Nonlinear Mathematics (Dover Books on Mathematics)

"We are surrounded and deeply involved, in the natural world, with nonlinear events which are not necessarily mathematical," the authors write. "For example … the nonlinear problem of pedaling a bicycle up and down a hillside. On a grand scale … the struggle for existence between two species, one of which preys exclusively on the other." This book is for mathematicians and researchers who believe that "nonlinear mathematics is the mathematics of today"; it is also for economists, engineers, operations analysts, "the reader who has been thus bemused into an artificially linear conception of the universe."
Nonlinear Mathematics is the first attempt to consider the widest range of nonlinear topic found in the scattered literature. Accessible to nonmathematics professionals as well as college seniors and graduates, it offers a discussion both particular and broad enough to stimulate research toward a unifying theory of nonlinear mathematics. Ideas are presented "according to existence and uniqueness theorems, characterization (e.g., stability and asymptotic behavior), construction of solutions, convergence, approximation, and errors." Contents include:
Linear and nonlinear transformations
Nonlinear algebraic and transcendental equations
Nonlinear optimization; nonlinear programming and systems of inequalities
Nonlinear ordinary differential equations
Introduction to automatic control and the Pontryagin principle
Linear and nonlinear prediction theory
Exercises are given as guides through theorem proofs and methodology. Extensive bibliographies are appended to each section.
"We have tried," write the authors, "to give more than a compilation of known techniques, to convey a spirit. In this spirit we integrate different topics of mathematics to show their interrelations, to stimulate inventiveness through variety, and to emphasize their origins from interaction with natural phenomena in physics, engineering, and economics." This examination of nonlinear mathematics gives a panoramic introduction (suitable for self-study) to a crucial field in contemporary research.