Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices

In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These result...

Full description

Autores:

Tipo de recurso:

Fecha de publicación:: 2025

Institución:: Universidad del Rosario

Repositorio:: Repositorio EdocUR - U. Rosario

Idioma:: eng

id	EDOCUR2_9ac62c88f94b4cae9c3b705279542e2c
oai_identifier_str	oai:repository.urosario.edu.co:10336/45741
network_acronym_str	EDOCUR2
network_name_str	Repositorio EdocUR - U. Rosario
repository_id_str
dc.title.none.fl_str_mv	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
dc.title.TranslatedTitle.none.fl_str_mv	Exponentes de Lyapunov para predecir el comportamiento del producto de matrices aleatorias
title	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
spellingShingle	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices Exponentes de Lyapunov Matrices aleatorios Caos Teoría ergódica Lyapunov Exponents Random Matrices Chaos Ergodic Theory
title_short	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
title_full	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
title_fullStr	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
title_full_unstemmed	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
title_sort	Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
dc.contributor.advisor.none.fl_str_mv	Artigiani, Mauro Martínez, Cristian
dc.subject.none.fl_str_mv	Exponentes de Lyapunov Matrices aleatorios Caos Teoría ergódica
topic	Exponentes de Lyapunov Matrices aleatorios Caos Teoría ergódica Lyapunov Exponents Random Matrices Chaos Ergodic Theory
dc.subject.keyword.none.fl_str_mv	Lyapunov Exponents Random Matrices Chaos Ergodic Theory
description	In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These results provide the tools to quantify exponential growth rates and directional behavior in random matrix products. To visualize our theoretical conclusions, we present a series of simulations that illustrate the emergence of Lyapunov exponents and their predictive power in practical settings.
publishDate	2025
dc.date.accessioned.none.fl_str_mv	2025-06-19T14:33:07Z
dc.date.available.none.fl_str_mv	2025-06-19T14:33:07Z
dc.date.created.none.fl_str_mv	2025-06-09
dc.type.none.fl_str_mv	bachelorThesis
dc.type.coar.fl_str_mv	http://purl.org/coar/resource_type/c_7a1f
dc.type.spa.none.fl_str_mv	Trabajo de grado
dc.identifier.uri.none.fl_str_mv	https://repository.urosario.edu.co/handle/10336/45741
url	https://repository.urosario.edu.co/handle/10336/45741
dc.language.iso.none.fl_str_mv	eng
language	eng
dc.rights.*.fl_str_mv	Attribution-NonCommercial-NoDerivatives 4.0 International
dc.rights.coar.fl_str_mv	http://purl.org/coar/access_right/c_abf2
dc.rights.acceso.none.fl_str_mv	Abierto (Texto Completo)
dc.rights.uri.*.fl_str_mv	http://creativecommons.org/licenses/by-nc-nd/4.0/
rights_invalid_str_mv	Attribution-NonCommercial-NoDerivatives 4.0 International Abierto (Texto Completo) http://creativecommons.org/licenses/by-nc-nd/4.0/ http://purl.org/coar/access_right/c_abf2
dc.format.extent.none.fl_str_mv	40 pp
dc.format.mimetype.none.fl_str_mv	application/pdf
dc.publisher.none.fl_str_mv	Universidad del Rosario
dc.publisher.department.none.fl_str_mv	Escuela de Ingeniería, Ciencia y Tecnología
dc.publisher.program.none.fl_str_mv	Programa de Matemáticas Aplicadas y Ciencias de la Computación - MACC
publisher.none.fl_str_mv	Universidad del Rosario
institution	Universidad del Rosario
