Soluciones periódicas en ecuaciones diferenciales singulares

: tablas, figuras

Autores:: Gutiérrez Gutiérrez, Alexánder

Tipo de recurso:: Book

Fecha de publicación:: 2017

Institución:: Universidad Tecnológica de Pereira

Repositorio:: Repositorio Institucional UTP

Idioma:: spa

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dc.title.none.fl_str_mv	Soluciones periódicas en ecuaciones diferenciales singulares
title	Soluciones periódicas en ecuaciones diferenciales singulares
spellingShingle	Soluciones periódicas en ecuaciones diferenciales singulares 510 - Matemáticas Matemáticas - Enseñanza Ecuaciones - Soluciones numéricas Números reales Ecuaciones diferenciales Ecuaciones diferenciales no lineales Aplicaciones (matemáticas) Números reales Electrostática Métodos de simulación
title_short	Soluciones periódicas en ecuaciones diferenciales singulares
title_full	Soluciones periódicas en ecuaciones diferenciales singulares
title_fullStr	Soluciones periódicas en ecuaciones diferenciales singulares
title_full_unstemmed	Soluciones periódicas en ecuaciones diferenciales singulares
title_sort	Soluciones periódicas en ecuaciones diferenciales singulares
dc.creator.fl_str_mv	Gutiérrez Gutiérrez, Alexánder
dc.contributor.author.none.fl_str_mv	Gutiérrez Gutiérrez, Alexánder
dc.subject.ddc.none.fl_str_mv	510 - Matemáticas
topic	510 - Matemáticas Matemáticas - Enseñanza Ecuaciones - Soluciones numéricas Números reales Ecuaciones diferenciales Ecuaciones diferenciales no lineales Aplicaciones (matemáticas) Números reales Electrostática Métodos de simulación
dc.subject.armarc.none.fl_str_mv	Matemáticas - Enseñanza Ecuaciones - Soluciones numéricas Números reales
dc.subject.proposal.spa.fl_str_mv	Ecuaciones diferenciales Ecuaciones diferenciales no lineales Aplicaciones (matemáticas) Números reales Electrostática Métodos de simulación
description	: tablas, figuras
publishDate	2017
dc.date.issued.none.fl_str_mv	2017
dc.date.accessioned.none.fl_str_mv	2024-10-16T19:39:11Z
dc.date.available.none.fl_str_mv	2024-10-16T19:39:11Z
dc.type.none.fl_str_mv	Libro
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dc.type.content.none.fl_str_mv	Text
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dc.identifier.isbn.none.fl_str_mv	978-958-722-280-7
dc.identifier.uri.none.fl_str_mv	https://hdl.handle.net/11059/15392
dc.identifier.eisbn.none.fl_str_mv	978-958-722-694-2
dc.identifier.doi.none.fl_str_mv	https://doi.org/10.22517/9789587226942
dc.identifier.instname.none.fl_str_mv	Universidad Tecnológica de Pereira
dc.identifier.reponame.none.fl_str_mv	Repositorio Universidad Tecnológica de Pereira
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identifier_str_mv	978-958-722-280-7 978-958-722-694-2 Universidad Tecnológica de Pereira Repositorio Universidad Tecnológica de Pereira
url	https://hdl.handle.net/11059/15392 https://doi.org/10.22517/9789587226942 https://repositorio.utp.edu.co/home
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dc.relation.references.none.fl_str_mv	[1] S. Ai y J. A. Pelesko. Dynamics of a canonical electrostatic MEMS/NEMS system, Journal of Dynamics and Differential Equations, vol 20, pp 609- 641, 2008. [2] C. Bereanu y J. Mawhin. Existence and multiplicity results for some nonlinear problems with singular φ-laplacian, Journal Differential Equations, 243(2):536-557, 2007. [3] L. E. J. Brouwer. Über abbildung von mannigfaltigkeiten, Mathematische Annalen, 71(1):97-115, 1911. [4] A. Capietto, J. Mawhin y F. Zanolin. Continuation theorems for periodic perturbations of autonomous