Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, or lack thereof, of each technique. These techniques are integral in applied mathematics and it is shown that correct employment allows us to see the behaviour of a differential equation when the exact solution may not be attainable. Finding and interpreting the solutions of differential equations is a central and essential part of applied mathematics. This book aims to enable the reader to develop the required skills needed for a thorough understanding of the subject. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds are discussed. Prepared with an eye toward the needs of applied mathematicians, engineers, and physicists, the treatment is equally valuable as a reference for professionals. The focuses of the book is developing analytical skills needed for research in this area. Designed for senior undergraduates and graduate students, as well as professionals the text emphasizes practical application and analytical techniques rather than abstract formulations.
Peter J. Olver is a mathematician known for his contributions to the study of hyperbolic differential equations and general relativity. He is currently the Eugene Higgins Professor of Mathematics at Princeton University, where he has been teaching since 1987. After attending the Petru Groza High School, he studied mathematics at the University of Bucharest from 1969 to 1974. For graduate studies he went to New York University, obtaining his Ph.D. in 1978. He is a member of the U.S. National Academy of Sciences (elected 2005), a foreign member of the French Academy of Sciences (elected 2002) and a Fellow of the American Academy of Arts and Sciences (elected 1996). In 2003, he was awarded the Romanian Order of Merit, Commander Rank. He was elected to the 2018 class of fellows of the American Mathematical Society.
| Preface Chapter 1. Introduction to Nonlinear Differential Equations Definition and Classification of Nonlinear Equations Differences Between Linear and Nonlinear Systems Historical Development of Nonlinear Theory Examples of Nonlinear Ordinary Differential Equations (ODEs) Sources of Nonlinearity in Physical Systems Chapter 2. First-Order Nonlinear Differential Equations Autonomous and Non-Autonomous First-Order Equations Direction Fields and Phase Portraits Isoclines and Qualitative Methods Bernoulli and Riccati Equations Chapter 3. Second-Order and Higher-Order Nonlinear ODEs Nonlinear Spring-Mass Systems Van der Pol Oscillator Pendulum Equation (Nonlinear Dynamics) Duffing Equation and Bifurcations Reduction of Higher-Order to First-Order Systems Chapter 4. Systems of Nonlinear Differential Equations Phase Plane Analysis Nullclines and Critical Points Predator-Prey Models (Lotka–Volterra) Competing Species and Coexistence Models Chemical Reaction Kinetics (Brusselator, Oregonator) Chapter 5. Stability Theory and Lyapunov Methods Equilibrium Points and Their Classification Lyapunov Direct Method LaSalle’s Invariance Principle Global vs Local Stability Chapter 6. Bifurcation Theory Saddle-Node Bifurcation Pitchfork Bifurcation (Supercritical and Subcritical) Hopf Bifurcation and Limit Cycles Fold and Transcritical Bifurcations Bifurcation Diagrams and Continuation Methods Catastrophe Theory and Nonlinear Dynamics |
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