dc.source.bibliographicCitation.none.fl_str_mv	Robert G. Bartle. The Elements of Integration and Lebesgue Measure. 1st ed. Wiley Clas- sics Library v.92. Hoboken: John Wiley & Sons, Incorporated, 1995. 1 p. isbn: 9780471042228 9781118164488. Walter Rudin. Real and complex analysis. 3. ed., internat. ed., [Nachdr.] McGraw-Hill international editions Mathematics series. New York, NY: McGraw-Hill, 2013. 416 pp. isbn: 978-0-07-100276-9 978-0-07-054234-1. Donald L. Cohn. Measure Theory: Second Edition. Birkhäuser Advanced Texts Basler Lehrbücher. New York, NY: Springer New York, 2013. isbn: 978-1-4614-6955-1 978-1- 4614-6956-8. doi: 10.1007/978-1-4614-6956-8. url: https://link.springer.com/ 10.1007/978-1-4614-6956-8 (visited on 08/14/2024). Elias M. Stein and Rami Shakarchi. Real analysis: measure theory, integration, and Hilbert spaces. Princeton lectures in analysis v. 3. Princeton, N.J: Princeton University Press, 2005. 402 pp. isbn: 9780691113869. Manfred Einsiedler and Thomas Ward. Ergodic Theory: with a view towards Number The- ory. London: Springer London, 2011. isbn: 978-0-85729-020-5 978-0-85729-021-2. doi: 10.1007/978- 0- 85729- 021- 2. url: https://link.springer.com/10.1007/978- 0-85729-021-2 (visited on 08/14/2024). Daniel W. Stroock. Probability theory: an analytic view. 2nd ed. Cambridge New York: Cambridge University Press, 2011. 1 p. isbn: 978-0-521-76158-1 978-0-511-97424-3 978-1- 139-01188-4 Sheldon M. Ross. Introduction to probability models. Tenth edition. Amsterdam Boston: Academic Press, an imprint of Elsevier, 2010. 1 p. isbn: 978-0-12-375686-2 978-0-12- 375687-9. Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to probability. 2nd ed. Optimiza- tion and computation series. Belmont: Athena scientific, 2008. isbn: 978-1-886529-23-6. Patrick Billingsley. Probability and measure. 3rd ed. Wiley series in probability and math- ematical statistics. New York: Wiley, 1995. 593 pp. isbn: 978-0-471-00710-4. Paul E. Pfeiffer. Probability for Applications. Springer Texts in Statistics. New York, NY: Springer New York, 1990. 679 pp. isbn: 978-1-4615-7676-1. doi: 10.1007/978-1-4615- 7676-1. Queen Mary University of London. Lecture 2: Second Language Acquisition. https:// qmplus . qmul . ac . uk / pluginfile . php / 2470175 / mod _ resource / content / 13 / L2 - 2022.pdf. Accessed: 2025-04-21. 2022. Alessandro Rinaldo. Lecture Notes: February 20, 2018. https://www.stat.cmu.edu/ ~arinaldo/Teaching/36752/S18/Scribed_Lectures/Feb20.pdf. Scribed lecture notes for 36-752: Advanced Probability Overview, Spring 2018, Carnegie Mellon University. 2018. David A. Stephens. The Borel-Cantelli Lemma. https://www.math.mcgill.ca/dstephens/ OldCourses/556-2006/Math556-BorelCantelli.pdf. Lecture notes for Math 556: Math- ematical Statistics I, McGill University. 2006. Jeffrey S. Rosenthal. A first look at rigorous probability theory. 2. ed., reprinted. New Jersey: World Scientific, 2010. 219 pp. isbn: 978-981-270-371-2. Marcelo Viana. Lectures on Lyapunov exponents. Cambridge studies in advanced math- ematics 145. Cambridge ; New York: Cambridge University Press, 2014. 202 pp. isbn: 978-1-107-08173-4. Peter Walters. An introduction to ergodic theory. 1. softcover printing. Graduate texts in mathematics 79. New York: Springer, 2000. 250 pp. isbn: 9780387951522. Artur Avila and Jairo Bochi. “On the Subadditive Ergodic Theorem”. In: Ergodic The- ory and Dynamical Systems 29.1 (Jan. 2009). Publisher: Cambridge University Press,pp. 1–16. doi: 10 . 