systems, Transactions Of The American Mathematical Society, vol. 329, pp 41-72, 1992. [5] G. Carapella y G. Costabile. Ratchet Effect: Demonstration of a Relativistic Fluxon Diode, Physical Review Letters, 87, 077002 (4pp)(7), August 2001. [6] P. B. Chu, P. R. Nelson, M. L. Tachiki y K. S. J. Pister. Dynamics of polysilicon parallel-plate electrostatic actuator, Sensors and Actuators A: Physical, vol. 52, pp. 216-220, 1996. [7] J. A. Cid y P. J. Torres. On the existence and stability of periodic solutions for pendulum-like equations with friction and φ-laplacian, Discrete and Continuous Dynamical Systems 33, 141-153, 2013. [8] K. L. Cooke y W. Huang. On the problem of linearization for state- dependent delay differential equations, Proceedings on the American Mathematical Society, 124(5):1417-1426, 1996 [9] C. De Coster y P. Habets. Two-Point Boundary Value Problems: Lower and Upper Solutions, vol. 205. Mathematics in Science and Engineering, 2006. [10] K. Deimling. Nonlinear Functional Analysis, Berlin, 1980. [11] J. P. Den Hartog. Mechanical Vibrations, New York, 1956. [12] H. Delavari, A. Asadbeigi y O. Heydarnia. Synchronization of MicroElectro-Mechanical-Systems in Finite Time, Discontinuity, Nonlinearity, and Complexity, 173, 2015. [13] R. D. Driver. A two-body problem of classical electrodynamics:the onedimensional case, Annals of Physics, vol. 21, pp 122-142, 1963. [14] M. Elwenspoek y W. Remco. Mechanical microsensors, Springer Science & Business Media, 2012. [15] C. Escudero, A. Rivera y P. Torres. Chemical Oscillations out the Chemical Noise, SIAM Journal Applied Dynamical System, 10, 960-986, 2011. [16] A. Fonda y F. Zanolin. Bounded solutions of nonlinear second order ordinary differential equations, Discrete And Continuous Dynamical Systems, 4(1):91-98, 1998. [17] J. Galán, D. Núñez y A. Rivera. Quantitative stability of certain families of periodic solutions in the Sitnikov problem, arXiv:1612.07254[math.DS] Dec. 21 (2016). [18] A. Gutiérrez. Ecuación de Lazer-Solimini con retraso dependiente del estado: una demostración alternativa, Revista Colombiana De Matemáticas, vol.47 fasc.1 pp 29-38, 2013. [19] A. Gutiérrez y P. J. Torres. Periodic solutions of Liénard equation with one or two weak singularities, Differential Equations & Applications, 3(3):375- 384, 2011. [20] A. Gutiérrez y P. J. Torres. Nonautonomuos Saddle-Node Bifurcation in a Canonical Electrostatic MEMS, International Journal of Bifurcation and Chaos, 19, 187, 2013. [21] A. Gutiérrez y P. J. Torres. The Lazer-Solimini equation with statedependent delay, Applied Mathematics Letters, vol 25, pp 643-647, 2012. [22] A. Gutiérrez, A. Rivera y Daniel Núñez. Effects of voltage change in the dynamics in a Comb-Driver finger of electrostatic actuator, Preprint 23] D. Hah. A design method of comb-drive actuators for linear tuning characteristics in mechanically tunable optical filters, Microsystem Technologies, 1-8, 2015. [24] R. Halk y P. J. Torres. On periodic solutions of second order differential equations with attractive repulsive singularities, Journal Differential Equations, 248:111-126, 2010. [25] R. Hakl y M. Zamora. On the open problems connected to the results of Lazer and Solimini, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 144, pp 109-118, 2014. 26] T. Hirano, T. Furuhata, K. J. Gabrie y H. Fujita. Design, fabrication and operation of sub-micron gap electrostatic comb-drive actuators, Journal of Microelectromechanical Systems, vol. 1, 52-9, 1992. [27] P. Korman. Global solution curves for several classes of singular periodic problems, Nonlinear Analysis, vol 34, pp 1-16, 2017. 