Journal articles: 'Chaos Theory (Mathematics)'

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Blackmore, Denis. "The mathematical theory of chaos." Computers & Mathematics with Applications 12, no. 3-4 (May 1986): 1039–45. http://dx.doi.org/10.1016/0898-1221(86)90439-6.

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Marion, Russ. "Chaos, Topology, and Social Organization." Journal of School Leadership 2, no. 2 (March 1992): 144–77. http://dx.doi.org/10.1177/105268469200200202.

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This paper attempts to apply chaos theory to social organization. It begins with a mathematical definition of chaos. Specifically, the geometric concept of attractor is explored; phase space and Poincaré maps are discussed and applied to the concept of attractor; and nonlinearity is conceptually defined. We then apply the mathematics of chaos to social systems by showing how a nonlinear equation might be used to describe organization and how data derived from a simple univariate equation can be converted into a multivariate Poincaré map. The conclusion section identifies three approaches to analyzing chaos in social organization: metaphorical analysis, mathematical modeling, and data collection. Finally, possible uses of chaos are explored, as are the shortcomings of such analysis.

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Y. Al-hami, Kifah. "A NOTE ON CHAOS THEORY." Advances in Mathematics: Scientific Journal 10, no. 4 (April 12, 2021): 2077–82. http://dx.doi.org/10.37418/amsj.10.4.22.

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Aref'eva, I. Ya, P. B. Medvedev, O. A. Rytchkov, and I. V. Volovich. "Chaos in M(atrix) theory." Chaos, Solitons & Fractals 10, no. 2-3 (February 1999): 213–23. http://dx.doi.org/10.1016/s0960-0779(98)00159-3.

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Smith, R. D. "Social Structures and Chaos Theory." Sociological Research Online 3, no. 1 (March 1998): 82–102. http://dx.doi.org/10.5153/sro.113.

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Up to this point many of the social-scientific discussions of the impact of Chaos theory have dealt with using chaos concepts to refine matters of prediction and control. Chaos theory, however, has far more fundamental consequences which must also be considered. The identification of chaotic events arise as consequences of the attempts to model systems mathematically. For social science this means we must not only evaluate the mathematics but also the assumptions underlying the systems themselves. This paper attempts to show that such social-structural concepts as class, race, gender and ethnicity produce analytic difficulties so serious that the concept of structuralism itself must be reconceptualised to make it adequate to the demands of Chaos theory. The most compelling mode of doing this is through the use of Connectionism. The paper will also attempt to show this effectively means the successful inclusion of Chaos theory into social sciences represents both a new paradigm and a new epistemology and not just a refinement to the existing structuralist models. Research using structuralist assumptions may require reconciliation with the new paradigm.

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Abu Elwan, Reda. "The Effect of Teaching "Chaos Theory and Fractal Geometry" on Geometric Reasoning Skills of Secondary Students." INTERNATIONAL JOURNAL OF RESEARCH IN EDUCATION METHODOLOGY 6, no. 2 (August 30, 2015): 804–15. http://dx.doi.org/10.24297/ijrem.v6i2.3876.

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Chaos theory and fractal geometry have begun to appear as an important issue in secondary school mathematics. Chaos theory is the qualitative study of unstable periods in deterministic nonlinear dynamical systems, chaos theory looks at how things evolve. Fractal geometry is a subject that has established connections with many areas of mathematics (including number theory, probability theory and dynamical systems). Fractal geometry, together with the broader fields of nonlinear dynamics and complexity, represented a large segment of modern science at the end of the 20th century; this paper investigate the concepts of chaos theory and fractal geometry as a conceptual transformation at secondary school level. This paper reports a study of the effects of teaching chaos theory and fractal geometry on geometric reasoning skills in geometry. Thirty of the tenth grade students of basic education participated in an experimental group, which was involved in working with chaos theory and fractal geometry activities, pre-treatment measures the geometric Reasoning skills. Teaching fractal geometry properties and examples were focused in the teaching activities. At the end of the teaching measuresÂ geometric reasoning skills were again obtained. Since the study was an exploration, the effectiveness of teaching chaos theory and fractal geometry, the exploratory data collected by the researcher was also considered to be an important part of the study.Â

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SENGUPTA, A. "TOWARD A THEORY OF CHAOS." International Journal of Bifurcation and Chaos 13, no. 11 (November 2003): 3147–233. http://dx.doi.org/10.1142/s021812740300851x.

