Chaotic Cycle Research Articles

Heteroclinic cycles and periodic orbits for the O(2)-equivariant 0:1:2 mode interaction

We study the quadratic normal form describing the generic interaction of Fourier modes of wavenumbers 0, 1 and 2 under the symmetry group O(2) of rotations and reflections, in the case that the homogeneous quadratic terms preserve ‘energy’: the sum of the squares of absolute values of the (complex) variables. This system is a generalization of the 1:2 mode interaction studied by Dangelmayr [G. Dangelmayr, Steady-state mode interactions in the presence of O(2)-symmetry, Dyn. Stab. Syst. 1 (2) (1986) 159–185], Armbruster et al. [D. Armbruster, J. Guckenheimer, P. Holmes, Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica D 29 (1988) 257–282] and others, and its restriction to the 1:2 subspace is a degenerate case of that system. It displays all the classes of fixed points, periodic orbits (standing and travelling waves), invariant tori (modulated travelling waves) and heteroclinic cycles found in the 1:2 interaction, as well as new heteroclinic cycles connecting pure and mixed modes, chaotic cycles, and ‘strange’ periodic orbits. We describe the key dynamical features, show that the degenerate 1:2 case possesses a second organizing center at which bifurcation curves coalesce, provide representative bifurcation sets and diagrams for the 1:2 and 0:1:2 systems, and use a conservative limit to understand the periodic orbits in the latter system.

Viral infection model with periodic lytic immune response

Abstract Dynamical behavior and bifurcation structure of a viral infection model are studied under the assumption that the lytic immune response is periodic in time. The infection-free equilibrium is globally asymptotically stable when the basic reproductive ratio of virus is less than or equal to one. There is a non-constant periodic solution if the basic reproductive ratio of the virus is greater than one. It is found that period doubling bifurcations occur as the amplitude of lytic component is increased. For intermediate birth rates, the period triplication occurs and then period doubling cascades proceed gradually toward chaotic cycles. For large birth rate, the period doubling cascade proceeds gradually toward chaotic cycles without the period triplication, and the inverse period doubling can be observed. These results can be used to explain the oscillation behaviors of virus population, which was observed in chronic HBV or HCV carriers.

Attractor merging crisis in chaotic business cycles

A numerical study is performed on a forced-oscillator model of nonlinear business cycles. An attractor merging crisis due to a global bifurcation is analyzed using the unstable periodic orbits and their associated stable and unstable manifolds. Characterization of crisis can improve our ability to forecast sudden major changes in economic systems.

Chaotic solar cycles modulate the incidence and severity of mental illness

This paper hypothesizes that the intensity of ultraviolet radiation (UVR) from the Sun predisposes humans to polygenic mutation fostering major mental illness (MMI) and other disorders of neurodevelopment. In addition, the variation in the intensity of this radiation acts to stress immune systems, possibly mediated by cytokines, resulting in variable clinical expressions of mental illness and autoimmune disorders. Organisms can adapt to chronic high-intensity UVR by producing melanin and by retaining various pigments. We found that 28% of 11-year solar cycles produce particularly severe solar flares during which UVR is 300% more intense and hence more damaging than normal. Out of a total of six severe cycles in the past 250 years, four have occurred in the past 55 years, possibly explaining the apparent increase in the incidence of MMI in recent decades. UVR is 10 times more mutagenic than ionizing radiation to nuclear DNA, and especially damaging to mitochondrial DNA. However, variable light as manifested by seasons stresses adaptability to UVR, possibly through an immune mechanism. We show that the region of the Earth having the most UVR, relative to the most variation in that light, is at 54±∼10° (N or S) latitude. Therefore, the most potential damage from sunlight occurs between the Equator and the Poles, not at the Equator itself. The human brain, our most important organ of adaptability, must be able to survive environmental variation, with successful matching to the environment resulting in adaptation. Unsuccessful adaptation to UVR (and possibly other types of radiation) results in mutation, which can produce neuro-chemical abnormalities manifested by MMI. We postulate that the combination of intensity and variation in UVR serves as a global modulator of MMI.

The Sun determines human longevity: teratogenic effects of chaotic solar radiation

An association between fertility and longevity has been known for many years, and considerable research has been focused on the mechanisms of ageing that ultimately determine longevity, which has remained essentially unchanged despite a near doubling of human life expectancy in the past 200 years. In this paper, the authors present evidence that the Sun determines the limits of longevity for the longest-living complex organisms. The Sun is a dynamical system and although solar cycles occur every 8–14 years (averaging ∼11.1 years), the authors show that 28% of these cycles exhibit chaotic features and irregularly release up to 300% more ultraviolet radiation than usual. These chaotic solar cycles create an environment mutagenic to DNA that must be largely avoided in order to pass uncorrupted genes to the next generation. This requirement determines the limits of fertility, e.g., menarche and menopause in humans, and sets longevity to ≈100 years.

Nutritional interventions for individuals with bulimia nervosa.

