgian ramp in the north Iberian Basin (NE Spain)

The aim of this paper is to summarize the present knowledge reached by the authors on the carbonate ramp which developed in the iberian basin during Kimmeridgian times. Our results were obtained from a combined field analysis and computer modelling carried out in the north Iberian Chain (NE Spain). Extensive field analysis in the Ricia area (Zaragoza, NE Spain), resulted in a detailed mapping of the transition from inner to outerramp facies on this carbonate rampo Three facies belts may be distinguished in this rampo The outer ramp facies consists of marls and mudstones rhythmic facies. The inner ramp facies, located aboye fair-weather wave base, are dominated by coral patch reef growing. The middle ramp facies are represented by marls and micrites bearing skeletal and oolitic tempestite levels which sharply grade into high-amplitude o'olitic sandwave. Factors such as resedimentation by storms, carbonate production and relative variation of sea level acting in the Kimmeridgian ramp are also quantiphied and discussed. Most of the mud accumulated in outer-ramp areas was produced in the coral «carbonate factory» located in inner areas. Off-shore resedimentation by storm was the main agent of basinward transport of this mudo The deduced accommodation curve consists of three elements: a linear rise which satisfactorily matches the normal subsidence figures observed in intracratonic basins; a third-order cycle, that may have a regional cause and higher order cycles in the Milanckovich band, that may be eustatic in origin.

High-Frequent Cycles and Sequence Stratigraphy: New Aspects for Hydrocarbon Field Development and Re-Evaluation

A new development in the application of sequence stratigraphic concepts in subsurface hydrocarbon exploration, development and production is the recognition of high-frequent cyclic patterns in rock successions. Numerous studies of subsurface rock successions from different geological periods show an intimate relationship between the high-frequent cycles and sequence organization. Comparisons between a simulation modeling and the Exxon's third order curve show that a major part of the curve is a composite of these higher-order cycles. Some of the results from our studies can be summarized as follows: An intimate relationship exists between climate changes and sea level fluctuations. Changes in climate within the time dimensions of the third-order Exxon chart are for a large part responsible for the sequence stratigraphic organization. Fourth and higher-order cycles (mainly Milankovith cycles) are the major controlling parameters for changes in sedimentary facies and/or subfacies. For continental basins, a climate-forcing model is used in sequence analysis, while in transitional and marine basins an integrated climate-forcing and sea level model is used. This interrelationship allows an accurate quantitative evaluation of each sequence and/or systems tract. The InterGeos' cyclicity analysis tool (MILABAR) was designed for well analysis using wireline logs. A zonation into sequences where the sequence boundaries more » can be calibrated as well as the net duration and net accumulation rate of each measured sequence can be estimated. The results are summarized in a Well Composite Chart, which include e.g. Interval Time Estimates (ITE), Net Accumulation Rate (NAR), etc. In correlations an accuracy of 500-100 Ka can be obtained. Vertical as well as lateral continuity of lithofacies (e.g. reservoirs, permeability barriers, etc.) can be computed and predicted. « less

High-Resolution Stratigraphic Forward Modeling: A Case Study of the Lower-Middle San Andres Formation, Permian Basin

