Paradigm Of Chaos Research Articles

Thermodynamics of chaotic relaxation processes.

The established thermodynamic formalism of chaotic dynamics, valid at statistical equilibrium, is here generalized to systems out of equilibrium that have yet to relax to a steady state. A relation between information, escape rate, and the phase-space average of an integrated observable (e.g., Lyapunov exponent, diffusion coefficient) is obtained for finite time. Most notably, the thermodynamic treatment may predict the phase-space profile of any integrated observable for finite time, from the leading and subleading eigenfunctions of the Perron-Frobenius or Koopman transfer operator. Examples of that equivalence are shown, and the theory is tested analytically on the Bernoulli map while numerically on the perturbed cat map, the Hénon map, and the Ikeda map, all paradigms of chaos.

Chaotic Traveling Wave Solutions in Coupled Chua’s Circuits

Coupled arrays of Chua’s circuits have been studied for many years. The existence of traveling wave solutions for such system was shown numerically in Perez-Munuzuri et al. (Traveling wave front and its failure in a one-dimensional array of Chua’s circuit. Chua’s circuit : a paradigm for Chaos. World Scientific, Singapore, pp 336–350, 1993). The existence of periodic traveling wave solutions has been proved recently (Chow et al. in J Appl Anal Comput 3:213–237, 2013). The purpose of this paper is to prove the existence of chaotic traveling wave solutions for such system. Using the method of singular perturbations, we show that the ODE system for the traveling waves can have a heteroclinic loop consisting of two traveling waves moving at the same speed. Moreover, at the equilibrium points of the heteroclinic loop, the dominant eigenvalues of the system are a pair of complex numbers with negative real parts. By a generalization of Shilnikov’s theorem of symbolic dynamics, the system can have chaotic behavior near the traveling heteroclinic orbits.

Rayleigh–Bènard convection in the generalized Oberbeck–Boussinesq system

The original Oberbeck–Boussinesq (OB) equations which are the coupled two dimensional Navier–Stokes(NS) and heat conduction equations have been investigated by E.N. Lorenz half a century ago with Fourier series and opened the way to the paradigm of chaos. In our former study—Chaos, Solitons and Fractals 78, 249 (2015)—we presented fully analytic solutions for the velocity, pressure and temperature fields with the aim of the self-similar Ansatz and gave a possible explanation of the Rayleigh–Bènard convection cells. Now we generalize the Oberbeck–Boussinesq hydrodynamical system, going beyond the first order Boussinesq approximation and consider a non-linear temperature coupling. We investigate more general, power law dependent fluid viscosity or heat conduction material equations as well. Our analytic results obtained via the self-similar Ansatz may attract the interest of various fields like meteorology, oceanography or climate studies.

From Cosmos to Chaos: Philosophical Aspects of Paradigm Shift

The article aims to highlight that the conceptual apparatus of the contemporary science (not only the problem-solving techniques but also the type of the raised issues) testify the shift of the paradigm, namely, the passage from the Cosmos paradigm to the Chaos paradigm. The significance of the old paradigm, especially considering its astonishing longevity, is related not only with its explanatory potential but also with the innate strife of humans to dwell in the aesthetically and ethically attractive and unconditionally intelligible world. Reductionist methodology became established in the framework of the mechanistic worldview. In this case, simplicity is seen as the main measure of intelligibility; however, the important detail of the elasticity (or, in other words, vagueness) of the simplicity concept is insufficiently considered. The shift towards the new paradigm of chaos is presented as a well-grounded effort of liberation from the universe of anthropocentrism and apriority.