1017 / S014338570800067X. url: https : / / doi . org / 10 . 1017 / S014338570800067X. Hillel Furstenberg and Harry Kesten. “Products of Random Matrices”. In: The Annals of Mathematical Statistics 31.2 (June 1960), pp. 457–469. issn: 0003-4851. doi: 10.1214/ aoms/1177705909. url: http://projecteuclid.org/euclid.aoms/1177705909 (visited on 08/14/2024). Walter Rudin. Functional analysis. 2nd ed. International series in pure and applied math- ematics. New York: McGraw-Hill, 1991. 424 pp. isbn: 9780070542365. Jean H. Gallier. Geometric methods and applications: for computer science and engineer- ing. 2nd ed. Texts in applied mathematics 38. New York: Springer, 2011. 680 pp. isbn: 9781441999603 9781441999610. Geon Ho Choe. Computational ergodic theory. Algorithms and computation in mathemat- ics v. 13. Berlin: Springer, 2005. 453 pp. isbn: 978-3-540-23121-9. Walter Gander. Algorithms for the QR-Decomposition. Tech. rep. 80-021. Re-typed in LATEX in October 2003, In memory of Prof. H. Rutishauser. Zürich, Switzerland: Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule, 1980. url: https: //people.inf.ethz.ch/gander/papers/qrneu.pdf.
dc.source.instname.none.fl_str_mv	instname:Universidad del Rosario
dc.source.reponame.none.fl_str_mv	reponame:Repositorio Institucional EdocUR
bitstream.url.fl_str_mv	https://repository.urosario.edu.co/bitstreams/d543af69-06e4-4768-87f8-daf367c03aad/download https://repository.urosario.edu.co/bitstreams/295020ba-9259-4319-827f-559fdf66e8d4/download https://repository.urosario.edu.co/bitstreams/52bc6b9e-619a-4133-95b8-0c01309b1882/download https://repository.urosario.edu.co/bitstreams/cf24de7a-b25b-4b0d-83d7-123055cce5e5/download https://repository.urosario.edu.co/bitstreams/1e33ff55-b93d-4f0c-abfd-1b14948cd1c3/download
bitstream.checksum.fl_str_mv	b2825df9f458e9d5d96ee8b7cd74fde6 3b6ce8e9e36c89875e8cf39962fe8920 3bafe456515dabaf1e15b69eb6601550 39d66c70ae1c63659844e55dfbe11464 4c720d7595ec9beace1df12a764d6083
bitstream.checksumAlgorithm.fl_str_mv	MD5 MD5 MD5 MD5 MD5
repository.name.fl_str_mv	Repositorio institucional EdocUR
repository.mail.fl_str_mv	edocur@urosario.edu.co
_version_	1837007685460951040
spelling	Artigiani, Mauro7b7040ca-f33d-4b80-bcbf-806c727430d5-1Martínez, Cristian73c3e106-5afb-4ee5-8229-c1f263876f29-1Bermúdez Guzmán, JulianaProfesional en Matemáticas Aplicadas y Ciencias de la ComputaciónProfesional en Matemáticas Aplicadas y Ciencias de la ComputaciónPregrado47eddce6-d6d7-44b2-86d5-d869861a22c6-12025-06-19T14:33:07Z2025-06-19T14:33:07Z2025-06-09In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These results provide the tools to quantify exponential growth rates and directional behavior in random matrix products. To visualize our theoretical conclusions, we present a series of simulations that illustrate the emergence of Lyapunov exponents and their predictive power in practical settings.In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These results provide the tools to quantify exponential growth rates and directional behavior in random matrix products. To visualize our theoretical conclusions, we present a series of simulations that illustrate the emergence of Lyapunov exponents and their predictive power in practical settings.40 ppapplication/pdfhttps://repository.urosario.edu.co/handle/10336/45741engUniversidad del RosarioEscuela de Ingeniería, Ciencia y TecnologíaPrograma de Matemáticas Aplicadas y Ciencias de la Computación - MACCAttribution-NonCommercial-NoDerivatives 4.0 InternationalAbierto (Texto Completo)http://creativecommons.org/licenses/by-nc-nd/4.0/http://purl.org/coar/access_right/c_abf2Robert G. Bartle. The Elements of Integration and Lebesgue Measure. 