28] A. C. Lazer y S. Solimini. On Periodic solutions of nonlinear differential equations with singularities, Proceedings Of The American Mathematical Society, vol 99, n 1, pp 109-114, 1987. [29] D. Lehotzky, T. Insperger y G. Stepan. Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays, Communications in Nonlinear Science and Numerical Simulation, Vol 35, pp 177?189, 2016 [30] M. Levi. Stability of the inverted pendulum-A topological explanation, SIAM Review, 30(4):639-644, Dic. 1985. [31] M. Levi y W. Weckesser. Stabilization of the inverted linearized pendulum by high frequency vibrations, SIAM Review, 37(2):219-223, Jun 1995 [32] J. Leray y J. Schauder. Topologie et équations fonctionnelles, Annales scientifiques de l’École Normale Supérieure, 51(3):45-78, 1934. [33] J. Llibre y R. Ortega. On the families of periodic orbits of the Sitnikov problem, SIAM Journal on Applied Dynamical Systems, 7, 561-576, 2008. [34] N. G. Lloyd. Degree theory, Cambridge Univ. Press, 1978 [35] S. Lu. A new result on the existence of periodic solutions for Liénard equations with a singularity of repulsive type, Journal of Inequalities and Applications, 37, 2017. [36] W. Magnus y S. Winkler. Hill’s Equation, Dover: New York, 1979. [37] J. Mawhin. Topological degree and boundary value problems for nonlinear differential equations, In Topological Methods for Ordinary Differential Equations, M. Furi y P. Zecca, Eds., volume 1537, pages 74-142. Lecture Notes in Mathematics, Springer, Berlin, Germany, 1993. [38] J. Mawhin. Equivalence Theorems for Nonlinear Operator Equations and Coincidence Degree Theory for Some Mappings in Locally Convex Topological Vector Spaces, J. Difference Equations Appl., vol 12, pp 610-636, 1972. [39] M. Nagumo. On the periodic solution of an ordinary differential equation of second order, Zenkoku Shijou Suugaku Danwakai, (1944), 54-61 (en japones). English translation in Mitio Nagumo collected papers, SpringerVerlag, 1993. [40] H. C. Nathanson, W. E. Newell, R. A: Wickstrom y J.R., Davis. The resonant gate transistor, IEEE Transactions on Electron Devices, vol.14, no.3, pp. 117-133, 1967. 41] H.C. Nathanson y R.A. Wickstrom. A resonant-gate silicon surface transistor with high-Q band-pass properties, Applied Physics Letters, vol.7, pp.84-86, 1965. 42] F. I. Njoku y P. Omari. Stability properties of periodic solutions of a Duffing equation in the presence of upper and lower solutions, Applied Mathematics and Computation, 135:471-490, 2003. [43] D. Núñez. The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Analysis, 51:1207-1222, 2002. [44] R. Ortega. Stability and index of periodic solutions of an equation of Duffing type, Bollettino dell’Unione Matematica Italiana, 7:533-546, 1989. 45] R. Ortega. Stability of a periodic problem of Ambrosetti-Prodi type, Differential and Integral Equations, 3:275-284, 1990. [46] R. Ortega. Topological degree and stability of periodic solutions for certain differential equations, Journal of the London Mathematical Society, 42:505-516, 1990. [47] R. Ortega. Some applications of the topological degree to stability theory, In Topological methods in differential equations and inclusions, A. Granas y M. Frigon, Eds., pages 377-409. Dordrecht: Kluwer, 1995. [48] R. Ortega y A. Rivera. Global bifurcations from the center of mass in the Sitnikov problem, Discrete and Continuous Dynamical Systems Series B. Vol 14, Septiembre, pp 719-732, 2010. [49] S. Pérez-Gonzáleza, J. Torregrosa y P.J. Torres. Existence and uniqueness of limit cycles for generalized φ-Laplacian Liénard equations, Journal of Mathematical Analysis and Applications, Vol 439, Issue 2, pp 745-765, 2016. [50] M. Protter y H. Weinberger. Maximum Principles in Differential Equations, Englewood Cliffs, NJ, 1967. 