Origin Of Chaos Research Articles

Ziyaüddin Sardar’ın Postnormal Zamanlar Teorisi Çerçevesinde Toplumsal Değişim Sürecinde Din Eğitimi Üzerine Bir Değerlendirme

Toplumsal değişim, hızı, etkisi ve kapsadığı alan değişmekle birlikte her dönemde gerçekleşmiş bir olgudur. Toplumsal değişim sürecinden etkilenen pek çok faktör arasında din ve eğitim de vardır. Dolayısıyla din eğitiminde de toplumsal değişimle birlikte farklılaşmalar görülmektedir. Postnormal zamanlar; belirsizlik, hızlı değişim, gücün yeniden düzenlenmesi, karışık ve kaotik davranışlarla karakterize edilir. Ziyaüddin Sardar, postnormal zamanların özelliklerini üç ‘K’ ile açıklamaktadır. Bunlar, karmaşa, kaos ve karşıtlıktır. Sardar’a göre ilişkide bulunduğumuz her şey karmaşık/komplikedir. Postnormal zamanların ikinci özelliği kaostur. Kaosun asıl sebebi ise iletişim ağının doğasının, kapsamının ve işlevinin değişmesidir. Sardar, karşıtlığı hızlı değişim ve bilgiye ulaşma, bilgiyi kullanma ve yorumlamada meydana gelen değişimle ilişkilendirerek bu üç husus çerçevesinde yaşadığımız çağın özelliklerinden bahsetmektedir. Bu makalede Ziyaüddin Sardar’ın toplumsal değişimin geldiği noktayı betimlerken kullandığı postnormal zamanlar teorisi çerçevesinde Türkiye’de örgün ve yaygın din eğitimi incelenerek bu teorinin ortaya koyduğu dönemsel özelliklere göre yorumlamak amaçlanmıştır. Teorik olan bu çalışmanın, toplumsal değişim sürecinde ortaya çıkan ihtiyaçlar karşısında, örgün ve yaygın din eğitiminin mahiyetini değerlendirmesi bakımından alana katkı sağlayacağı düşünülmektedir.

Origin of chaos near three-dimensional quantum vortices: A general Bohmian theory.

We provide a general theory for the structure of the quantum flow near three-dimensional (3D) nodal lines, i.e., one-dimensional loci where the 3D wave function becomes equal to zero. In suitably defined coordinates (comoving with the nodal line) the generic structure of the flow implies the formation of 3D quantum vortices. We show that such vortices are accompanied by nearby invariant lines of the comoving quantum flow, called X lines, which are normally hyperbolic. Furthermore, the stable and unstable manifolds of the X lines produce chaotic scatterings of nearby quantum (Bohmian) trajectories, thus inducing an intricate form of the quantum current in the neighborhood of each 3D quantum vortex. Generic formulas describing the structure around 3D quantum vortices are provided, applicable to an arbitrary choice of 3D wave function. We also give specific numerical examples as well as a discussion of the physical consequences of chaos near 3D quantum vortices.

Rulers, businessmen, suspicious people, institutions and corruption: the origin of chaos (Notas de Ensino)

DOI: 10.12660/gvcasosv5n2n10http://dx.doi.org/10.12660/gvcasosv5n2n10

Rulers, businessmen, suspicious people, institutions and corruption: the origin of chaos

Through the comparison between the actions and accomplishments of the Baron of Mauá at the time of the Brazilian Empire and some current events, like the links among suspected citizens, politicians and businessmen, this teaching case aims to highlight that institutions, whether formal or informal, have been playing a significant role in society and have, over time, been influenced by several events and people. The case serves as a pedagogical instrument to show, in practice, the applicability of theoretical concepts in disciplines related to Theory of Chaos, entry into new markets, Strategy (Resource-Based View, Industry-Based View, Institutions–Based View) and Governance.DOI: 10.12660/gvcasosv5n2c10http://dx.doi.org/10.12660/gvcasosv5n2c10

Chaotic dynamics of stellar spin driven by planets undergoing Lidov-Kozai oscillations: resonances and origin of chaos