This study has attempted to calibrate Shell's two-dimensional (2-D) basin modeling program as an exploration tool by simulating the stratigraphy of a mixed carbonate/clastic third-order depositional sequence. The lower-middle San Andres Formation was selected because available log, core, and outcrop data from the Northwest Shelf area, Permian basin, provided an excellent calibration set. A regional stratigraphic cross section from the Cato-Chaveroo to the Wasson fields was constructed delineating lithology and porosity distribution. Approximately 10 shoaling-upward depositional cycles were interpreted. A higher frequency, five-in-one cyclicity was also interpreted based on core and outcrop data. The observed stratigraphy was simulated using a composite eustasy consisting of third-order (2,000,000 yr), fourth-order (100,000 yr), and fifth-order (20,000 yr) sinusoids each at five-meter amplitudes. Subsidence input was constrained by back-stripped tectonic subsidence curves calculated from well data. Sedimentation parameters were interactively derived. New empirically based algorithms were used to model Dunham lithofacies, environmental facies, and sabkha anhydrite distribution. Synthetic log and 2-D synthetic seismic profiles were constructed from the simulation output. The simulation results suggest that (1) relative sea level is the dominant control on the observed depositional cyclicity, (2) the distribution of regional seal facies (anhydrite) reflects falling sea level and prolonged exposure, (3)more » limestone-dolomite trends on the shelf are grossly related to environment and (4) the distribution of grainstones and packstones (potential reservoirs) occurs as fourth- and fifth-order offlapping and aggradational pods. The synthetic log signatures compared to [open quotes]real[close quotes] logs substantiate the interpreted depositional cyclicity, but also point out the difficulty in interpreting high-order cycles based on log data alone.« less

Forced regressions in a large wave- and storm-dominated anoxic lake, Rhaetian-Sinemurian Kap Stewart Formation, East Greenland

During Rhaetian-Sinemurian time a large wave- and storm-dominated lake was situated in the Jameson Land basin, East Greenland. Lake deposits consist of alternating black unfossiliferous mudstones and sheet sandstones. Anoxic conditions dominated at the lake bottom during deposition of the muds, and the water column was probably stratified. The sandstones were deposited by progradation of wave- and storm-dominated deltas in a water depth of less than 15 m. Sequence-stratigraphic interpretation suggests that the mudstones were deposited in periods of rising and very high stands of lake level, whereas progradation of the deltaic sheet sandstones took place during forced regressions caused by significant falls. The lake thus underwent a large number of fairly high amplitude changes in level, probably caused by climatic fluctuations. The high-order cycles can be grouped into several long-period cycles that show the same number of major fluctuations as published eustatic sea-level curves. This similarity suggests a causal link between eustasy and long-period variations in the lake. The Kap Stewart Formation represents one of the few ancient examples of a large wave- and storm-dominated lake, and it is probably the first documented case of abundant well-developed lacustrine forced regressions.

Trajectory scaling functions for the circle map and the quasiperiodic route to chaos.

There is a certain amount of arbitrariness in defining a trajectory scaling function for the circle map for the various quasiperiodic routes to chaos. We show how one may exploit this freedom to select trajectory scaling functions to suit a particular purpose. For comparison with experiment, one might seek a trajectory scaling function that is relatively flat, is smooth, clearly distinguishes between the subcritical and critical regimes of the quasiperiodic route to chaos, and is quite resilient to noise. To allow for experimental difficulties in obtaining (consecutive) exact cycles, we have also developed a trajectory scaling function that is constructed from subcycles within some higher-order cycle. We consider the quasiperiodic routes to chaos with golden-mean and silver-mean (\ensuremath{\surd}2 -1) rotation winding numbers.

High-order cycles in the logistic map or centers of cardioids in the mandelbrot set

Asymptotic expressions for the positions of the centres (along the real axis) of high-order cycle cardioids closest to the limit point −2 in the Mandelbrot set are obtained by a combination of numerical and analytical methods.

Lacustrine sedimentary sequences in the polish miocene lignite-bearing basins — Facies distribution and sedimentary development

Abstract Lacustrine sediments are typically associated with the widespread Miocene lignite formations of north and west Poland, where packets of these sediments alternate with fluvial and paludal deposits. They are developed most completely in structural depressions of tectonic, or, more rarely, karstic origin, both on the Palaeozoic platform and in intramontane depressions. The lacustrine deposits can be divided into characteristic lithofacies related to open-lake environments, nearshore peat-bog vegetation, river and fan deltas, and, subordinately, areas of fluvial sedimentation within the peat-bogs. Microfacies examination of the lignites allows the recognition within lignite seams of members corresponding to various peat-bog vegetation sub-zones (e.g. peat-bog “taluss”, reed sub-zone, swamp-forest sub-zone). In sequences within tectonic depressions the lacustrine sediments display a marked cyclicity. First-order cyclicity is mainly connected with allocyclic (gravitational tectonic movements) and hypautocyclic mechanisms (compaction of underlying beds). Higher-order cyclicity is primarily related to autocyclic mechanisms (migration of river channels), although compactional processes may also generate some higher-order cycles. A model cyclic sequence following tectonic (or compactional) lowering of the basin floor shows the sedimentary evolution of the basin from river-delta to open-lake environment, with swamp-related phytogenic deposition completing the sequence.

