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Relevant bibliographies by topics / Chaos Theory (Mathematics)

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Academic literature on the topic 'Chaos Theory (Mathematics)'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 January 2023

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Journal articles on the topic "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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Dissertations / Theses on the topic "Chaos Theory (Mathematics)"

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Koperski, Jeffrey David. "Defending chaos: An examination and defense of the models used in chaos theory /." The Ohio State University, 1997. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487945015616055.

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Krcelic, Khristine M. "Chaos and Dynamical Systems." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1364545282.

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Monte, Brent M. "Chaos and the stock market." CSUSB ScholarWorks, 1994. https://scholarworks.lib.csusb.edu/etd-project/860.

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Filler, Andreas. "Modelling in Mathematics and Informatics: How Should the Elevators Travel so that Chaos Will Stop?" Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79751.

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Didactic proposals on modelling in mathematics education mostly give priority to models which describe, explain as well as partially forecast and provide mathematical solutions to real situations. A view of the modelling concept of informatics, which also initiates rapidly generalised deliberations of models, can also make a contribution to the spectrum of models, which are treated in a meaningful sense in mathematics lessons so as to expand some interesting aspects. In this paper, this is illustrated by means of conceptual design models – and, here, especially of process models – using the example of elevator organisation in a multi-storey construction.

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Webb, Mary A. "RESISTING IN THE MIDST OF CHAOS: ONE REVOLUTIONARY EDUCATOR’S CURRERE JOURNEY." Miami University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=miami1448470679.

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Werndl, Charlotte. "Philosophical aspects of chaos : definitions in mathematics, unpredictability, and the observational equivalence of deterministic and indeterministic descriptions." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/226754.

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This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts. First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important. Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant. Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence. By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence.

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Karytinos, Aristotle D. "A chaos theory and nonlinear dynamics approach to the analysis of financial series : a comparative study of Athens and London stock markets." Thesis, University of Warwick, 1999. http://wrap.warwick.ac.uk/51481/.

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This dissertation presents an effort to implement nonlinear dynamic tools adapted from chaos theory in financial applications. Chaos theory might be useful in explaining the dynamics of financial markets, since chaotic models are capable of exhibiting behaviour similar to that observed in empirical financial data. In this context, the scope of this research is to provide an insight into the role that nonlinearities and, in particular, chaos theory may play in explaining the dynamics of financial markets. From a theoretical point of view, the basic features of chaos theory, as well as, the rationales for bringing chaos theory to the attention of financial researchers are discussed. Empirically, the fundamental issue of determining whether chaos can be observed in financial time series is addressed. Regarding the latter, empirical literature has been controversial. A quite exhaustive analysis of the existing literature is provided, revealing the inadequacies in terms of methodology and the testing framework adopted, so far. A new "multiple testing" methodology is developed combining methods and techniques from the fields of both Natural Sciences and the Economics, most of which have not been applied to financial data before. A serious effort has been made to fill, as much as possible, the gap which results from the lack of a proper statistical framework for the chaotic methods. To achieve this the bootstrap methodology is adopted. The empirical part of this work focuses on the comparison of two markets with different levels of maturity; the Athens Stock Exchange (ASE), an emerging market, and London Stock Exchange (LSE). Our aim is to determine whether structural differences exist in these markets in terms of chaotic dynamics. In the empirical level we find nonlinearities in both markets by the use of the BDS test. R/S analysis reveals fractality and long term memory for the ASE series only. Chaotic methods, such as the correlation dimension (and related methods and techniques) and the largest Lyapunov exponent estimation, cannot rule out a chaotic explanation for the ASE market, but no such indication could be found for the LSE market. Noise filtering by the SVD method does not alter these findings. Alternative techniques based on nonlinear nearest neighbour forecasting methods, such as the "piecewise polynomial approximation" and the "simplex" methods, support our aforementioned conclusion concerning the ASE series. In all, our results suggest that, although nonlinearities are present, chaos is not a widespread phenomenon in financial markets and it is more likely to exist in less developed markets such as the ASE. Even then, chaos is strongly mixed with noise and the existence of low-dimensional chaos is highly unlikely. Finally, short-term forecasts trying to exploit the dependencies found in both markets seem to be of no economic importance after accounting for transaction costs, a result which supports further our conclusions about the limited scope and practical implications of chaos in Finance.

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Taylor, S. Richard. "Probabilistic Properties of Delay Differential Equations." Thesis, University of Waterloo, 2004. http://hdl.handle.net/10012/1183.