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This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.

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Nazzal, Jamal M., and Ammar N. Natsheh. "Chaos control using sliding-mode theory." Chaos, Solitons & Fractals 33, no. 2 (July 2007): 695–702. http://dx.doi.org/10.1016/j.chaos.2006.01.071.

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Ye, Fred Y. "From chaos to unification: U theory vs. M theory." Chaos, Solitons & Fractals 42, no. 1 (October 2009): 89–93. http://dx.doi.org/10.1016/j.chaos.2008.10.030.

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Bedford, Crayton W. "The Case for Chaos." Mathematics Teacher 91, no. 4 (April 1998): 276–81. http://dx.doi.org/10.5951/mt.91.4.0276.

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Chaos theory and fractal geometry have begun to appear as separate units in the mathematics curriculum of many high schools. Many more, however, have found these units appealing but impossible to squeeze into an already packed program. Yet a number of students at these same schools complete a precalculus sequence before the end of their senior year and find themselves, for one reason or another, not ready to go right on to the calculus. Typically, schools offer such students semester courses in probability, statistics, discrete mathematics or computer programming. In this article, I make a case for expanding this offering to include a semester of chaos theory that pulls together all those appealing units that may be scattered through the curriculum. After a brief overview of chaos theory and fractal geometry, I outline eight units that could constitute such a course and include a list of useful resources. For a number of years, I have taught a course based on this outline to precalculus students whose enthusiastic response is the motivation for this article.

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WANG, QIUDONG, and ALI OKSASOGLU. "RANK ONE CHAOS: THEORY AND APPLICATIONS." International Journal of Bifurcation and Chaos 18, no. 05 (May 2008): 1261–319. http://dx.doi.org/10.1142/s0218127408021002.

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The main purpose of this tutorial is to introduce to a more application-oriented audience a new chaos theory that is applicable to certain systems of differential equations. This new chaos theory, namely the theory of rank one maps, claims a comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain systems of differential equations, the existence of the indicated phenomenon of chaos can be verified through a well-defined computational process. Applications to the well-known Chua's and MLC circuits employing controlled switches are also presented to demonstrate the usefulness of the theory. We try to introduce this new chaos theory by using a balanced combination of examples, numerical simulations and theoretical discussions. We also try to create a standard reference for this theory that will hopefully be accessible to a more application-oriented audience.

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KOCAREV, LJUPCO, and ZARKO TASEV. "CHAOS AND CONTROL OF TRANSIENT CHAOS IN TURBO-DECODING ALGORITHMS." International Journal of Bifurcation and Chaos 14, no. 03 (March 2004): 1147–53. http://dx.doi.org/10.1142/s0218127404009740.

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We suggest a link between coding theory (iterative algorithms) and chaos theory. A whole range of phenomena known to occur in nonlinear systems, like the existence of multiple fixed points, oscillatory behavior, bifurcations, chaos and transient chaos are found in turbo-decoding algorithms. We develop a simple technique to control transient chaos in turbo-decoding algorithms and improve the performance of the classical turbo codes.

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Kaddoum, G., Anthony J. Lawrance, P. Chargé, and D. Roviras. "Chaos Communication Performance: Theory and Computation." Circuits, Systems, and Signal Processing 30, no. 1 (October 14, 2010): 185–208. http://dx.doi.org/10.1007/s00034-010-9217-1.

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Et.al, Nor Zila Abd Hamid. "A Pilot Study Using Chaos Theory to Predict Temperature Time Series in Malaysian Semi Urban Area." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 3 (April 11, 2021): 997–1003. http://dx.doi.org/10.17762/turcomat.v12i3.834.

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Chaos theory draws more attention since it has been widely used in modeling various time series. Changes in temperature can cause bad effect on health and lead to death. Therefore, in this study, chaos theory was applied to the temperature time series. The temperature time series is observed hourly in one of Malaysiansemi urban area namely Tanjong Malim,located in the state of Perak. This pilot study begins by detecting the chaos nature in time series through phase space approach and Cao method. Next, the time series was predicted through the local approximation method, a method based on chaos theory. This study resulted that the nature of the observed temperature time series was chaos. Prediction through the local approximation method was success with correlation coefficient value 0.9138. This shows that there exist a strong relationship between the predicted and observed temperature time series. Therefore, chaos theorywas a good approach that can be used to determine the nature and predict temperature time series in the semi urban area. In implication, this findingwas expected to serve stakeholders such as Ministry of Higher Education, Meteorological Department as well as Department of Environment in temperature time series management.