Many physical and psychological effects of bulimia nervosa are caused by the patient's partial starvation and chaotic nutritional cycle. Attention should thus be initially directed to correcting nutritional deficiencies and abnormal eating patterns, and providing dietary counselling. Nevertheless, very little has been written about the nutritional management of this eating disorder. Nutritional counselling for bulimia patients is reviewed in this paper. Current knowledge about nutritional therapy and its efficacy, goals and objectives is presented, along with recommendations used in treatment programmes. Lastly, the key steps of nutritional management are summarised.

The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model

We study dynamical behavior and bifurcation structure of a multi-strain SIR epidemiological model with seasonal forcing in transmission rate. The conventional single strain SIR dynamics with seasonal forcing is known to show a cascade of bifurcations as the strength of seasonality increases. In this study with an extension to multiple strains of pathogens, we first investigate the bifurcation patterns of the two strain SIR model. In the two strain SIR model, a new parameter called cross-immunity between strains plays a key role in the dynamical behavior. As analogous to the single strain model, we found that the period doubling bifurcation occurs as the strength of seasonal forcing is increased. However, bifurcation patterns differ greatly both in their subharmonic periods and the relative phase of cycles between strains, depending mostly on the degree of the cross-immunity between strains. With strong and weak cross-immunities, the period doubling cascade proceeds gradually toward chaotic cycles. On the other hand, with intermediate cross-immunity, the loss of stability of annual cycle attractor is immediately followed by chaotic cycles. Asynchronous cycles of two strains are robust outcome when annual cycles lose stability, but the population can converge to perfectly synchronized biennial cycles if two strains are antigenically distant from each other. Second, we found that there are simultaneously stable multiple attractors in two strain SIR model. Biennial and chaotic attractors, e.g., occasionally coexist at the same degree of seasonal forcing, and the trajectories converge to one of them depending on the initial conditions. Such multiple attractors are widely seen between biennial and annual cycles, or among the different types of biennial cycles, at the same strength of the seasonality. Finally, we found that the population may switch from one attractor to another by introducing a small random noise in seasonally varying transmission rate.

Dynamic neural activity as chaotic itinerancy or heteroclinic cycles?

I question whether chaotic itinerancy is anything new or different to existing research on heteroclinic cycles (cycling-chaos), and blow-out bifurcations (attractor-bubbling) that provide more detailed and better definition for nonlinear phenomena occurring in neural systems. I give a brief description of this research for comparison and expansion, and see it as an important component in dynamical models of neural activity.

From Arrow to cycles, instability, and chaos by untying alternatives

From remarkably general assumptions, Arrow's Theorem concludes that a social intransitivity must afflict some profile of transitive individual preferences. It need not be a cycle, but all others have ties. If we add a modest tie-limit, we get a chaotic cycle, one comprising all alternatives, and a tight one to boot: a short path connects any two alternatives. For this we need naught but (1) linear preference orderings devoid of infinite ascent, (2) profiles that unanimously order a set of all but two alternatives, and with a slightly fortified tie-limit, (3) profiles that deviate ever so little from singlepeakedness. With a weaker tie-limit but not (2) or (3), we still get a chaotic cycle, not necessarily tight. With an even weaker one, we still get a dominant cycle, not necessarily chaotic (every member beats every outside alternative), and with it global instability (every alternative beaten). That tie-limit is necessary for a cycle of any sort, and for global instability too (which does not require a cycle unless alternatives are finite in number). Earlier Arrovian cycle theorems are quite limited by comparison with these.

Direct Machine Translation Systems as Dynamical Systems: The Iterative Semantic Processing (ISP) Paradigm

Direct machine translation systems are usually implemented as uni-directional mappings between a source and target sentence, where the accuracy of the translation is evaluated with respect to the consistency of meaning of the target with respect to the source. However, this approach does not allow us to understand how comparable the semantic representations in target and source really are, since both are described by words in different (and possibly semantically incompatible) languages. In this paper, we generalise the common method of inverting translations to generate a mapping between target and source, by proposing a new, iterative translation methodology which is based on dynamical systems theory, the Iterative Semantic Processing (ISP) Paradigm. The basic premise of this paradigm is that the preservation (or loss) of semantic representations during successive iterative translations between source and target, target and source, source and target, and so on, can be best described as one of three possible dynamical system states: point attractors (i.e., invariant semantic representations), limit cycles (i.e., steady-state variant but predictable cross-language mappings) and chaotic cycles (i.e., variant mappings with rapid short-term information loss). The mathematical properties of each of these system states as a measure of semantic stability are described in this study, and quantitative measures of semantic information loss are derived, and applied to semantic-mis-translations from a freely-available direct translation system.

The 1: [formula omitted] Hopf/steady-state mode interaction in three-dimensional magnetoconvection

The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151–176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D 4⋉T 2 -equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435–1455). We also find new phenomena in the normal form for a Hopf bifurcation with D 4⋉T 2 symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063–1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.