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Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, i. e. in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, i. e. develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.

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Roger, Mikaël. "Propriétés stochastiques de systèmes dynamiques et théorèmes limites : deux exemples." Phd thesis, Université Rennes 1, 2008. http://tel.archives-ouvertes.fr/tel-00362479.

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Ce travail met en jeu plusieurs systèmes dynamiques sur des tores en dimension finie, pour lesquels on sait établir des théorèmes limites, qui permettent de préciser leur comportement stochastique. On généralise d'abord le théorème limite local usuel sur un sous-shift de type fini, en ajoutant un terme de perturbation, en reprenant la preuve classique, par des techniques d'opérateurs. On en déduit un théorème limite local pour les sommes de « Riesz-Raïkov unitaires étendues », et des observables höldériennes. Pour cela, on reprend une méthode employée par Bernard Petit, en utilisant des codages symboliques, et le théorème limite local avec perturbation. Puis, on présente plusieurs situations de composées d'automorphismes hyperboliques du tore en dimension deux pour lesquelles on sait établir un théorème limite central quelque soit le choix de la composée. En particulier, on aborde le cas des matrices à coefficients entiers positifs.

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Passey, Jr David Joseph. "Growing Complex Networks for Better Learning of Chaotic Dynamical Systems." BYU ScholarsArchive, 2020. https://scholarsarchive.byu.edu/etd/8146.

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This thesis advances the theory of network specialization by characterizing the effect of network specialization on the eigenvectors of a network. We prove and provide explicit formulas for the eigenvectors of specialized graphs based on the eigenvectors of their parent graphs. The second portion of this thesis applies network specialization to learning problems. Our work focuses on training reservoir computers to mimic the Lorentz equations. We experiment with random graph, preferential attachment and small world topologies and demonstrate that the random removal of directed edges increases predictive capability of a reservoir topology. We then create a new network model by growing networks via targeted application of the specialization model. This is accomplished iteratively by selecting top preforming nodes within the reservoir computer and specializing them. Our generated topology out-preforms all other topologies on average.

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Books on the topic "Chaos Theory (Mathematics)"

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Tsonis, Anastasios A. Chaos: From Theory to Applications. Boston, MA: Springer US, 1992.

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Li, Zhong. Integration of fuzzy logic and chaos theory. Berlin [u.a.]: Springer, 2010.

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Chan, Kung-sik. Chaos: A Statistical Perspective. New York, NY: Springer New York, 2001.

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Hurt, Norman E. Quantum Chaos and Mesoscopic Systems: Mathematical Methods in the Quantum Signatures of Chaos. Dordrecht: Springer Netherlands, 1997.

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Mori, Hazime. Dissipative structures and chaos. Berlin: Springer, 1998.

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Mori, Hazime. Dissipative structures and chaos. Berlin: Springer, 1998.

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S, Mahavier William, and SpringerLink (Online service), eds. Inverse Limits: From Continua to Chaos. New York, NY: Springer Science+Business Media, LLC, 2012.

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Kelsey, Robert Bruce. Chaos and complexity in software. Commack, N.Y: Nova Science Publishers, 1999.

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Workshop on Chaos and Complexity (1987 Turin, Italy). Workshop on Chaos and Complexity, Torino, October 5-11, 1987. Edited by Livi Roberto and Institute for Scientific Interchange. Singapore: World Scientific, 1988.

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Blümel, R. Advanced quantum mechanics the classical-quantum connection. Sudbury, Mass: Jones and Bartlett Publishers, 2011.

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Book chapters on the topic "Chaos Theory (Mathematics)"

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Bungartz, Hans-Joachim, Stefan Zimmer, Martin Buchholz, and Dirk Pflüger. "Chaos Theory." In Springer Undergraduate Texts in Mathematics and Technology, 291–314. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-39524-6_12.

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Buzzi, Jérôme. "Chaos and Ergodic Theory." In Mathematics of Complexity and Dynamical Systems, 63–87. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-1806-1_6.

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Howland, James S. "Random perturbation theory and quantum chaos." In Lecture Notes in Mathematics, 197–204. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0080597.

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Sinai, Ya G. "Mathematical problems in the theory of quantum chaos." In Lecture Notes in Mathematics, 41–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0089214.