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Sharma, Vijay. "Deterministic Chaos and Fractal Complexity in the Dynamics of Cardiovascular Behavior: Perspectives on a New Frontier." Open Cardiovascular Medicine Journal 3, no. 1 (September 10, 2009): 110–23. http://dx.doi.org/10.2174/1874192400903010110.

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Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts.

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Li, Jian, and Xiang Dong Ye. "Recent development of chaos theory in topological dynamics." Acta Mathematica Sinica, English Series 32, no. 1 (February 15, 2015): 83–114. http://dx.doi.org/10.1007/s10114-015-4574-0.

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Gonchenko, Sergey, Alexey Kazakov, Dmitry Turaev, and Andrey L. Shilnikov. "Leonid Shilnikov and mathematical theory of dynamical chaos." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 1 (January 2022): 010402. http://dx.doi.org/10.1063/5.0080836.

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ŠPÁNY, VIKTOR, PAVOL GALAJDA, MILAN GUZAN, LADISLAV PIVKA, and MARTIN OLEJÁR. "CHUA'S SINGULARITIES: GREAT MIRACLE IN CIRCUIT THEORY." International Journal of Bifurcation and Chaos 20, no. 10 (October 2010): 2993–3006. http://dx.doi.org/10.1142/s0218127410027544.

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In the work [Chua, 1992], a deep intuition of its author gave rise to the choice of singularities corresponding to Chua's circuit. Therefore, it is the only one probably exhibiting three saddle points named Chua's singularities in this paper. One of the singularities is a saddle in forward time (dt > 0) of integration, whereas the other two are saddles in backward time (dt < 0) of integration. In the following, the term Chua's Chaos denotes chaos related to Chua's singularities. These singularities are the source of all special surfaces that are the subject of this contribution. We named the surface to which all other surfaces are bound as the Double-Arm Stable Manifold (DASM). The beauty and multifunctionality of this surface represents the unfathomable Intelligence in the sense of [Tolle, 2003]. The presence of the DASM in the state space is a sufficient condition for the generation of Chua's chaos or corresponding periodic windows. Since Chua's singularities are not limited by circuit morphology or the order of state equations, the research on Chua's chaos seems to be still very promising.

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Rasedee, A. F. N., M. H. Abdul Sathar, N. Mohd Najib, T. J. Wong, and L. F. Koo. "Numerical analysis on chaos attractors using a backward difference formulation." Mathematical Modeling and Computing 9, no. 4 (2022): 898–908. http://dx.doi.org/10.23939/mmc2022.04.898.

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The chaos attractor is a system of ordinary differential equations which is known for having chaotic solutions for certain parameter values and an initial condition. Research conducted in the current work establishes a backward difference algorithm to study these chaos attractors. Different types of chaos mapping, namely the Lorenz chaos, 'sandwich' chaos and 'horseshoe' chaos will be analyzed. Compared to other numerical methods, the proposed backward difference algorithm will show to be an efficient tool for analyzing solutions for the chaos attractors.

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Lonie, Isla. "Chaos Theory: A New Paradigm for Psychotherapy?" Australian & New Zealand Journal of Psychiatry 25, no. 4 (December 1991): 548–60. http://dx.doi.org/10.3109/00048679109064449.

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Thomas Kuhn's concept of paradigm as central to the functioning of a mature science is linked with Johnson-Abercrombie's recognition that perception itself is shaped by the schemata available to the subject. The rapidly advancing field of non-linear mathematics, in offering conceptual forms to represent complex events, may provide a useful framework in which to place various psychodynamic formulations about the development of the personality, and suggests the possibility of a new approach to research concerning the efficacy of psychotherapy. Dan Stern's latest concept of “moments” as the basic unit in structuring the personality, leading to the complex representational patterns and feed-back loops he terms “RIGS” may be viewed in this context. The paradigm may be extended to include such concepts as Peterfreund's linkage of psychodynamic theorising with aspects of information theory generated by the study of computers, and with Sullivan's concepts of repetitive patterns of behaviour recognisable, and changing, throughout the course of a therapy.