51] N. R. Quintero, R. Alvarez-Nodarse y J. A Cuesta. Ratchet effect on a relativistic particle driven by external forces, Journal of Physics A, 44, 425205 (10pp), 2011. [52] A. Rivera. Periodic solutions in the generalized Sitnikov (N + 1)-body Problem, SIAM Journal on Applied Dynamical Systems, 12, 1515-1540, 2013. [53] J. B. Starr. Squeeze-film damping in solid-state accelerometers, IEEE 4th Technical Digest on Solid-State Sensor and Actuator Workshop, pp. 44-47, 1990. [54] G.I. Taylor. The coalescence of closely spaced drops when they are at different electric potentials, Proceedings of the Royal Society A, 306, 423-434, 1968. [55] P. J. Torres. Nondegeneracy of the periodically forced Liénard differential equation with φ-laplacian, Communications in Contemporary Mathematics, 13:283-292, 2011. [56] P. J. Torres. Mathematical Models with Singularities: A zoo of singular creatures, Atlantis Press:Paris, 124p, 2015. [57] S. P. Travis. A one-dimensional two-body problem of classical electrodynamics, SIAM Journal on Applied Mathematics, vol 28, n 3, pp 611-632, 1975. [58] H.-O. Walther, F. Hartung, T. Krisztin y J. Wu. Functional differential equations with state-dependent delay: theory and applications, Handbook of Differential Equations: Ordinary Differential Equations, A. Canada, P. Drbek, A. Fonda (Eds.), Elsevier: North-Holand, vol 3, pp 435-545, 2006. [59] Y. Xin y Z. Cheng. Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument, Advances in Difference Equations, 41, 2016. [60] E. Yakaboylu, M. Klaiber, H. Bauke, K. Hatsagortsyan y C. H. Keitel. Relativistic features and time delay of laser-induced tunnel ionization, Physical Review A, vol 88, n 6, pp 063421, 2013. [61] M. I. Younis. MEMS Linear and Nonlinear Statics and Dynamics, Springer: New York, 452p, 2011. [62] W.M. Zhang, H. Yan, Z.K. Peng, y G. Meng. Electrostatic pull-in instability in MEMS/NEMS: A review., Sensors and Actuators A: Physical, 214, pp 719-732, 2014. [63] W. Zhang y G. Meng. Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS, 30th Annual Conference of IEEE Industrial Electronics Society, 2004. IECON 2004, 119:291-299, 2005. [64] Z. Zhang y R. Yuan. Existence of positive periodic solutions for the Liénard differential equations with weakly repulsive singularity, Acta Applicandae Mathematicae, 111(2):171-178, 2010. [65] A. Zitan y R. Ortega. Existence of asymptotically stable periodic solutions of a forced equation of Liénard type, Nonlinear Analysis, 22:993-1003, 1994
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spelling	Atribución-NoComercial-CompartirIgual 4.0 Internacional (CC BY-NC-SA 4.0)Manifiesto (Manifestamos) en este documento la voluntad de autorizar a la Biblioteca Jorge Roa Martínez de la Universidad Tecnológica de Pereira la publicación en el Repositorio institucional (http://biblioteca.utp.edu.co), la versión electrónica de la OBRA titulada: La Universidad Tecnológica de Pereira, entidad académica sin ánimo de lucro, queda por lo tanto facultada para ejercer plenamente la autorización anteriormente descrita en su actividad ordinaria de investigación, docencia y publicación. La autorización otorgada se ajusta a lo que establece la Ley 23 de 1982. Con todo, en mi (nuestra) condición de autor (es) me (nos) reservo (reservamos) los