Differential equations, dynamical systems, and an introduction to chaos (Third Edition) / / Morris W. Hirsch, Stephen Smale, Robert L. Devaney

• Record Nr.: UNINA9910965872603321
• Autore: Hirsch Morris W. <1933->
• Pubbl/distr/stampa: Waltham, MA, : Academic Press, c2013
• ISBN: 0-12-382011-1
• Edizione: [3rd ed.]
• Descrizione fisica: 1 online resource (xiv, 418 p. ) : ill.
• Altri autori (Persone): SmaleStephen <1930->; DevaneyRobert L. <1948->
• Disciplina: 515/.35
• Soggetti: Differential equations; Algebras, Linear; Chaotic behavior in systems; Electronic books.
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Previous ed.: 2003.
• Nota di bibliografia: Includes bibliographical references (p. 411-413) and index
• Nota di contenuto: Cover -- Differential Equations, Dynamical Systems, and an Introduction to Chaos -- Copyright -- Table of Contents -- Preface to Third Edition -- Preface -- 1 First-Order Equations -- 1.1 The Simplest Example -- 1.2 The Logistic Population Model -- 1.3 Constant Harvesting and Bifurcations -- 1.4 Periodic Harvesting and Periodic Solutions -- 1.5 Computing the Poincaré Map -- 1.6 Exploration: A Two-Parameter Family -- Exercises -- 2 Planar Linear Systems -- 2.1 Second-Order Differential Equations -- 2.2 Planar Systems -- 2.3 Preliminaries from Algebra -- 2.4 Planar Linear Systems -- 2.5 Eigenvalues and Eigenvectors -- 2.6 Solving Linear Systems -- 2.7 The Linearity Principle -- Exercises -- 3 Phase Portraits for Planar Systems -- 3.1 Real Distinct Eigenvalues -- 3.2 Complex Eigenvalues -- 3.3 Repeated Eigenvalues -- 3.4 Changing Coordinates -- Exercises -- 4 Classification of Planar Systems -- 4.1 The Trace-Determinant Plane -- 4.2 Dynamical Classification -- Case 1 -- Case 2 -- Case 3 -- 4.3 Exploration: A 3D Parameter Space -- Exercises -- 5 Higher-Dimensional Linear Algebra -- 5.1 Preliminaries from Linear Algebra -- 5.2 Eigenvalues and Eigenvectors -- 5.3 Complex Eigenvalues -- 5.4 Bases and Subspaces -- 5.5 Repeated Eigenvalues -- 5.6 Genericity -- ; Exercises -- 6 Higher-Dimensional Linear Systems -- 6.1 Distinct Eigenvalues -- 6.2 Harmonic Oscillators -- 6.3 Repeated Eigenvalues -- 6.4 The Exponential of a Matrix -- 6.5 Nonautonomous Linear Systems -- Exercises -- 7 Nonlinear Systems -- 7.1 Dynamical Systems -- 7.2 The Existence and Uniqueness Theorem -- 7.3 Continuous Dependence of Solutions -- 7.4 The Variational Equation -- 7.5 Exploration: Numerical Methods -- 7.6 Exploration: Numerical Methods and Chaos -- Exercises -- 8 Equilibria in Nonlinear Systems -- 8.1 Some Illustrative Examples -- 8.2 Nonlinear Sinks and Sources -- 8.3 Saddles.; 8.4 Stability -- 8.5 Bifurcations -- 8.6 Exploration: Complex Vector Fields -- Exercises -- 9 Global Nonlinear Techniques -- 9.1 Nullclines -- 9.2 Stability of Equilibria -- 9.3 Gradient Systems -- 9.4 Hamiltonian Systems -- 9.5 Exploration: The Pendulum with Constant Forcing -- Exercises -- 10 Closed Orbits and Limit Sets -- 10.1 Limit Sets -- 10.2 Local Sections and Flow Boxes -- 10.3 The Poincaré Map -- 10.4 Monotone Sequences in Planar Dynamical Systems -- 10.5 The Poincaré-Bendixson Theorem -- 10.6 Applications of Poincaré-Bendixson -- 10.7 Exploration: Chemical Reactions that Oscillate -- Exercises -- 11 Applications in Biology -- 11.1 Infectious Diseases -- 11.2 Predator-Prey Systems -- 11.3 Competitive Species -- 11.4 Exploration: Competition and Harvesting -- 11.5 Exploration: Adding Zombies to the SIR Model -- Exercises -- 12 Applications in Circuit Theory -- 12.1 An RLC Circuit -- 12.2 The Liénard Equation -- 12.3 The van der Pol Equation -- 12.4 A Hopf Bifurcation -- 12.5 Exploration: Neurodynamics -- Exercises -- 13 Applications in Mechanics -- 13.1 Newton's Second Law -- 13.2 Conservative Systems -- 13.3 Central Force Fields -- 13.4 The Newtonian Central Force System -- 13.5 Kepler's First Law -- 13.6 The Two-Body Problem -- 13.7 Blowing up the Singularity -- 13.8 Exploration: Other Central Force Problems -- 13.9 Exploration: Classical Limits of Quantum Mechanical Systems -- 13.10 Exploration: Motion of a Glider -- Exercises -- 14 The Lorenz System -- 14.1 Introduction -- 14.2 Elementary Properties of the Lorenz System -- 14.3 The Lorenz Attractor -- 14.4 A Model for the Lorenz Attractor -- 14.5 The Chaotic Attractor -- 14.6 Exploration: The Rössler Attractor -- Exercises -- 15 Discrete Dynamical Systems -- 15.1 Introduction -- 15.2 Bifurcations -- 15.3 The Discrete Logistic Model -- 15.4 Chaos -- 15.5 Symbolic Dynamics.; 15.6 The Shift Map -- 15.7 The Cantor Middle-Thirds Set -- 15.8 Exploration: Cubic Chaos -- 15.9 Exploration: The Orbit Diagram -- Exercises -- 16 Homoclinic Phenomena -- 16.1 The Shilnikov System -- 16.2 The Horseshoe Map -- 16.3 The Double Scroll Attractor -- 16.4 Homoclinic Bifurcations -- 16.5 Exploration: The Chua Circuit -- Exercises -- 17 Existence and Uniqueness Revisited -- 17.1 The Existence and Uniqueness Theorem -- 17.2 Proof of Existence and Uniqueness -- 17.3 Continuous Dependence on Initial Conditions -- 17.4 Extending Solutions -- 17.5 Nonautonomous Systems -- 17.6 Differentiability of the Flow -- Exercises -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W -- Z.
• Sommario/riassunto: This text is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. It has become the standard textbook for graduate courses in this area.