Many exoplanetary systems containing hot Jupiters are found to possess significant misalignment between the spin axis of the host star and the planet's orbital angular momentum axis. A possible channel for producing such misaligned hot Jupiters involves Lidov-Kozai oscillations of the planet's orbital eccentricity and inclination driven by a distant binary companion. In a recent work (Storch, Anderson & Lai 2014), we have shown that a proto-hot Jupiter undergoing Lidov-Kozai oscillations can induce complex, and often chaotic, evolution of the spin axis of its host star. Here we explore the origin of the chaotic spin behavior and its various features in an idealized non-dissipative system where the secular oscillations of the planet's orbit are strictly periodic. Using Hamiltonian perturbation theory, we identify a set of secular spin-orbit resonances in the system, and show that overlaps of these resonances are responsible for the onset of wide-spread chaos in the evolution of stellar spin axis. The degree of chaos in the system depends on the adiabaticity parameter $\epsilon$, proportional to the ratio of the Lidov-Kozai nodal precession rate and the stellar spin precession rate, and thus depends on the planet mass, semi-major axis and the stellar rotation rate. For systems with zero initial spin-orbit misalignment, our theory explains the occurrence (as a function of $\epsilon$) of "periodic islands" in the middle of a "chaotic ocean" of spin evolution, and the occurrence of restricted chaos in middle of regular/periodic spin evolution. Finally, we discuss a novel "adiabatic resonance advection" phenomenon, in which the spin-orbit misalignment, trapped in a resonance, gradually evolves as the adiabaticity parameter slowly changes. This phenomenon can occur for certain parameter regimes when tidal decay of the planetary orbit is included.

Magnetic field induced dynamical chaos

In this article, we have studied the dynamics of a particle having charge in the presence of a magnetic field. The motion of the particle is confined in the x-y plane under a two dimensional nonlinear potential. We have shown that constant magnetic field induced dynamical chaos is possible even for a force which is derived from a simple potential. For a given strength of the magnetic field, initial position, and velocity of the particle, the dynamics may be regular, but it may become chaotic when the field is time dependent. Chaotic dynamics is very often if the field is time dependent. Origin of chaos has been explored using the Hamiltonian function of the dynamics in terms of action and angle variables. Applicability of the present study has been discussed with a few examples.

Four dimensional chaos and intermittency in a mesoscopic model of the electroencephalogram

The occurrence of so-called four dimensional chaos in dynamical systems represented by coupled, nonlinear, ordinary differential equations is rarely reported in the literature. In this paper, we present evidence that Liley's mesoscopic theory of the electroencephalogram (EEG), which has been used to describe brain activity in a variety of clinically relevant contexts, possesses a chaotic attractor with a Kaplan-Yorke dimension significantly larger than three. This accounts for simple, high order chaos for a physiologically admissible parameter set. Whilst the Lyapunov spectrum of the attractor has only one positive exponent, the contracting dimensions are such that the integer part of the Kaplan-Yorke dimension is three, thus giving rise to four dimensional chaos. A one-parameter bifurcation analysis with respect to the parameter corresponding to extracortical input is conducted, with results indicating that the origin of chaos is due to an inverse period doubling cascade. Hence, in the vicinity of the high order, strange attractor, the model is shown to display intermittent behavior, with random alternations between oscillatory and chaotic regimes. This phenomenon represents a possible dynamical justification of some of the typical features of clinically established EEG traces, which can arise in the case of burst suppression in anesthesia and epileptic encephalopathies in early infancy.

Chaos, Artificial Life and Risk Mathematics

The current work addresses the importance of chaos theory and artificial life as scientific bases for the development of a mathematical research of risk in the systems, within the framework of risk science and risk mathematics. It is argued, in particular, that a combination of chaos theory and artificial life may provide for a theoretical basis for the understanding of systemic situations of risk. It is presented, within the context of category computation theory, a notion of morphic incompressibility and it is addressed how statistical laws emerge in chaotic dynamics, which becomes morphically compressible at the statistical level, while being morphically incompressible at the individual orbits' level. The issue of the systemic origins of chaos is addressed in connection with this dual (orbit) incompressibility/(statistical) compressibility. Through the introduction of a one-dimensional game of life, that combines a coupled map lattice and a cellular automaton, in a model of an artificial ecosystem, it is, then, shown that artificial life can be effectively combined with chaos theory, in order to support the development of a mathematical theory of risk.