derechos morales de la OBRA antes citada con arreglo al artículo 30 de la Ley 23 de 1982. En concordancia suscribo (suscribimos) este documento en el momento mismo que hago (hacemos) entrega de mi (nuestra) OBRA a la Biblioteca “Jorge Roa Martínez” de la Universidad Tecnológica de Pereira. Manifiesto (manifestamos) que la OBRA objeto de la presente autorizaciónhttps://creativecommons.org/licenses/by-nc-sa/4.0/http://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessGutiérrez Gutiérrez, Alexánder2024-10-16T19:39:11Z2024-10-16T19:39:11Z2017978-958-722-280-7https://hdl.handle.net/11059/15392978-958-722-694-2https://doi.org/10.22517/9789587226942Universidad Tecnológica de PereiraRepositorio Universidad Tecnológica de Pereirahttps://repositorio.utp.edu.co/home: tablas, figurasEste libro es un compendio del trabajo de los últimos seis años y los modelos matemáticos tienen en común dos cosas: singularidades no lineales y coeficientes periódicos, se entiende por singularidad el límite al infinito del término no lineal cuando la variable de estado se acerca a un punto. El estudio de los modelos con singularidades es importante debido a que modelan procesos del mundo real que surgen en forma natural al considerar, por ejemplo, fuerzas gravitatorias, electromagnéticas o intermoleculares. En particular, centramos nuestra atención en el estudio de la existencia, unicidad y estabilidad de soluciones periódicas. Las herramientas matemáticas elegidas para abordar el estudio son amplias y comprende técnicas del análisis no lineal como son: teoremas de punto fijo, método de sub y super soluciones, y topológicas como: grado topológico, grado de coincidencia. Los resultados obtenidos se han logrado gracias a la colaboración de mis colegas: Profesor Pedro J. Torres de la Universidad de Granada y a los profesores Andrés M. Rivera y Daniel Núñez de la Universidad Javeriana, sede Cali.Capítulo I--Soluciones periódicas de la ecuación en Liénard con una o dos singularidades débiles--Esquema de la demostración del resultado principal--Soluciones periódicas asintóticamente estables--Ejemplos y comparación de resultados--Comentarios finales--Capítulo II--Ecuación de Lazer-Solimini con retardo--Esquemas de demostración de los resultados principales--Demostración alternativa--Comentarios finales--Capítulo III--Modelo de accionador electrostático canónico--Bifurcación nodo-silla en MEMS--Comentarios finales--Capítulo IV--Modelo de accionador electrostático tipo peine--El modelo del accionador tipo peine--Soluciones periódicas persistentes--Comentarios finales--Apéndice A--Teoría del grado--Grado de coincidencia--Principio de continuación global de Leray-Schauder--Estabilidad e índice de soluciones periódicas--Sub-super soluciones90 páginasapplication/pdfspaUniversidad Tecnológica de PereiraPereira510 - MatemáticasMatemáticas - EnseñanzaEcuaciones - Soluciones numéricasNúmeros realesEcuaciones diferencialesEcuaciones diferenciales no linealesAplicaciones (matemáticas)Números realesElectrostáticaMétodos de simulaciónSoluciones periódicas en ecuaciones diferenciales singularesLibroinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_2f33Textinfo:eu-repo/semantics/book[1] S. Ai y J. A. Pelesko. Dynamics of a canonical electrostatic MEMS/NEMS system, Journal of Dynamics and Differential Equations, vol 20, pp 609- 641, 2008.[2] C. Bereanu y J. Mawhin. Existence and multiplicity results for some nonlinear problems with singular φ-laplacian, Journal Differential Equations, 243(2):536-557, 2007.