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Tanaka, Yosuke. "Space–time symmetry, chaos and E infinity theory." Chaos, Solitons & Fractals 23, no. 2 (January 2005): 335–49. http://dx.doi.org/10.1016/j.chaos.2004.05.035.

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Christie, J. R., K. Gopalsamy, and Jibin Li. "Chaos in sociobiology." Bulletin of the Australian Mathematical Society 51, no. 3 (June 1995): 439–51. http://dx.doi.org/10.1017/s000497270001426x.

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It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

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LAWRANCE, A. J., and N. BALAKRISHNA. "STATISTICAL DEPENDENCY IN CHAOS." International Journal of Bifurcation and Chaos 18, no. 11 (November 2008): 3207–19. http://dx.doi.org/10.1142/s0218127408022366.

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This paper is concerned with the statistical dependency effects in chaotic map processes, both before and after their discretization at branch boundaries. The resulting processes are no longer chaotic but are left with realizable statistical behavior. Such processes have appeared over several years in the electronic engineering literature. Informal but extended mathematical theory that facilitates the practical calculation of autocorrelation of such statistical behavior, is developed. Both the continuous and discretized cases are treated further by using Kohda's notions of equidistribution and constant-sum to maps which are not onto. Some particularly structured chaotic map processes, and also well-known maps are examined for their statistical dependency, with the tailed shift map family from chaotic communications receiving detailed attention. Several parts of the paper form a brief review of existing theory.

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Kumar, Arunachalam. "CHAOS THEORY: IMPACT ON AND APPLICATIONS IN MEDICINE." Journal of Health and Allied Sciences NU 02, no. 04 (December 2012): 93–99. http://dx.doi.org/10.1055/s-0040-1703623.

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AbstractThe advent of a revolutionary and exciting theory that proposes non-linear models as effective adjuncts to linear mathematics in interpreting long-held scientific tenets has provided novel and innovative designs and methodologies that help the medical world to better understand inferences from laboratory investigations, physiological processes, pharmarmaco-therapeutics and clinical diagnostics.This overview outlines a few salient areas in medicine that have successfully applied the principles of the chaos theory. Chaotic systems have been shown to operate in quite a few physiological processes. The impact and implications of the new science on the future course of medical diagnosticsand health science systems as a whole cannot be overstressed.

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BRAUER, ECKART, STEFAN BLOCHWITZ, and HORST BEIGE. "PERIODIC WINDOWS INSIDE CHAOS — EXPERIMENT VERSUS THEORY." International Journal of Bifurcation and Chaos 04, no. 04 (August 1994): 1031–39. http://dx.doi.org/10.1142/s0218127494000745.

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A series resonance circuit that consists of a linear inductance, a nonlinear capacitance and a sinusoidal driving is investigated. The nonlinearity arises from a ferroelectric crystal. We observed the Feigenbaum scenario, crises, periodic windows inside chaos that show a period adding behaviour and coexisting attractors of different symmetry. We conclude a Duffing equation to cover all significant properties of our dynamical system.

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J Zweedijk, Rene. "Osteopathy in the Cranial Field from a Systems Theory Perspective." Journal of Alternative, Complementary & Integrative Medicine 7, no. 5 (November 19, 2021): 1–9. http://dx.doi.org/10.24966/acim-7562/100197.

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A proposition is made for scientific substantiation of “Primary respiration” and related concepts, including suggestions for future research. For research and support, the field of mathematics, artificial intelligence, chaos theory and complex systems thinking can be of fundamental and essential value.

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Glendinning, P. "The anharmonic route to chaos: kneading theory." Nonlinearity 6, no. 3 (May 1, 1993): 349–67. http://dx.doi.org/10.1088/0951-7715/6/3/001.

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BUB, GIL, and LEON GLASS. "BIFURCATIONS IN A DISCONTINUOUS CIRCLE MAP: A THEORY FOR A CHAOTIC CARDIAC ARRHYTHMIA." International Journal of Bifurcation and Chaos 05, no. 02 (April 1995): 359–71. http://dx.doi.org/10.1142/s0218127495000302.