On the Origin of Chaos in the Belousov-Zhabotinsky Reaction in Closed and Unstirred Reactors

We investigate the origin of deterministic chaos in the Belousov–Zhabotinsky (BZ) reaction carried out in closed and unstirred reactors (CURs). In detail, we develop a model on the idea that hydrodynamic instabilities play a driving role in the transition to chaotic dynamics. A set of partial differential equations were derived by coupling the two variable Oregonator–diffusion system to the Navier–Stokes equations. This approach allows us to shed light on the correlation between chemical oscillations and spatial–temporal dynamics. In particular, numerical solutions to the corresponding reaction-diffusion-convection (RDC) problem show that natural convection can change the evolution of the concentration distribution as well as oscillation patterns. The results suggest a new way of perceiving the BZ reaction when it is conducted in CURs. In conflict with the common experience, chemical oscillations are no longer a mere chemical process. Within this framework the evolution of all dynamical observables are demonstrated to converge to the regime imposed by the RDC coupling: chemical and spatial–temporal chaos are genuine manifestations of the same phenomenon.

Origin of chaos near critical points of quantum flow

The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie-Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function psi there is an unstable point (called the X point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the "nodal-point- X -point complex." We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point- X -point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point- X -point complex. It is also an inverse power of the distance of a trajectory from the X point's stable manifold far from the complex. This distance plays the role of an effective "impact parameter." The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point- X -point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.

Using Mechanistic Models to Understand Synchrony in Forest Insect Populations: The North American Gypsy Moth as a Case Study

In many forest insects, subpopulations fluctuate concurrently across large geographical areas, a phenomenon known as population synchrony. Because of the large spatial scales involved, empirical tests to identify the causes of synchrony are often impractical. Simple models are, therefore, a useful aid to understanding, but data often seem to contradict model predictions. For instance, chaotic population dynamics and limited dispersal are not uncommon among synchronous forest defoliators, yet both make it difficult to achieve synchrony in simple models. To test whether this discrepancy can be explained by more realistic models, we introduced dispersal and spatially correlated stochasticity into a mechanistic population model for the North American gypsy moth Lymantria dispar. The resulting model shows both chaotic dynamics and spatial synchrony, suggesting that chaos and synchrony can be reconciled by the incorporation of realistic dynamics and spatial structure. By relating alterations in model structure to changes in synchrony levels, we show that the synchrony is due to a combination of spatial covariance in environmental stochasticity and the origins of chaos in our multispecies model.

Origin of chaos in soft interactions and signatures of nonergodicity

The emergence of chaotic motion is discussed for hard-point like and soft collisions between two particles in a one-dimensional box. It is known that ergodicity may be obtained in hard-point like collisions for specific mass ratios gamma=m(2)/m(1) of the two particles and that Lyapunov exponents are zero. However, if a Yukawa interaction between the particles is introduced, we show analytically that positive Lyapunov exponents are generated due to double collisions close to the walls. While the largest finite-time Lyapunov exponent changes smoothly with gamma , the number of occurrences of the most probable one, extracted from the distribution of finite-time Lyapunov exponents over initial conditions, reveals details about the phase-space dynamics. In particular, the influence of the integrable and pseudointegrable dynamics without Yukawa interaction for specific mass ratios can be clearly identified and demonstrates the sensitivity of the finite-time Lyapunov exponents as a phase-space probe. Being not restricted to two-dimensional problems such as Poincaré sections, the number of occurrences of the most probable Lyapunov exponents suggests itself as a suitable tool to characterize phase-space dynamics in higher dimensions. This is shown for the problem of two interacting particles in a circular billiard.

Origin of chaos in the spherical nuclear shell model: Role of symmetries

To elucidate the mechanism by which chaos is generated in the shell model, we compare three random-matrix ensembles: the Gaussian orthogonal ensemble, French's two-body embedded ensemble, and the two-body random ensemble (TBRE) of the shell model. Of these, the last two take account of the two-body nature of the residual interaction, and only the last, of the existence of conserved quantum numbers like spin, isospin, and parity. While the number of independent random variables decreases drastically as we follow this sequence, the complexity of the (fixed) matrices which support the random variables, increases even more. In that sense we can say that in the TBRE, chaos is largely due to the existence of (an incomplete set of) symmetries.

