Ordinary differential equations and dynamical systems (Q2849542)

This book provides an introduction to the subject of ordinary differential equations for advanced undergraduate or beginning graduate students. The style is concise, mathematically rigorous and requires a solid background in analysis and linear algebra.NEWLINENEWLINEThe book starts with a brief introduction to the concept of a differential equation. Chapter 1 presents the theory of linear systems. The basic existence and uniqueness results for initial value problems are included in Chapter 2 as well as the following topics: continuous dependence, extensibility, flows and stability. Next, the non-autonomous linear systems and Floquet theory are considered in Chapter 3.NEWLINENEWLINEA brief overview of some results from Functional Analysis is given in Chapter 5: Banach spaces and examples, Fréchet differentiation, the Contraction Mapping Principle, the Implicit Function Theorem and the Lyapunov-Schmidt Method. The next chapters are devoted to the qualitative and geometric theory. First, the dependence on initial conditions and parameters is considered in Chapter 6, followed by topological conjugacy, Hartman-Grobman theorem, invariant manifolds (Chapter 7), existence and orbital stability of periodic solutions, and the Poincaré-Bendixson Theorem (Chapter 8). Then, the bifurcation theory is included in Chapter 9 with special attention to the center manifold, normal forms and the Lyapunov-Schmidt method.NEWLINENEWLINEChapter 10 concerns the homoclinic solutions and the Smale-Birkhoff theorem. Newton's equation \(\ddot{\varphi}+g(\varphi)=0\) is used as an example illustrating the main points in the theory. The connection between the existence of a transverse homoclinic point and chaotic dynamics is briefly summarized in the final section.NEWLINENEWLINEIn conclusion, the text is focused on the commonly used techniques rather than the detailed treatment of a large number of topics. Its concise style makes the book convenient for the preparation of lecture notes and it may be very useful for any course in ordinary differential equations.