AN EFFECTIVE HYBRID SUPPORT VECTOR REGRESSION WITH CHAOS-EMBEDDED BIOGEOGRAPHY-BASED OPTIMIZATION STRATEGY FOR PREDICTION OF EARTHQUAKE-TRIGGERED SLOPE DEFORMATIONS

Abstract. Earthquake can pose earth-shattering health hazards to the natural slops and land infrastructures. One of the chief consequences of the earthquakes can be land sliding, which is instigated by durable shaking. In this research, an efficient procedure is proposed to assist the prediction of earthquake-originated slope displacements (EIDS). New hybrid SVM-CBBO strategy is implemented to predict the EIDS. For this purpose, first, chaos paradigm is combined with initialization of BBO to enhance the diversification and intensification capacity of the conventional BBO optimizer. Then, chaotic BBO is developed as the searching scheme to investigate the best values of SVR parameters. In this paper, it will be confirmed that how the new computing approach is effective in prediction of EIDS. The outcomes affirm that the SVR-BBO strategy with chaos can be employed effectively as a predicting tool for evaluating the EIDS.

Analytic self-similar solutions of the Oberbeck–Boussinesq equations

In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtonian–Navier–Stokes — with Boussinesq approximation — and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh–Bénard convection cells. Finally, it is discussed how our result may be related to nonlinear or chaotic dynamical regimes.

Nowy paradygmat chaosu w życiu i twórczości Andrzeja Urbanowicza

Andrzej Urbanowicz (28.04.1938, Wilno (Vilnius) – 22.08.2011, Szklarska Poręba) was an artist associated, among others, with works referring to magic, alchemy and philosophy of the East, containing manifold symbols. The motif of chaos plays a significant role in them.The paper attempts to present this theme basing on pictures from different periods of the painter’s career and demonstrating various techniques, as it was present in the oeuvre of Urbanowicz during all his creative years. He referred to it not only in titles and subjects, but also by using specific methods of artistic activity, i.e. automatic writing, accidental effects and conjoining numerous heterogeneous pictorial elements.Espousing of the new chaos paradigm is particularly characteristic for Urbanowicz. Associated usually with disorder and confusion, chaos in his works is an expression of creative, feminine energy. He perceives it in theological sense, frequently defining it with symbolism pertaining to the beginning of the world and primeval matter.Additional consideration is given to quotations from the artist’s statements about chaos, which disclose a personality not only engrossed by a fascinating issue, but also one for whom it became a way of life and activity.Andrzej Urbanowicz (28.04.1938, Wilno (Vilnius) – 22.08.2011, Szklarska Poręba) was an artist associated, among others, with works referring to magic, alchemy and philosophy of the East, containing manifold symbols. The motif of chaos plays a significant role in them.The paper attempts to present this theme basing on pictures from different periods of the painter’s career and demonstrating various techniques, as it was present in the oeuvre of Urbanowicz during all his creative years. He referred to it not only in titles and subjects, but also by using specific methods of artistic activity, i.e. automatic writing, accidental effects and conjoining numerous heterogeneous pictorial elements.Espousing of the new chaos paradigm is particularly characteristic for Urbanowicz. Associated usually with disorder and confusion, chaos in his works is an expression of creative, feminine energy. He perceives it in theological sense, frequently defining it with symbolism pertaining to the beginning of the world and primeval matter.Additional consideration is given to quotations from the artist’s statements about chaos, which disclose a personality not only engrossed by a fascinating issue, but also one for whom it became a way of life and activity.

사회과학연구 패러다임 변화대안으로서의 카오스 이론

Social phenomena require the fundamentals of novel scientific epistemology and methodology due to the continual change by complex variables. A systematic understanding of paradigm shifts is also necessary for analyzing the problems currently faced in society. This study presents chaos theory, which was developed from the natural sciences, to discuss the abnormality of modern society that underscores the importance of the chaos paradigm. Chaos theory is applicable to diverse social phenomena that stretches across politics, economics, and sociology. If social science research can appropriately and effectively develop and apply chaos theory, the theory can be utilized as a useful paradigm to reinforce the fundamentals of epistemology and methodology in exploring social phenomena.