[3] L. E. J. Brouwer. Über abbildung von mannigfaltigkeiten, Mathematische Annalen, 71(1):97-115, 1911.[4] A. Capietto, J. Mawhin y F. Zanolin. Continuation theorems for periodic perturbations of autonomous systems, Transactions Of The American Mathematical Society, vol. 329, pp 41-72, 1992.[5] G. Carapella y G. Costabile. Ratchet Effect: Demonstration of a Relativistic Fluxon Diode, Physical Review Letters, 87, 077002 (4pp)(7), August 2001.[6] P. B. Chu, P. R. Nelson, M. L. Tachiki y K. S. J. Pister. Dynamics of polysilicon parallel-plate electrostatic actuator, Sensors and Actuators A: Physical, vol. 52, pp. 216-220, 1996.[7] J. A. Cid y P. J. Torres. On the existence and stability of periodic solutions for pendulum-like equations with friction and φ-laplacian, Discrete and Continuous Dynamical Systems 33, 141-153, 2013.[8] K. L. Cooke y W. Huang. On the problem of linearization for state- dependent delay differential equations, Proceedings on the American Mathematical Society, 124(5):1417-1426, 1996[9] C. De Coster y P. Habets. Two-Point Boundary Value Problems: Lower and Upper Solutions, vol. 205. Mathematics in Science and Engineering, 2006.[10] K. Deimling. Nonlinear Functional Analysis, Berlin, 1980.[11] J. P. Den Hartog. Mechanical Vibrations, New York, 1956.[12] H. Delavari, A. Asadbeigi y O. Heydarnia. Synchronization of MicroElectro-Mechanical-Systems in Finite Time, Discontinuity, Nonlinearity, and Complexity, 173, 2015.[13] R. D. Driver. A two-body problem of classical electrodynamics:the onedimensional case, Annals of Physics, vol. 21, pp 122-142, 1963.[14] M. Elwenspoek y W. Remco. Mechanical microsensors, Springer Science & Business Media, 2012.[15] C. Escudero, A. Rivera y P. Torres. Chemical Oscillations out the Chemical Noise, SIAM Journal Applied Dynamical System, 10, 960-986, 2011.[16] A. Fonda y F. Zanolin. Bounded solutions of nonlinear second order ordinary differential equations, Discrete And Continuous Dynamical Systems, 4(1):91-98, 1998.[17] J. Galán, D. Núñez y A. Rivera. Quantitative stability of certain families of periodic solutions in the Sitnikov problem, arXiv:1612.07254[math.DS] Dec. 21 (2016).[18] A. Gutiérrez. Ecuación de Lazer-Solimini con retraso dependiente del estado: una demostración alternativa, Revista Colombiana De Matemáticas, vol.47 fasc.1 pp 29-38, 2013.[19] A. Gutiérrez y P. J. Torres. Periodic solutions of Liénard equation with one or two weak singularities, Differential Equations & Applications, 3(3):375- 384, 2011.[20] A. Gutiérrez y P. J. Torres. Nonautonomuos Saddle-Node Bifurcation in a Canonical Electrostatic MEMS, International Journal of Bifurcation and Chaos, 19, 187, 2013.[21] A. Gutiérrez y P. J. Torres. The Lazer-Solimini equation with statedependent delay, Applied Mathematics Letters, vol 25, pp 643-647, 2012.[22] A. Gutiérrez, A. Rivera y Daniel Núñez. Effects of voltage change in the dynamics in a Comb-Driver finger of electrostatic actuator, Preprint23] D. Hah. A design method of comb-drive actuators for linear tuning characteristics in mechanically tunable optical filters, Microsystem Technologies, 1-8, 2015.[24] R. Halk y P. J. Torres. On periodic solutions of second order differential equations with attractive repulsive singularities, Journal Differential Equations, 248:111-126, 2010.[25] R. Hakl y M. Zamora. On the open problems connected to the results of Lazer and Solimini, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 144, pp 109-118, 2014.26] T. Hirano, T. Furuhata, K. J. Gabrie y H. Fujita. Design, fabrication and operation of sub-micron gap electrostatic comb-drive actuators, Journal of Microelectromechanical Systems, vol. 1, 52-9, 1992.