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Broer, Henk W., and Heinz Hanßmann. "Hamiltonian Perturbation Theory (and Transition to Chaos)." In Mathematics of Complexity and Dynamical Systems, 657–82. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-1806-1_41.

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Rosser, J. Barkley. "The Mathematics of Discontinuity." In From Catastrophe to Chaos: A General Theory of Economic Discontinuities, 9–68. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-017-1613-0_2.

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Rosser, J. Barkley. "The Mathematics of Discontinuity." In From Catastrophe to Chaos: A General Theory of Economic Discontinuities, 7–35. Boston, MA: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4613-3796-6_2.

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Kasman, Alex. "Uses of Chaos Theory and Fractal Geometry in Fiction." In The Palgrave Handbook of Literature and Mathematics, 129–47. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-55478-1_8.

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van Wyk, M. A., and W. H. Steeb. "Controlling Chaos." In Mathematical Modelling: Theory and Applications, 291–339. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8921-5_7.

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Mitkowski, Paweł J. "Chaos and Ergodic Theory." In Mathematical Structures of Ergodicity and Chaos in Population Dynamics, 19–40. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-57678-3_4.

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Conference papers on the topic "Chaos Theory (Mathematics)"

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Pauline, Ong, Ong Kok Meng, and Sia Chee Kiong. "An improved flower pollination algorithm with chaos theory for function optimization." In PROCEEDINGS OF THE 24TH NATIONAL SYMPOSIUM ON MATHEMATICAL SCIENCES: Mathematical Sciences Exploration for the Universal Preservation. Author(s), 2017. http://dx.doi.org/10.1063/1.4995922.

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Karpukhin, Aleksandr, Ludmila Kirichenko, Dmitrij Gritsiv, and Aleksandr Tkachenko. "Mathematical modelling of infocommunication systems by means of chaos theory methods." In 2014 First International Scientific-Practical Conference Problems of Infocommunications Science and Technology (PIC S&T`2014). IEEE, 2014. http://dx.doi.org/10.1109/infocommst.2014.6992283.

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Pineda Paz, Eduardo, and Alejandro Guerrero Torrenegra. "Vialidad, conectividad y fractales." In Seminario Internacional de Investigación en Urbanismo. Barcelona: Maestría en Planeación Urbana y Regional. Pontificia Universidad Javeriana de Bogotá, 2014. http://dx.doi.org/10.5821/siiu.6075.

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La morfología urbana es posible analizarla mediante ecuaciones no lineales que aparentemente reflejan el comportamiento del hombre. La teoría del caos, la incertidumbre y los fractales, aportan nuevas posibilidades al planificador urbano. El estudio es descriptivo y analítico, siguiendo pautas fenomenológicas, combinando teoría y práctica urbanística, con matemática sencilla. La parroquia Olegario Villalobos de Maracaibo es el caso de estudio. La investigación abordó la dimensión longitudinal, representada por la Vialidad. Se aplicó a las vías principales, teoría y factor de escala para develar posibles patrones fractales que indiquen conectividad. Este método es el primer paso de un proceso de indagación que permitirá aplicar en próximas investigaciones, la ley de potencias mediante ecuaciones diferenciales y el uso del computador. Estas teorías no demostradas aún, pueden verificarse, comparando la realidad observada con los resultados cuantitativos. Es un método distinto para planificar, diseñar y realizar intervenciones urbanísticas. Urban morphology is apparently possible by analyzing human behavior reflect nonlinear equations. Chaos theory , the uncertainty and fractals , bring new opportunities to the urban planner. The study is descriptive and analytical, phenomenological following patterns , combining theory and practice of urban development , with simple mathematics. The parish Olegario Villalobos from Maracaibo is the case study. The research addressed the longitudinal dimension , represented by the Roads . Was applied to the main roads , and scale factor theory to uncover possible fractal patterns that indicate connectivity. This method is the first step in a process of inquiry that will apply in future research , the power law with differential equations and computer use . Have not yet proven , theories can be verified by comparing the observed reality with the quantitative results . It is a different approach to plan, design and carry out urban interventions method.

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Agrwal, Saurabh Kumar, Bhanu Pratap Singh, and Rajesh Kumar. "Chaos theory based mathematical modelling as manifested from scalp EEG using frequency analysis." In 2013 IEEE Conference on Information & Communication Technologies (ICT). IEEE, 2013. http://dx.doi.org/10.1109/cict.2013.6558170.