A Mimetic Psyche

Apollo come dance with me examined by my interpretation and art practice. the expression of chaos and order as a creative paradigm in the (tragic) theater of ancient Greece, the (black-figure) Ancient Greek vase painting and in my painting The project includes three series of paintings as well as an interpretation that examines the mimetic, and metamorphic metaphorical expression that adheres to these three art forms. These designs include detailed methods, methodology, philosophy and psychology of visual expression of this art form. Dealing with the creative paradigm of chaos and order in the tragic drama arose from the use of portraits of me and my family to characters of the Theban plays of Sophocles portray. This illustrates my contemporary art, the individual and society with regard to the investigation of human experience in the tragic theater: the social, psychological and cognitive relationships between the characters with each other and with the external world.

Dynamical instability in kicked Bose-Einstein condensates

Bose-Einstein condensates subject to short pulses (``kicks'') from standing waves of light represent a nonlinear analog of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (i.e., exponential proliferation of noncondensate particles) suggested that the transition to instability might be associated with a transition to chaos. Here we conclude instead that instability is due to resonant driving of Bogoliubov modes. We investigate the Bogoliubov spectrum for both the quantum kicked rotor (QKR) and a variant, the double kicked rotor (QKR-2). We present an analytical model, valid in the limit of weak impulses which correctly gives the scaling properties of the resonances and yields good agreement with mean-field numerics.

Bright Matter-Wave Soliton Collisions in a Harmonic Trap: Regular and Chaotic Dynamics

Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated and shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, presenting a paradigm for chaos in wave mechanics.

Considerations on the application of the chaos paradigm to describe the postural sway

Time-series of statokinesigram (SKG) of healthy subjects and parkinsonians are investigated and compared. This is done by employing the chaos paradigm in order to obtain the main characteristics of the SKG. The interpretation of our findings is twofold: when a proper Theiler window is not used we find a virtual invariance of the chaos parameters when healthy subjects and parkinsonians are compared but a discrepancy of our values (correlation dimension equals to 1.4) with those found in previous works; when a proper Theiler window is used (more) appropriately, the SKGs do not show a convergence of the fractal dimension estimates; therefore nothing can be said in terms of chaoticity of system.

Chaotic quantum ratchets and filters with cold atoms in optical lattices: Analysis using Floquet states

Recently, cesium atoms in optical lattices subjected to cycles of unequally spaced pulses have been found to show interesting behavior: they represent an experimental demonstration of a Hamiltonian ratchet mechanism, and they show strong variability of the dynamical localization lengths as a function of initial momentum. The behavior differs qualitatively from corresponding atomic systems pulsed with equal periods, which are a textbook implementation of a well-studied quantum chaos paradigm, the quantum $\ensuremath{\delta}$-kicked rotor ($\ensuremath{\delta}$-QKR). We investigate here the properties of the corresponding eigenstates (Floquet states) in the parameter regime of the recent experiments and compare them with those of the eigenstates of the $\ensuremath{\delta}$-QKR at similar kicking strengths. We show that by studying the properties of the Floquet states we can shed light on the form of the observed ratchet current, as well as variations in the dynamical localization length.

Posture as a chaotic system and an application to the Parkinson’s disease

In this work we investigate and compare a number of time series of stabilograms of healthy subjects and Parkinsonians. This is carried out by means of the chaos paradigm through the preliminary computation of the first minimum of the mutual information function and the embedding dimension (using false nearest neighbours) in order to obtain the correlation dimension as well as the largest Lyapunov exponent . We show that the postural act is indeed chaotic and especially that the latter two parameters do not allow to discriminate healthy subjects from parkinsonians. Moreover we report a discrepancy of our values with those found in previous works.

Chaos and non-linear dynamics

In the last 20 years, chaos and non-linear dynamics have become new scientific disciplines in the macrosystems theory. They have provided new theoretical, conceptual and methodological tools that allow us to capture, understand and link together the surprisingly complex behaviors of simple, dissipative systems, the type of behavior called chaos, in every field of contemporary science. This article offers the most recent thought in applying the chaos paradigm to the social sciences. The author explores different methodological approaches and techniques that can be used to diagnose and describe all chaotic processes and identify the strengths and limitations of a variety of approaches.