ORIGIN OF CHAOS IN A TWO-DIMENSIONAL MAP MODELING SPIKING-BURSTING NEURAL ACTIVITY

Origin of chaos in a simple two-dimensional map model replicating the spiking and spiking-bursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: stable fixed point (replicating silence) and superstable limit cycle (replicating spikes). Coupling this subsystem with the slow subsystem leads to the generation of periodic or chaotic spiking-bursting behavior. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases.

Origin of chaos in the Prometheus–Pandora system

We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting, resulting in resonance overlap which causes the chaos. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 and 3.1 deg yr −1, respectively. A model with 1.5 degrees of freedom captures the essential features of the chaotic dynamics. We use it to show that the Lyapunov exponent of 0.3 yr −1 arises because the critical argument of the dominant member of the resonant quartet makes approximately two separatrix crossings every 6.2 year precessional cycle.

Lorenz-like systems and classical dynamical equations with memory forcing: An alternate point of view for singling out the origin of chaos

An alternate view for the emergence of chaos in Lorenz-like systems is presented in this paper. For such purpose, the Lorenz problem is reformulated in a classical mechanical form and it turns out to be equivalent to the problem of a damped and forced one-dimensional motion of a particle in a two-well potential, with a forcing term depending on the "memory" of the particle past motion. The dynamics of the original Lorenz system in the proposed particle phase space can then be rewritten in terms of a one-dimensional first-exit-time problem. The emergence of chaos turns out to be due to the discontinuous solutions of the transcendental equation ruling the time for the particle to cross the intermediate potential wall. The whole problem is tackled analytically deriving a piecewise linearized Lorenz-like system that preserves all the essential properties of the original model.

The origin of chaos in the outer solar system

Classical analytic theories of the solar system indicate that it is stable, but numerical integrations suggest that it is chaotic. This disagreement is resolved by a new analytic theory. The theory shows that the chaos among the jovian planets results from the overlap of the components of a mean motion resonance among Jupiter, Saturn, and Uranus, and provides rough estimates of the Lyapunov time (10(7) years) and the dynamical lifetime of Uranus (10(18) years). The jovian planets must have entered the resonance after all the gas and most of the planetesimals in the protoplanetary disk were removed.

On the Origin of Chaos in the Asteroid Belt

We consider the effect of gravitational perturbations from Jupiter on the dynamics of asteroids, when Jupiter is itself perturbed by Saturn. The presence of Saturn introduces a number of additional frequencies into Jupiter's orbit. These frequencies in turn produce chaos in narrow regions on either side of the chaotic zones associated with the mean motion resonances between the asteroids and Jupiter. The resonant arguments of these three-body resonances contain the longitudes of Jupiter and the asteroid together with either the secular frequency g6, or the longitude of Saturn. Resonances involving the longitude of Saturn are analogs of the Laplace resonance in the Jovian satellite system. We show that many three-body resonances involving the longitude of Saturn are chaotic. We give simple expressions for the width of the chaotic region and the associated Lyapunov time. In some cases the chaos can produce a diffusive growth in the eccentricity of the asteroid that leads to ejection of the asteroid on times shorter than the age of the solar system. We give simple estimates for the diffusion time. Finally, we present the results of numerical integrations testing the theory.

Celestial Encounters: The Origins of Chaos and Stability. Florin Diacu , Philip Holmes

<i>Celestial Encounters: The Origins of Chaos and Stability</i>. Florin Diacu , Philip Holmes

The Impact of chaos on science and society

Chaotic dynamics chaos and politics - applications of nonlinear dynamics to socio-political issues is the EEG a strange attractor? the impact of chaos on theory chaos and physics irreversability and quantum chaos impact of high-dimensional chaos - a further step towards a dynamical complexity the impact of chaos on biology - promising directions for research dynamical disease - the impact of nonlinear dynamics and chaos on cardiology and medicine the impact of chaos on meterology the concept of chaos in the problem of earthquake prediction the impact of chaos on engineering the impact of chaos on economic theory chaos in society - reflections on the impact of chaos on sociology strange attractors and the origin of chaos.