Ordinary Differential Equations: Linear And Nonlinear Systems, Dynamical Systems And Applications

The theory of ordinary differential equations is addressed in detail in this textbook, and is split into three sections: linear equations and systems, the general theory of nonlinear systems, and the theory of dynamical systems. These topics can be taken together or studied independently.

In addition to standard materials on the theory of ordinary differential equations, this textbook specialises in covering non-standard materials related to this theory including the theory of linear equation and systems with holomorphic coefficients; Kneser's theorem on the complexity of the set of solutions in the absence of uniqueness; the method of sub- and supersolutions for cooperative systems; and a detailed construction of the global bifurcation diagrams for some parametric classes of one-dimensional boundary value problems, which are pivotal for applications of the theory.

This is a self-contained, rigorous treatment of ordinary differential equations that is complemented by a variety of illustrating examples of the theory in practice. Many of these examples are related to models in Physics and Applied Sciences, making them suitable for students in Physics, Chemistry, Engineering, Mathematical Biology, Economics and Ecology as well as in Mathematics. Each chapter contains exercises to test students' understanding of the topic and concludes with some historical notes and further discussions.

Julián López-Gómez is an advisor of 15 PhD theses, author of around 220 research papers, most published in leading journals, and 10 books, and editor of a further five, among them the Proceedings of the 10th AIMS Conference (Madrid 2014, the largest conference on Differential Equations ever held, with over 2,300 participants). He has delivered invited talks at 88 conferences, co-organized 33 international scientific events, served as member of the editorial boards of Advanced Nonlinear Studies, Nonlinear Analysis, RIMUT of Trieste, Abstract and Applied Analysis, Journal of Applied Mathematics, Mathematics, and Discrete and Continuous Dynamical Systems, and has elaborated reports for over 80 international institutions and scientific interdisciplinary journals. His research interests encompass a large variety of fields including PDEs, ODEs, Numerical Analysis, Nonlinear Analysis, Operator Theory, Complex Analysis, Topology, Geometry, as well as a number of applications in Chemistry, Physics, Engineering and Population Dynamics.

He has been guest professor and delivered Master's courses at the Mathematical Institute of Züversity, the National Center for Theoretical Sciences of Taiwan at Hsinchu, the Technical University of Xi'an (PRC), the Department of Mathematics of the Universities of Trieste (Italy) and Brasília (Brazil), the National Institute for the Development of the Chemical Industry at Santa Fe, the Universities of Buenos Aires and Bahía Blanca (Argentina), as well as at the University of Los Andes at Mérida (Venezuela). He has done scientific tours through Argentina (1988, Santa Fe, Córdoba, San Juan, Rosario, Buenos Aires, La Plata), South Korea (2011, Seoul, Pohang, Pusan), Taiwan (2013, Taipei, Hsinchu, Kaohsiung), Japan (2023, Tokyo, Ibaraki, Meiji, Waseda, Hiroshima and Fukuoka), Brazil (2024, Río de Janeiro, Salvador de Bahía, Belém do Pará and Brasília), and has delivered seminars at Mathematical Institutes worldwide: Kyoto, Hiroshima, Kitakyushu, Fukuoka, Beijing, Xi'an, Tianjin, Taiyuan, Banff, Toronto, Winnipeg, Madison, San Antonio, Miami, Orlando, Dallas, Sydney, and the Courant Institute of New York University, where he had the opportunity to discuss mathematics with L Nirenberg and S R S Varadhan, Abel prize recipients.

He has collaborated with many influential mathematicians, like H Amann, E N Dancer, P Omari, P Pucci, F Zanolin, L Véron and P H Rabinowitz, former President of the Section of Mathematics of the National Academy of Sciences (USA). His current Research Team, funded by a Research Grant from the Ministry of Science and Innovation of Spain since December 2021, includes P Omari, F Zanolin and P H Rabinowitz.

He has been included in the list of 2% most influential scientists elaborated by the Standford University, and, according to the words of the organizers of his last course in Brasília: "He is an influential mathematician, internationally recognized for his profound contributions to Nonlinear Analysis, Bifurcation Theory and their applications in Partial Differential Equations."

Andrea Tellini is a Ramón y Cajal Research Fellow at Polytechnic University of Madrid, where he also teaches a course on Ordinary Differential Equations I in the Bachelor Degree in Mathematics. After completing his Bachelor and Master's studies at the University of Udine (Italy) as a student of the Scuola Superiore of the same university, he obtained his PhD at Complutense University of Madrid in 2013, under the supervision of Julián López-Gómez and Marcela Molina-Meyer. Following this, he occupied post-doc positions in Paris at the Centre d'Analyse et de Mathématique Sociales (France), under the supervision of Henri Berestycki, and at Autónoma University of Madrid, under the supervision of Carlos Mora-Corral, before obtaining a position at Polytechnic University of Madrid, where he has been working since 2018.

He has authored 18 research papers on differential equations, both ordinary and partial, where he has investigated the existence and multiplicity of solutions for reaction-diffusion problems related to variants of the logistic equation, complementing the analytical results with numerical simulations. He has supervised several bachelor and master's theses related to the analysis and numerical simulations of models involving differential equations and, currently, he is co-supervising two PhD students together with Julián López-Gómez.