Jane Cronin - Böcker

Designed for a rigorous first course in ordinary differential equations, Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition includes basic material such as the existence and properties of solutions, linear equations, autonomous equations, and stability as well as more advanced topics in periodic solutions of nonlinear equations. Requiring only a background in advanced calculus and linear algebra, the text is appropriate for advanced undergraduate and graduate students in mathematics, engineering, physics, chemistry, or biology.This third edition of a highly acclaimed textbook provides a detailed account of the Bendixson theory of solutions of two-dimensional nonlinear autonomous equations, which is a classical subject that has become more prominent in recent biological applications. By using the Poincaré method, it gives a unified treatment of the periodic solutions of perturbed equations. This includes the existence and stability of periodic solutions of perturbed nonautonomous and autonomous equations (bifurcation theory). The text shows how topological degree can be applied to extend the results. It also explains that using the averaging method to seek such periodic solutions is a special case of the use of the Poincaré method.

This book is an introduction to the study of mathematical models of electrically active cells, which play an essential role in, for example, nerve conduction and cardiac functions. This is an important and vigorously researched field. In the book, Dr Cronin synthesizes and reviews this material and provides a detailed discussion of the Hodgkin-Huxley model for nerve conduction, which forms the cornerstone of this body of work. Her treatment includes a derivation of the Hodgkin-Huxley model, which is a system of four nonlinear differential equations; a discussion of the validity of this model; and a summary of some of the mathematical analysis carried out on this model. Special emphasis is placed on singular perturbation theory, and arguments, both mathematical and physiological, for using the perturbation viewpoint are presented.

This book is an introduction to the study of mathematical models of electrically active cells, which play an essential role in, for example, nerve conduction and cardiac functions. This is an important and vigorously researched field. In the book, Dr Cronin synthesizes and reviews this material and provides a detailed discussion of the Hodgkin-Huxley model for nerve conduction, which forms the cornerstone of this body of work. Her treatment includes a derivation of the Hodgkin-Huxley model, which is a system of four nonlinear differential equations; a discussion of the validity of this model; and a summary of some of the mathematical analysis carried out on this model. Special emphasis is placed on singular perturbation theory, and arguments, both mathematical and physiological, for using the perturbation viewpoint are presented.