1st ed. Wiley Clas- sics Library v.92. Hoboken: John Wiley & Sons, Incorporated, 1995. 1 p. isbn: 9780471042228 9781118164488.Walter Rudin. Real and complex analysis. 3. ed., internat. ed., [Nachdr.] McGraw-Hill international editions Mathematics series. New York, NY: McGraw-Hill, 2013. 416 pp. isbn: 978-0-07-100276-9 978-0-07-054234-1.Donald L. Cohn. Measure Theory: Second Edition. Birkhäuser Advanced Texts Basler Lehrbücher. New York, NY: Springer New York, 2013. isbn: 978-1-4614-6955-1 978-1- 4614-6956-8. doi: 10.1007/978-1-4614-6956-8. url: https://link.springer.com/ 10.1007/978-1-4614-6956-8 (visited on 08/14/2024).Elias M. Stein and Rami Shakarchi. Real analysis: measure theory, integration, and Hilbert spaces. Princeton lectures in analysis v. 3. Princeton, N.J: Princeton University Press, 2005. 402 pp. isbn: 9780691113869.Manfred Einsiedler and Thomas Ward. Ergodic Theory: with a view towards Number The- ory. London: Springer London, 2011. isbn: 978-0-85729-020-5 978-0-85729-021-2. doi: 10.1007/978- 0- 85729- 021- 2. url: https://link.springer.com/10.1007/978- 0-85729-021-2 (visited on 08/14/2024).Daniel W. Stroock. Probability theory: an analytic view. 2nd ed. Cambridge New York: Cambridge University Press, 2011. 1 p. isbn: 978-0-521-76158-1 978-0-511-97424-3 978-1- 139-01188-4Sheldon M. Ross. Introduction to probability models. Tenth edition. Amsterdam Boston: Academic Press, an imprint of Elsevier, 2010. 1 p. isbn: 978-0-12-375686-2 978-0-12- 375687-9.Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to probability. 2nd ed. Optimiza- tion and computation series. Belmont: Athena scientific, 2008. isbn: 978-1-886529-23-6.Patrick Billingsley. Probability and measure. 3rd ed. Wiley series in probability and math- ematical statistics. New York: Wiley, 1995. 593 pp. isbn: 978-0-471-00710-4.Paul E. Pfeiffer. Probability for Applications. Springer Texts in Statistics. New York, NY: Springer New York, 1990. 679 pp. isbn: 978-1-4615-7676-1. doi: 10.1007/978-1-4615- 7676-1.Queen Mary University of London. Lecture 2: Second Language Acquisition. https:// qmplus . qmul . ac . uk / pluginfile . php / 2470175 / mod _ resource / content / 13 / L2 - 2022.pdf. Accessed: 2025-04-21. 2022.Alessandro Rinaldo. Lecture Notes: February 20, 2018. https://www.stat.cmu.edu/ ~arinaldo/Teaching/36752/S18/Scribed_Lectures/Feb20.pdf. Scribed lecture notes for 36-752: Advanced Probability Overview, Spring 2018, Carnegie Mellon University. 2018.David A. Stephens. The Borel-Cantelli Lemma. https://www.math.mcgill.ca/dstephens/ OldCourses/556-2006/Math556-BorelCantelli.pdf. Lecture notes for Math 556: Math- ematical Statistics I, McGill University. 2006.Jeffrey S. Rosenthal. A first look at rigorous probability theory. 2. ed., reprinted. New Jersey: World Scientific, 2010. 219 pp. isbn: 978-981-270-371-2.Marcelo Viana. Lectures on Lyapunov exponents. Cambridge studies in advanced math- ematics 145. Cambridge ; New York: Cambridge University Press, 2014. 202 pp. isbn: 978-1-107-08173-4.Peter Walters. An introduction to ergodic theory. 1. softcover printing. Graduate texts in mathematics 79. New York: Springer, 2000. 