Dividing the Yarmouk's waters: Jordan's treaties with Syria and Israel

Seismic cycles emerge in a broad range of rupture styles, from slow-slip events to pulse-like earthquake sequences. Meanwhile, large earthquakes in paleoseismic and instrumental catalogues exhibit various recurrence patterns going from periodic to chaotic cycles with characteristic or dissimilar ruptures. The potential connection between these observations is still poorly known. Here, we investigate the link between rupture styles and recurrence patterns in quasi-dynamic models of seismicity in two-dimensional faults embedded in a compliant zone, exploring a wide range of frictional and fault zone properties. The recurrence patterns evolve from purely periodic to multiple-periodic time- and slip-predictable cycles, to chaotic sequences of super-cycles with full and partial ruptures with an increasing number of aftershocks. This transition is accompanied by changes of rupture styles from crack-like to pulse-like ruptures. These behaviors can be obtained either by a more compliant fault zone or a reduced characteristic weakening distance of friction. The effects of the compliant zone and other physical characteristics can be conflated into a single non-dimensional number, such that seismic cycles with similar behaviors can be obtained with or without a compliant fault zone in quasi-dynamic simulations. The connection between recurrence patterns and rupture styles implies that the paleoseismic record can bring useful constraints to rupture characteristics and fault zone properties.

The Age of Confusion: Why So Many Teens are Getting Pregnant, Turning to Welfare and Ending Up Homeless

Abstract The ongoing debate over the flaws in the nation's welfare system centers primarily on two highly charged social ills: teen-age pregnancy and long-term dependence on public assistance. For the poorest and fastest growing segment of the welfare population—homeless mothers—these problems are both severe and inextricably linked. At an alarmingly young age, these women are becoming trapped in a chaotic cycle that offers little structure and few alternatives.

A Random Walk or Color Chaos on the Stock Market? Time-Frequency Analysis of S&P Indexes

The random-walk (white-noise) model and the harmonic model are two polar models in linear systems. A model in between is color chaos, which generates irregular oscillations with a narrow frequency (color) band. Time-frequency analysis is introduced for evolutionary time-series analysis. The deterministic component from noisy data can be recovered by a time-variant filter in Gabor space. The characteristic frequency is calculated from the Wigner decomposed distribution series. It is found that about 70 percent of fluctuations in Standard & Poor stock price indexes, such as the FSPCOM and FSDXP monthly series, detrended by the Hodrick-Prescott (HP) filter, can be explained by deterministic color chaos. The characteristic period of persistent cycles is around three to four years. Their correlation dimension is about 2.5. The existence of persistent chaotic cycles reveals a new perspective of market resilience and new sources of economic uncertainties. The nonlinear pattern in the stock market may not be wiped out by market competition under nonequilibrium situations with trend evolution and frequency shifts. The color-chaos model of stock-market movements may establish a potential link between business-cycle theory and asset-pricing theory.

Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology.

Recently, biological preparations which are thought to be chaotic have been controlled using algorithms based on the detection and manipulation of periodic unstable points. The dynamics of these systems are, however, contaminated with noise; thus detection becomes a statistical process. Here we show that low dimensional chaos can be reliably detected with large noise contamination and distinguished from noisy limit cycles. We also examine a purely chaotic high dimensional system.

Investigation of a Chaotic Mechanism for Cycle-to-cycle Variations

Abstract A simple thermodynamic model of combustion in Otto cycle engines has been published and shown to exhibit chaotic cycle to cycle variations in certain regimes. However, these chaotic regimes are far from any physical limit. When the model is modified to predict observables within their physical ranges, the chaotic behaviour disappears. It is not possible from these results to claim that the fuel thermo-chemistry alone can induce chaotic behaviour. By elimination, this would support the view that cycle-to-cycle variations are caused by the effects of turbulence on fluid flow and combustion in engines, and not by chaotic phenomena arising from the non-linearity of fuel thermo-chemistry.

Phenomenological approach to construction of attractors

Simple phenomenological principles permitting the construction of systems of differential equations that lead to chaotic attractors and complicated cycles are proposed. The general principles are illustrated by specific examples.

CHAOS THEORY AND THE NEW KEYNESIAN ECONOMICS*

The possibility of endogenous chaotic cycles in rational expectations models undermines New Classical arguments that macroeconomic stabilization policies will be either ineffective or inefficient. However, the possibility of chaos also weakens the ability of New Keynesian stabilization policies because of the associated forecasting difficulties. Although numerous kinds of models can generate chaotic dynamics, empirical evidence of the existence of such dynamics in economics is ambiguous. Copyright 1990 by Blackwell Publishers Ltd and The Victoria University of Manchester

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.

Regular and chaotic cycling in models of ecological genetics

A model of density-dependent selection is investigated for the cyclical behavior associated with the analogous nonlinear models of population growth. If the population size regulating mechanism reacts too sharply to perturbations in population size, regular and chaotic limit cycles may result. It is established analytically that the population may converge to fixation or an invariant polymorphic gene frequency while the population size undergoes regular or chaotic oscillations. The possibility of joint limit cycles in both the population size and gene frequency is demonstrated and investigated numerically. Such cycles may occur even though one or both fixation equilibria are locally stable. In the context of equilibrium cycles it is found that overdominance in carrying capacity is not necessary for the maintenance of genetic variation in the population. Furthermore, the genetic system appears able to exert a stabilizing influence on the overall system.