Ordinary Differential Equations with Applications
Springer Science & Business Media, 1999 - 561 síđur
This book developed over 20 years of the author teaching the course at his own university. It serves as a text for a graduate level course in the theory of ordinary differential equations, written from a dynamical systems point of view. It contains both theory and applications, with the applications interwoven with the theory throughout the text. The author also links ordinary differential equations with advanced mathematical topics such as differential geometry, Lie group theory, analysis in infinite-dimensional spaces and even abstract algebra. The second edition incorporates corrections and improvements of the original text. New material includes a proof of the Grobman-Hartman theorem for flows based on the Lie derivative, more extensive treatment of the Euler-Lagrange equation and its applications, a proof of Noether's theorem on the existence of first integrals in the presence of symmetries and a new section on dynamic bifurcation with a proof of Pontryagin's formula. The impressive array of existing exercises has been more than doubled in size and further enhanced in scope, providing mathematics, physical science and engineering graduate students with a thorough introduction to the theory and application of ordinary differential equations. Reviews of the first edition: ``As an applied mathematics text on linear and nonlinear equations, the book by Chicone is written with stimulating enthusiasm. It will certainly appeal to many students and researchers.'' -- F. Verhulst, SIAM Review ``The author writes lucidly and in an engaging conversational style. His book is wide-ranging in its subject matter, thorough in its presentation, and written at a generally high level of generality, detail, and rigor.'' -- D. S. Shafer, Mathematical Reviews.
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apply approximation averaging Banach space bifurcation bifurcation theory boundary bounded called complete compute consider constant contained continuous coordinates corresponding curve deﬁned deﬁnition denote depends derivative determine differential diﬀerential equation direction eigenvalues equivalent example Exercise existence fact Figure ﬁrst ﬂow follows force formula function fundamental given Hamiltonian hyperbolic identity important initial condition initial value integral interval invariant invertible limit cycle linear linear system manifold matrix method Moreover motion multiple norm obtain operator origin oscillator parameter partial particular periodic orbit periodic solution perturbed phase portrait plane positive proof Proposition prove resonant respect rest point result simple smooth solution space stable subset Suppose tangent term theorem theory tion transformation unique unperturbed unstable usual value problem variables vector ﬁeld zero