[27] P. Korman. Global solution curves for several classes of singular periodic problems, Nonlinear Analysis, vol 34, pp 1-16, 2017.28] A. C. Lazer y S. Solimini. On Periodic solutions of nonlinear differential equations with singularities, Proceedings Of The American Mathematical Society, vol 99, n 1, pp 109-114, 1987.[29] D. Lehotzky, T. Insperger y G. Stepan. Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays, Communications in Nonlinear Science and Numerical Simulation, Vol 35, pp 177?189, 2016[30] M. Levi. Stability of the inverted pendulum-A topological explanation, SIAM Review, 30(4):639-644, Dic. 1985.[31] M. Levi y W. Weckesser. 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Existence of asymptotically stable periodic solutions of a forced equation of Liénard type, Nonlinear Analysis, 22:993-1003, 1994PublicationLICENSElicense.txtlicense.txttext/plain; charset=utf-815543https://repositorio.utp.edu.co/bitstreams/e8753270-d640-46e5-a3d7-63c28134ca79/download73a5432e0b76442b22b026844140d683MD51ORIGINALSoluciones periódicas en ecuaciones diferenciales singulares (1).pdfapplication/pdf2419587https://repositorio.utp.edu.co/bitstreams/f9df6e4e-c30e-4034-bc53-cd82d5d0475e/downloadec6b80b34feb941986c6ed136ef3d9cfMD56THUMBNAILImagen10.pngimage/png105872https://repositorio.utp.edu.co/bitstreams/cbebf042-b293-4ad6-9624-64b5cbef2378/download325c8da600d830824ef80c9948f887d1MD53Soluciones periódicas en ecuaciones diferenciales singulares.pdf.jpgSoluciones periódicas en ecuaciones diferenciales singulares.pdf.jpgGenerated Thumbnailimage/jpeg9292https://repositorio.utp.edu.co/bitstreams/bf4dc779-8b2f-4272-b6ac-dcf38c2cb58e/download3fed7b3de1fa71076019c773f6402ee5MD55Soluciones periódicas en ecuaciones diferenciales singulares (1).pdf.jpgSoluciones periódicas en ecuaciones diferenciales singulares (1).pdf.jpgGenerated Thumbnailimage/jpeg9292https://repositorio.utp.edu.co/bitstreams/0e0869b6-9d71-42dc-9d6b-455f4bd63798/download3fed7b3de1fa71076019c773f6402ee5MD58TEXTSoluciones periódicas en ecuaciones diferenciales singulares.pdf.txtSoluciones periódicas en ecuaciones diferenciales singulares.pdf.txtExtracted texttext/plain101226https://repositorio.utp.edu.co/bitstreams/9e4c39b5-581f-481f-b305-dfae723accad/downloadcfae76c9c96ed73fef94d4b100a303acMD54Soluciones periódicas en ecuaciones diferenciales singulares (1).pdf.txtSoluciones periódicas en ecuaciones diferenciales singulares (1).pdf.txtExtracted texttext/plain101227https://repositorio.utp.edu.co/bitstreams/35072678-8153-41f7-ae0e-2df06190de98/downloadcc2067d11de85e973aab1cc3a71403e1MD5711059/15392oai:repositorio.utp.edu.co:11059/153922024-10-23 09:35:20.323https://creativecommons.org/licenses/by-nc-sa/4.0/Manifiesto (Manifestamos) en este documento la voluntad de autorizar a la Biblioteca Jorge Roa Martínez de la Universidad Tecnológica de Pereira la publicación en el Repositorio institucional (http://biblioteca.utp.edu.co), la versión electrónica de la OBRA titulada: La Universidad Tecnológica de Pereira, entidad académica sin ánimo de lucro, queda por lo tanto facultada para ejercer plenamente la autorización anteriormente descrita en su actividad ordinaria de investigación, docencia y publicación. 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Manifiesto (manifestamos) que la OBRA objeto de la presente autorizaciónopen.accesshttps://repositorio.utp.edu.coRepositorio de la Universidad Tecnológica de 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