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The dynamics of discontinuous circle maps are investigated in the context of modulated parasystole, a cardiac arrhythmia in which there is an interaction between normal (sinus) and abnormal (ectopic) pacemaking sites in the heart. A class of noninvertible discontinuous circle maps with slope greater than 1 displays banded chaos under certain conditions. Banded chaos in these maps is characterized by a zero rotation interval width in the presence of a positive Lyapunov exponent. The bifurcations of a simple piecewise linear circle map are investigated. Parameters that result in banded chaos are organized into discrete, nonoverlapping zones in the parameter space. We apply these results to analyze a circle map that models modulated parasystole. Analysis of the model is complicated by the fact that the map has slope less than 1 for part of its domain. However, numerical simulations indicate that the modulated parasystole map displays banded chaos over a wide range of parameters. Banded chaos in this map produces rhythms with a relatively constant sinus-ectopic coupling interval, long trains of uninterrupted sinus beats, and patterns of successive sinus beats between ectopic beats characteristic of those found clinically.

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Еськов and V. Eskov. "Third global paradigm in medicine, mathematics and philosophy." Complexity. Mind. Postnonclassic 4, no. 1 (August 23, 2015): 6–12. http://dx.doi.org/10.12737/10859.

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Realizing the realities of the systems of third type – STT is gradually approaching the society of “skeptical” scientists, i.e. firm supporters of determinism and stochastic. Therefore the need for a brief presentation of the third paradigm of theory of chaos and self-organization in all their diversity increases many times. It is possible to express the hope that this will lead to a change in not only science but also in consciousness (view of the natural world). The proof of differences of theory of chaos and self-organization (TCS), objects (STT) from objects of deterministic and stochastic approaches (paradigms) – DSP is needed. Furthermore, a demonstration of pointlessness use of methods and theories of DSP in describing the STT. These all requires the synergy of mutual tolerance by all scientists. We must move away from the revolution in T. Kuhn’s paradigm shift to synergy and parallel existence of three global paradigms (deterministic, stochastic and third – syn-ergetic). As in nature, there are (and will be) three types of systems and in science there should be three different approaches, three paradigms. V.S. Stepin talked about these things representing postnonclassic.

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Mukhamedov, A. M. "Meso-structures of dynamical chaos and E-infinity theory." Chaos, Solitons & Fractals 41, no. 4 (August 2009): 1930–38. http://dx.doi.org/10.1016/j.chaos.2008.07.050.

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Postavaru, O., S. R. Anton, and A. Toma. "COVID-19 pandemic and chaos theory." Mathematics and Computers in Simulation 181 (March 2021): 138–49. http://dx.doi.org/10.1016/j.matcom.2020.09.029.

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Morales Rangel, Milagros Del Valle. "Applied mathematics in crisis scenarios (Covid-19)." Revista EDUCARE - UPEL-IPB - Segunda Nueva Etapa 2.0 24, no. 2 (August 12, 2020): 353–66. http://dx.doi.org/10.46498/reduipb.v24i2.1335.

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Applied mathematics is part of undergraduate and postgraduate university education. From this perspective, this essay aims to study the psychological effects, economic and, educational effects upon the population caused by a crisis scenario as COVID-19. The mathematical theories developed in this essay are Chaos Theory, Markov Chains, and Nash Theory. COVID-19 has spread throughout the world, affecting populations, and countries, without distinction of race, economic, political, or socio-cultural position. The impact that COVID-19 has caused on the world population could be measured, in the medium and long term, through changes in psychological behavior, social, health, economic and educational habits. This impact will leave deep traces and moral dilemmas that will permit prioritize which areas address and the political effort directed to each one.

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Yokoo, Masanori, and Junichiro Ishida. "Misperception-driven chaos: Theory and policy implications." Journal of Economic Dynamics and Control 32, no. 6 (June 2008): 1732–53. http://dx.doi.org/10.1016/j.jedc.2007.06.013.

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Seliverstova, Anna. "The theory of dynamic chaos in the socio-philosophical and social studies." Науково-теоретичний альманах "Грані" 22, no. 2 (April 22, 2019): 40–47. http://dx.doi.org/10.15421/171921.

