Elements of Applied Bifurcation Theory

9780387944180

0387944184

This book is about nonlinear dynamical systems and their bifurcations under parameter variation. It aims to provide a reader with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems. Special attention is given to efficient numerical implementations of the developed techniques. Several examples from recent research papers are used as illustrations. The book is designed for advanced undergraduate or graduate students in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. Basic results of bifurcation theory, such as Andronov-Hopf, planar homoclinic, period-doubling, and Neimark-Sacker bifurcations, are presented in much detail, including self-contained proofs not available in existing textbooks. Several recent results are also discussed, like generic two-parameter bifurcations of equilibria and fixed points, bifurcations of homoclinic orbits to non-hyperbolic equilibria, and one-parameter bifurcations of limit cycles in systems with reflectional symmetry, which are hardly covered in graduate-level textbooks. The presented material is sufficient to perform quite complex bifurcation analysis of dynamical systems and to understand basic methods, results, and terminology used in modern applied mathematics literature.