Higher Order Cycles, Inferred Chronostratigraphy, and Impact of Multiple Lowstand Systems Tracts on Hydrocarbon Exploration, Pletmos Basin, Southern Offshore, South Africa: ABSTRACT

Higher Order Cycles, Inferred Chronostratigraphy, and Impact of Multiple Lowstand Systems Tracts on Hydrocarbon Exploration, Pletmos Basin, Southern Offshore, South Africa: ABSTRACT

Unscrambling the Concept of Chaos through Thick and Thin: Reply

Melese and Transue rightly distinguish two types of chaos: that in which the scrambled set has positive measure (observable chaos); and that when it has zero measure (unobservable chaos). In a series of studies that began with the reference they cite, Wayne Shafer and I have explored the existence of observable chaos for a Keynesian business cycle model [Day and Shafer, 1985a,b; Day, 1985]. On the basis of this work it is clear that observable chaos can exist and be robust for a continuum of parameter values for a large class of difference equations that arise quite naturally in economics. The functional forms for this class involve piece-wise smooth functions-in effect, switching regimes. Readers concerned with these issues may also be interested to learn of extensive computational studies, carried out by Pianigiani, of the classic quadratic difference equation that I used as a special case of a more general example and to which Melese and Transue confine their attention. These simulations, which were shown to me in Siena and which augment similar studies by James Yorke (both of these scholars are among the pioneers in the field), show that throughout the range of values 3.57 < m < 4.00, behavior switches from stable, relatively loworder cycles (with thin chaos) to very high-order cycles or thick chaos. The high-order cycles appear to be nonperiodic for finite series shorter than their periodicity. The latter could be in the hundreds, thousands, millions, or higher! One should also be reminded that Lorenz [1963], in one of his seminal papers, provided evidence that the asymptotic behavior of solutions changed erratically within what is now known to be the chaos zone. Putting all this together, we see that (1) thick chaos can be robust in important classes of models; (2) thin chaos may frequently be associated with stable, but high-order cycles that are practically indistinguishable from chaos; and (3) the long-run behavior of nonlinear difference equations typically switches frequently and sometimes erratically with parameter shifts. In a nutshell, complex, essentially unpredictable behavior is a structurally stable feature of nonlinear difference equations through

Trajectory scaling function for bifurcations in area-preserving maps on the plane

The trajectory scaling function for area-preserving maps on the plane is found using a calculation of the unstable manifold for the renormalization group operator R · T = Λ · T 2 · Λ -1 with Λ= α 0 0 β . Internal self-similarities of high order cycles and of power spectra are deduced.

A Comparison of Refrigerants When Used in Vapor Compression Cycles Over an Extended Temperature Range

Abstract An important current engineering problem involves providing apparatus to meet the cooling requirements of supersonic aircraft. Because the time is already passing when simple air cycles can render adequate performance due to their inability to extend over a sufficient temperature range, vapor compression cycles are being considered. However, the maximum temperature range over which a single refrigerant can operate successfully in a vapor cycle is also relatively limited. Since the total required temperature range increases rapidly with the aircraft flight Mach number, series combination or “stacking” of vapor cycles leading to binary and higher order cycles may become necessary. This paper gives methods for selecting the refrigerant, or series combination of refrigerants, which exhibits optimum thermodynamic performance when employed in vapor cycles operating between arbitrarily selected temperatures.