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The article discusses the application of the theory of dynamic chaos to the study of social phenomena. Appeal to the origins of the creation of the theory of dynamic chaos in natural science (A. Poincaré, I. Prigogine, E. Lorenz, and others) revealed nonlinear dynamic systems in the natural environment (turbulent flows, atmosphere, biological populations, etc.). The category of “chaos” is now firmly established in the arsenal of the social sciences and humanities, although only recently it referred exclusively to natural science knowledge (the theory of chaos in mathematics, physics, biology, etc.). In synergetics, for the first time, the description of self-organization processes as a mutual transition of order and chaos was proposed by I.R. Prigogine.But in the social sciences such systems are society, its economic, political and other spheres, which have the properties of non-closure, instability and non-linear development. In Ukrainian philosophical thought, one of the first works in which the problem of the development of nonlinear self-developing systems was highlighted was the work of I.S. Dobronravova (1991). Scientific monograph I.V. Yershova-Babenko (1992) also had a significant impact on the development of studies of complex non-linear systems, since for the first time the system of the human psyche was considered as a non-linear self-organized system. The psycho-synergetic model of social reality is based on the fact that social reality is a psychomeric environment, i.e. a complex nonlinear system consisting of other complex nonlinear integrity, which are determined by phase transitions between different states of chaos and order. The application of chaos theory is also possible at the micro and macro levels of social research, which is presented by Ukrainian researchers in synergetics (I. S. Dobronravova, L. Finkel) and in psychosynergetics (I.V. Yershova-Babenko), L. Bevzenko and others.

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White, D. "A mathematical model for the f0(600) based on chaos theory." Journal of Interdisciplinary Mathematics 10, no. 5 (October 2007): 625–35. http://dx.doi.org/10.1080/09720502.2007.10700523.

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Esquível, Manuel L. "On the asymptotic behavior of the second moment of the Fourier transform of a random measure." International Journal of Mathematics and Mathematical Sciences 2004, no. 63 (2004): 3423–34. http://dx.doi.org/10.1155/s0161171204210183.

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The behavior at infinity of the Fourier transform of the random measures that appear in the theory of multiplicative chaos of Mandelbrot, Peyrière, and Kahane is an area quite unexplored. For context and further reference, we first present an overview of this theory and then the result, which is the main objective of this work, generalizing a result previously announced by Kahane. We establish an estimate for the asymptotic behavior of the second moment of the Fourier transform of the limit random measure in the theory of multiplicative chaos. After looking at the behavior at infinity of the Fourier transform of some remarkable functions and measures, we prove a formula essentially due to Frostman, involving the Riesz kernels.

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DE FEO, OSCAR. "TRANSFORMING SUBHARMONIC CHAOS TO HOMOCLINIC CHAOS SUITABLE FOR PATTERN RECOGNITION." International Journal of Bifurcation and Chaos 15, no. 10 (October 2005): 3345–57. http://dx.doi.org/10.1142/s0218127405014106.

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Various forms of chaotic synchronization have been proposed as ways of realizing associative memories and/or pattern recognizers. To exploit this kind of synchronization phenomena in temporal pattern recognition, a chaotic dynamical system representing the class of signals that are to be recognized must be established. As shown recently [De Feo, 2003], this system can be determined by means of identification techniques where chaos emerges by itself to model the diversity of nearly periodic signals. However, the emerging chaotic behavior is subharmonic, i.e. period doubling-like, and therefore, as explained in [De Feo, 2004a, 2004b], it is not suitable for a synchronization-based pattern recognition technique. Nevertheless, as shown here, bifurcation theory and continuation techniques can be combined to modify a subharmonic chaotic system and drive it to homoclinic conditions; obtaining in this way a model suitable for synchronization-based pattern recognition.

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Chen, Guanrong, and Yuming Shi. "Introduction to anti-control of discrete chaos: theory and applications." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364, no. 1846 (July 28, 2006): 2433–47. http://dx.doi.org/10.1098/rsta.2006.1833.

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In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.

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Pellegrini, L., C. Tablino Possio, and G. Biardi. "An Example of How Nonlinear Dynamics Tools Can be Successfully Applied to A Chemical System." Fractals 05, no. 03 (September 1997): 531–47. http://dx.doi.org/10.1142/s0218348x97000425.