250 pp. isbn: 9780387951522.Artur Avila and Jairo Bochi. “On the Subadditive Ergodic Theorem”. In: Ergodic The- ory and Dynamical Systems 29.1 (Jan. 2009). Publisher: Cambridge University Press,pp. 1–16. doi: 10 . 1017 / S014338570800067X. url: https : / / doi . org / 10 . 1017 / S014338570800067X.Hillel Furstenberg and Harry Kesten. “Products of Random Matrices”. In: The Annals of Mathematical Statistics 31.2 (June 1960), pp. 457–469. issn: 0003-4851. doi: 10.1214/ aoms/1177705909. url: http://projecteuclid.org/euclid.aoms/1177705909 (visited on 08/14/2024).Walter Rudin. Functional analysis. 2nd ed. International series in pure and applied math- ematics. New York: McGraw-Hill, 1991. 424 pp. isbn: 9780070542365.Jean H. Gallier. Geometric methods and applications: for computer science and engineer- ing. 2nd ed. Texts in applied mathematics 38. New York: Springer, 2011. 680 pp. isbn: 9781441999603 9781441999610.Geon Ho Choe. Computational ergodic theory. Algorithms and computation in mathemat- ics v. 13. Berlin: Springer, 2005. 453 pp. isbn: 978-3-540-23121-9.Walter Gander. Algorithms for the QR-Decomposition. Tech. rep. 80-021. Re-typed in LATEX in October 2003, In memory of Prof. H. Rutishauser. Zürich, Switzerland: Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule, 1980. url: https: //people.inf.ethz.ch/gander/papers/qrneu.pdf.instname:Universidad del Rosarioreponame:Repositorio Institucional EdocURExponentes de LyapunovMatrices aleatoriosCaosTeoría ergódicaLyapunov ExponentsRandom MatricesChaosErgodic TheoryLyapunov Exponents to Predict the Behavior of the Product of Random MatricesExponentes de Lyapunov para predecir el comportamiento del producto de matrices aleatoriasbachelorThesisTrabajo de gradohttp://purl.org/coar/resource_type/c_7a1fEscuela de Ingeniería, Ciencia y TecnologíaBogotáLICENSElicense.txtlicense.txttext/plain1483https://repository.urosario.edu.co/bitstreams/d543af69-06e4-4768-87f8-daf367c03aad/downloadb2825df9f458e9d5d96ee8b7cd74fde6MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8899https://repository.urosario.edu.co/bitstreams/295020ba-9259-4319-827f-559fdf66e8d4/download3b6ce8e9e36c89875e8cf39962fe8920MD54ORIGINALLyapunov_Exponents_to_Predict_Bermudez_Guzman_Juliana.pdfLyapunov_Exponents_to_Predict_Bermudez_Guzman_Juliana.pdfapplication/pdf667354https://repository.urosario.edu.co/bitstreams/52bc6b9e-619a-4133-95b8-0c01309b1882/download3bafe456515dabaf1e15b69eb6601550MD55TEXTLyapunov_Exponents_to_Predict_Bermudez_Guzman_Juliana.pdf.txtLyapunov_Exponents_to_Predict_Bermudez_Guzman_Juliana.pdf.txtExtracted texttext/plain74196https://repository.urosario.edu.co/bitstreams/cf24de7a-b25b-4b0d-83d7-123055cce5e5/download39d66c70ae1c63659844e55dfbe11464MD56THUMBNAILLyapunov_Exponents_to_Predict_Bermudez_Guzman_Juliana.pdf.jpgLyapunov_Exponents_to_Predict_Bermudez_Guzman_Juliana.pdf.jpgGenerated Thumbnailimage/jpeg2623https://repository.urosario.edu.co/bitstreams/1e33ff55-b93d-4f0c-abfd-1b14948cd1c3/download4c720d7595ec9beace1df12a764d6083MD5710336/45741oai:repository.urosario.edu.co:10336/457412025-06-20 03:02:40.496http://creativecommons.org/licenses/by-nc-nd/4.0/Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttps://repository.urosario.edu.coRepositorio institucional EdocURedocur@urosario.edu.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

Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices

Publicaciones similares