Research Catalog

Title

Differential equations, dynamical systems, and linear algebra [by] Morris W. Hirsch and Stephen Smale.

Author

Hirsch, Morris W., 1933-

Publication

New York, Academic Press [1974]

Items in the Library & Off-site

Details

Additional Authors

Smale, Stephen, 1930-

Description

xi, 358 pages illustrations; 25 cm.

Summary

This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces the authors have included a self-contained treatment of that subject.

Series Statement

Pure and applied mathematics; a series of monographs and textbooks ; v. 60

Uniform Title

Pure and applied mathematics (Academic Press) ; 60.

Subject

• Differential equations
• Algebras, Linear
• Mathematics
• Mathematics
• Mathématiques
• Algèbre
• Algebras, Linear
• Differential equations
• Differentiaalvergelijkingen
• Dynamische systemen
• Lineaire algebra
• Dynamique différentiable
• Opérateurs, Théorie des
• Systèmes dynamiques
• Algèbre linéaire
• Equation différentielle
• Système dynamique

Bibliography (note)

• Bibliography: p. [341]-342.

Contents

First examples : The simplest examples ; Linear systems with constant coefficients -- Newton’s equation and Kepler’s law : Harmonic oscillators ; Some calculus background ; Conservative force fields ; Central force fields ; States ; Elliptical planetary orbits -- Linear systems with constant coefficients and real eigenvalues : Basic linear algebra ; Real eigenvalues ; Differential equations with real, distinct eigenvalues ; Complex eigenvalues -- Linear systems with constant coefficients and complex eigenvalues : Complex vector spaces ; Real operators with complex eigenvalues ; Application of complex linear algebra to differential equations -- Linear systems and exponentials of operators : Review of topology in Rn ; New norms for old ; Exponentials of operators ; Homogeneous linear systems ; A nonhomogeneous equation ; Higher order systems -- Linear systems and canonical forms of operators : The primary decomposition ; The S + N decomposition ; Nilpotent canonical forms ; Jordan and real canonical forms ; Canonical forms and differential equations ; Higher order linear equations ; Operators on function spaces -- Contractions and generic properties of operators : Sinks and sources ; Hyperbolic flows ; Generic properties of operators ; The significance of genericity -- Fundamental theory : Dynamical systems and vector fields ; The fundamental theorem ; Existence and uniqueness ; Continuity of solutions in initial conditions ; On extending solutions ; Global solutions ; The flow of a differential equation -- Stability of equilibria : Nonlinear sinks ; Stability ; Liapunov functions ; Gradient systems ; Gradients and inner products -- Differential equations for electrical circuits : An RLC circuit ; Analysis of the circuit equations ; Van der Pol’s equation ; Hopf bifurcation ; More general circuit equations -- The Poincaré-Bendixson theorem : Limit sets ; Local sections and flow boxes ; Monotone sequences in planar dynamical systems ; The Poincaré-Bendixson theorem ; Applications of the Poincaré-Bendixson theorem -- Ecology : One species ; Predator and prey ; Competing species -- Periodic attractors : Asymptotic stability of closed orbits ; Discrete dynamical systems ; Stability and closed orbits -- Classical mechanics : The n-body problem ; Hamiltonian mechanics -- Nonautonomous equations and differentiability of flows : Existence, uniqueness, and continuity for nonautonomous differential equations ; Differentiability of the flow of autonomous equations -- Perturbation theory and structural stability : Persistence of equilibria ; Persistence of closed orbits ; Structural stability.

ISBN

• 0123495504
• 9780123495501

LCCN

73018951

OCLC

• ocm00796918
• 796918
• SCSB-2115024

Owning Institutions

Princeton University Library