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The nonlinear dynamic behavior of a Proportional-Integral controlled Continuously Stirred Tank Reactor (CSTR) is analyzed in depth progressing from chaos characterization, through the high codimension bifurcation theory, up to the application of Controlling Chaos techniques. All these tools can be successfully applied to recognize, to avoid and to use chaos in practical applications, so that the nonlinear dynamic theory turns out to be an indispensable science to constrain dynamic systems to work in the most suitable operative conditions.

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Rodríguez, Ricardo Osés. "Chaos Theory of Mathematics as seen from a New Perspective for Weather Forecasting." Bioscience Biotechnology Research Communications 15, no. 3 (September 30, 2022): 390–98. http://dx.doi.org/10.21786/bbrc/15.3.4.

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In this work, 8 meteorological variables were modeled in the Yabú station, Cuba, for which the daily database of this meteorological station was used, where the meteorological variables were taken into account are: extreme temperatures, extreme humidity and its average value, precipitation, wind force and cloudiness corresponding to the period 1977 to 2021. A linear mathematical model was obtained using the Objective Regressive Regression (ORR) methodology for each variable, which explains its behavior according to these variables, 15, 13, 10 and 8 years in advance. The calculation of the mean error with respect to the persistence forecast in temperatures, wind strength and cloudiness, as well as the persistence model was better with respect to humidity, this allows having valuable long-term information of the weather in a locality, which results in better decision making in the different aspects of the economy and society that are impacted by the weather forecast. It is concluded that these models allow long-term weather forecasting, opening a new possibility for forecasting, so that weather chaos can be overcome if this way of forecasting is used; moreover, it is the first time that an ORR model is applied to weather forecasting processes for a specific day so many years in advance.

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Martínez-Giménez, Félix, Alfred Peris, and Francisco Rodenas. "Chaos on Fuzzy Dynamical Systems." Mathematics 9, no. 20 (October 18, 2021): 2629. http://dx.doi.org/10.3390/math9202629.

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Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

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Wang, Lei, XiaoSong Yang, WenJie Hu, and Quan Yuan. "Horseshoe Chaos in a Simple Memristive Circuit." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/546091.

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A simple memristive circuit model is revisited and the stability analysis is to be given. Furthermore, we resort to Poincaré section and Poincaré map technique and present rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoe theory.

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Mitkowski, Paweł, and Wojciech Mitkowski. "Ergodic theory approach to chaos: Remarks and computational aspects." International Journal of Applied Mathematics and Computer Science 22, no. 2 (June 1, 2012): 259–67. http://dx.doi.org/10.2478/v10006-012-0019-4.

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Ergodic theory approach to chaos: Remarks and computational aspectsWe discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.

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Li, Yin, and Chun-long Zheng. "The Complex Network Synchronization via Chaos Control Nodes." Journal of Applied Mathematics 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/823863.

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We investigate chaos control nodes of the complex network synchronization. The structure of the coupling functions between the connected nodes is obtained based on the chaos control method and Lyapunov stability theory. Moreover a complex network with nodes of the new unified Loren-Chen-Lü system, Coullet system, Chee-Lee system, and the New system is taken as an example; numerical simulations are used to verify the effectiveness of the method.

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MAISTRENKO, YURI L., OLEKSANDR V. POPOVYCH, and PETER A. TASS. "CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL." International Journal of Bifurcation and Chaos 15, no. 11 (November 2005): 3457–66. http://dx.doi.org/10.1142/s0218127405014155.

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The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.

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G., W., P. Fischer, and William R. Smith. "Chaos, Fractals, and Dynamics." Mathematics of Computation 46, no. 174 (April 1986): 768. http://dx.doi.org/10.2307/2008026.

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MIN, LEQUAN, YAN MENG, and LEON O. CHUA. "APPLICATIONS OF LOCAL ACTIVITY THEORY OF CNN TO CONTROLLED COUPLED OREGONATOR SYSTEMS." International Journal of Bifurcation and Chaos 18, no. 11 (November 2008): 3233–97. http://dx.doi.org/10.1142/s021812740802255x.