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Aihara, Kazuyuki. "IWCFTA2012 Keynote Speech II - Mathematical Theory for Complex Systems Modelling and its Applications in Science and Technology." In 2012 5th International Workshop on Chaos-Fractals Theories and Applications (IWCFTA). IEEE, 2012. http://dx.doi.org/10.1109/iwcfta.2012.9.

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Sapini, Muhamad Luqman, Nor Syahira Adam, Nursyahirah Ibrahim, Nursyaziella Rosmen, and Norliza Muhamad Yusof. "The presence of chaos in rainfall by using 0-1 test and correlation dimension." In PROCEEDINGS OF THE 13TH IMT-GT INTERNATIONAL CONFERENCE ON MATHEMATICS, STATISTICS AND THEIR APPLICATIONS (ICMSA2017). Author(s), 2017. http://dx.doi.org/10.1063/1.5012259.

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Yao, Weiguang, Pei Yu, and Chris Essex. "Estimation of Chaotic Parameter Regimes via Generalized Competitive Modes Approach." In ASME 2001 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/imece2001/de-23224.

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Abstract A procedure is proposed to estimate the parameter regimes of chaos in nonlinear systems by implementing a mathematical version of mode competition. The idea is that for a system to be chaotic there must exist at least two generalized competitive modes in the system. The Lorenz system and a thin plate in flow-induced vibrations system are analyzed to find chaotic regimes by this procedure.

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Wu, Shunhai, Haitao Hu, Changping Chen, and Liming Dai. "The Application of P-R Criterion in Chaos Identification of Micro Piezoelectric Laminated Beam." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-62900.

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In this paper, based on Euler-Bernoulli assumptions and piezoelectric theory, taken electric excitation loads generated by AC and DC’s interactions, control voltage of piezoelectric layer and geometric nonlinearity into consideration, the nonlinear differential governing equation of piezoelectric laminated microbeam is established. The equation is solved by Galerkin method and mathematical tools. Making use of P-R criterion method, the analysis of single parameter bifurcation is carried on firstly to understand the characteristics of system, and the effects of multiple parameters variation on nonlinear dynamic characteristics of system are discussed in detail. The results can be used for the design of such micro-structures.

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Salloum, Maher, Ronghui Ma, and Liang Zhu. "Applying Polynomial Chaos Expansions to Evaluate the Effect of Tissue Non-Homogeneous Properties in Biotransport." In ASME 2013 Summer Bioengineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/sbc2013-14174.

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Modeling drug transport in tissues has recently gained a lot of attention in the bioengineering community due its vast areas of applications [1]. Such injections often employ a positive pressure infusion directly in the target tissue. It is often referred to as convection enhanced delivery. There are several studies that addressed this problem in the literature. These studies rely on mathematical models of flow in porous media (Darcy, Brinkman...) and are successful in accounting for the existence of capillaries, tissue metabolism, etc. However, these models rely on the assumption that the tissue properties (e.g. permeability) are uniform inside tumors. MicroCT imaging following nanofluid infusion often reveals highly irregular distributions due to the spatial heterogeneity of the tissue [2].

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Khatoon, Sufia, Jyoti Phirani, and Supreet Singh Bahga. "Polynomial Chaos Based Solution to Inverse Problems in Petroleum Reservoir Engineering." In ASME-JSME-KSME 2019 8th Joint Fluids Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/ajkfluids2019-5291.

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Abstract In reservoir simulations, model parameters such as porosity and permeability are often uncertain and therefore better estimates of these parameters are obtained by matching the simulation predictions with the production history. Bayesian inference provides a convenient way of estimating parameters of a mathematical model, starting from a probable range of parameter values and knowing the production history. Bayesian inference techniques for history matching require computationally expensive Monte Carlo simulations, which limit their use in petroleum reservoir engineering. To overcome this limitation, we perform accelerated Bayesian inference based history matching by employing polynomial chaos (PC) expansions to represent random variables and stochastic processes. As a substitute to computationally expensive Monte Carlo simulations, we use a stochastic technique based on PC expansions for propagation of uncertainty from model parameters to model predictions. The PC expansions of the stochastic variables are obtained using relatively few deterministic simulations, which are then used to calculate the probability density of the model predictions. These results are used along with the measured data to obtain a better estimate (posterior distribution) of the model parameters using the Bayes rule. We demonstrate this method for history matching using an example case of SPE1CASE2 problem of SPEs Comparative Solution Projects. We estimate the porosity and permeability of the reservoir from limited and noisy production data.

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