Exploring phase space localization of chaotic eigenstates via parametric variation

In a previous paper [Phys. Rev. Lett. 77, 4158 (1996)], a new correlation measure was introduced that sensitively probes phase space localization properties of eigenstates. It is based on a system's response to varying an external parameter. The measure correlates level velocities with overlap intensities between the eigenstates and some localized state of interest. Random matrix theory predicts the absence of such correlations in chaotic systems whereas in the stadium billiard, a paradigm of chaos, strong correlations were observed. Here, we develop further the theoretical basis of that work, extend the stadium results to the full phase space, study the Planck's over 2pi dependence, and demonstrate the agreement between this measure and a semiclassical theory based on homoclinic orbits.

PIECEWISE-LINEAR CHAOTIC SYSTEMS WITH A SINGLE EQUILIBRIUM POINT

As a unique paradigm for chaos, the various versions of Chua's circuits and equations consists of a three-dimensional autonomous system with a three-segment piecewise-linear function which gives rise to three equilibrium points. This paper considers the possibility of simplifying the system configurations of piecewise-linear chaotic systems based on the structures of Chua's systems. We study a new class of piecewise-linear three-dimensional autonomous system with a three-segment piecewise-linear function. However, unlike Chua's systems, the systems we study in this paper have only single equilibrium points. To find chaotic attractors from this class of systems, we use a systematic random-search process to search the parameter space. The searching process consists of three stages. For the first stage, we simply count the number of points on a Poincaré section and find candidates for chaotic attractors. At the second stage, Lyapunov exponents are calculated for selecting chaotic attractors from the candidates. Finally, bifurcation diagrams constructed around the located chaotic attractors are used to find different types of chaotic attractors. Many qualitatively different chaotic attractors of this class of systems had been found and presented in this paper. Another method to simplify the configurations of a piecewise-linear chaotic system is to reduce the number of segments of the piecewise-linear function. We have developed some chaotic systems with a two-segment piecewise-linear function and which gives rise to two equilibrium points. Many color illustrations of chaotic attractors and bifurcation diagrams are presented.

Neural network chaos analysis

The analysis of a chaos theory, in general, and a neural network chaos paradigm with a concrete interpretation by a simple neural network, in particular, allowed us to set up a new aspect in an artificial neuronal approach of methodology: an analogy between natural chaos experimentally observed in neural systems of the brain and artificial neural network chaos phenomena has been considered. The significance of asymmetry and nonlinearity, which were increased on introducing a restricted N-shaped synaptical relation in a two-mod dynamic model, is emphasized. There are illustrated the diferent computational examples of the neural network properties which are expressed by the equillibrium point, stable cycle or chaotic behaviour in strong nonlinear neural systems.

Chaos: Generating Complexity from Simplicity

The most commonly used mapping to illustrate the phenomenon of chaos is the map x → 2x mod (1). This map is known as the 'unilateral shift' because, in the binary number system this map shifts all digits to the left by one decimal place, and truncates the integer. The second most commonly used paradigm of chaos is the Smale horseshoe whose complexity is essentially the bilateral shift obtained when we simply shift without truncation in some symbol system. Neither of these paradigms fully explains chaos since shifts cannot generate complex orbits from simple (rational) initial conditions. How chaos generates complexity from simplicity is an essential part that needs explanation. Providing this explanation is the objective of this paper.

The nature of Nature: conflict and consensus in fisheries management

Using the example of the New England Regional Fishenes Managcmcnt Council and its attempts tu manage the fisheries in its area, this paper argues that disferences in the way Nature operates constitutes a major source of divisivcncss between, on the one hand, those members (and other managers in, say, government) who use linear modeling systems and, on the other hand, the majority of those in the commercial fishing sector who, intuitively, havc cast nature in those non-linear tcrms currently being explored in the Chaos paradigm. Since thesc two different models have not bcen explicitly addressed, there is a history of members on each side accusing those on the othcr of bargaining in poor faith and lacking an adequate understanding of the resource. The thesis of the paper is that making explicit these underlying cognitive modes would provide more common ground for addressing management problems in such co-management arenas as the fishery management councils and elsewhere.