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The study of chemical reactions with oscillating kinetics has drawn increasing interest over the last few decades because it also contributes towards a deeper understanding of the complex phenomena of temporal and spatial organizations in biological systems. The Cellular Nonlinear Network (CNN) local activity principle introduced by Chua [1997, 2005] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. Recently, Yang and Epstein proposed a reaction–diffusion Oregonator model with five variables for mimicking the Belousov–Zhabotinskii reaction. The Yang–Epstein model can generate oscillatory Turing patterns, including the twinkling eye, localized spiral and concentric wave structures. In this paper, we first propose a modified Yang–Epstein's Oregonator model by introducing a controller, and then map the revised Oregonator reaction–diffusion system into a reaction–diffusion Oregonator CNN. The Oregonator CNN has two cell equilibrium points Q1 = (0, 0, 0, 0, 0) and Q2, representing the "original" equilibrium point and an additional equilibrium point, respectively. The bifurcation diagrams of the Oregonator CNN are calculated using the analytical criteria for local activity. The bifurcation diagrams of the Oregonator CNN at Q1 have only locally active and unstable regions; and the ones at Q2 have locally passive regions, locally active and unstable regions, and edge of chaos regions. The calculated results show that the parameter groups of the Oregonator CNN which generate complex patterns are located on the edge of chaos regions, or on locally active unstable regions near the edge of chaos boundary. Numerical simulations show also that the Oregonator CNNs can generate similar dynamics patterns if the parameter groups are selected the same as those of the Yang–Epstein model. In particular, the parameters of the Yang–Epstein model which exhibit twinkling-eye patterns, and pinwheel patterns are located on the edges of chaos regions near the boundaries of locally active unstable regions with respect to Q2. The parameters of the Yang–Epstein models which exhibit labyrinthine stripelike patterns are located on the locally active unstable regions near the boundaries of the edge of chaos regions with respect to Q2. However the parameter group of the Yang–Epstein model with isolated spiral patterns is in the locally passive region near the boundary with edge of chaos with respect to Q2, whose trajectories tend to the equilibrium point Q2. Choosing a kind of triggering initial conditions given in [Chua, 1997], and the parameters of the Oregonator equations with the twinkling-eye patterns, pinwheel patterns, labyrinthine stripelike patterns, and isolated spiral patterns, three kinds of new spiral waves generated by the Oregonator CNNs were observed by numerical simulations. They seem to be essentially different patterns to those generated by the Oregonator CNNs with initial conditions consisting of equilibrium points plus small random perturbations. Our results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or near the edge of chaos region.

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GANDHI, GAURAV, GYÖRGY CSEREY, JOHN ZBROZEK, and TAMÁS ROSKA. "ANYONE CAN BUILD CHUA'S CIRCUIT: HANDS-ON-EXPERIENCE WITH CHAOS THEORY FOR HIGH SCHOOL STUDENTS." International Journal of Bifurcation and Chaos 19, no. 04 (April 2009): 1113–25. http://dx.doi.org/10.1142/s0218127409023536.

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Chaos is a physical and mathematical phenomenon discovered by E. Lorenz in 1963. The first simple electronic implementation had been invented by L. O. Chua in 1984. This electronic circuit, called Chua's circuit was designed for ease of implementation. In the current brief we will explain chaos by building Chua's chaotic circuit using our Chua's circuit kit with inexpensive components. For readers without access to an oscilloscope, this paper proposes the use of a laptop/Personal Computer to capture the voltage waveforms generated from the circuit and plot the waveforms on a computer screen using a virtual oscilloscope software provided by the authors. The kit is available, the software is downloadable.

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Li, Y. Charles. "Major open problems in chaos theory, turbulence and nonlinear dynamics." Dynamics of Partial Differential Equations 10, no. 4 (2013): 379–92. http://dx.doi.org/10.4310/dpde.2013.v10.n4.a5.

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INABA, NAOHIKO, MUNEHISA SEKIKAWA, TETSURO ENDO, and TAKASHI TSUBOUCHI. "REVEALING THE TRICK OF TAMING CHAOS BY WEAK HARMONIC PERTURBATIONS." International Journal of Bifurcation and Chaos 13, no. 10 (October 2003): 2905–15. http://dx.doi.org/10.1142/s0218127403008272.

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Taming chaos by weak harmonic perturbations has been a hot topic in recent years. In this paper, the authors investigate a scenario for the mechanism of taming chaos via bifurcation theory, and assert that this phenomenon is caused by a slight shift of the saddle node bifurcation curves.

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Journal articles on the topic 'Chaos Theory (Mathematics)'

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