Published November 1999
by Springer-Verlag Telos .
Written in English
|Contributions||Zhangxin Chen (Editor), Kaitai Li (Editor)|
|The Physical Object|
|Number of Pages||227|
“This book is a valuable resource for mathematicians working in the areas of Nonlinear Analysis and/or Differential Equations. This book is intended for advanced graduate students, for specialists in Bifurcation Theory and for researchers in related areas willing to master the subject. this is a great reference book on the subject of Cited by: While self-contained and suitable for independent study, the book is also written with users in mind and is an invaluable reference for practitioners. Part I focuses on theory, providing a systematic presentation of bifurcations of fixed points and cycles of finite-dimensional maps, up to and including cases with two control by: 3. In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known : Springer-Verlag New York. Abstract. In this chapter we shall describe some of the basic techniques used in the numerical analysis of dynamical systems. We assume that low-level numerical routines like those for solving linear systems, finding eigenvectors and eigenvalues, and performing numerical integration of .
Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. Using qualitative methods of bifurcation theory, the behavior of the system is characterized for various parameter combinations. Inparticular, the catalog of system behaviors showing qualitative differences can be identified, together with the regions in parameter space where thedifferent behaviors occur. Bifurcations delimit such regions. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and ﬁnite diﬀerence methods. These results are used to pro- vide a convergence theory for the mixed ﬁnite element method applied to the steady incompressible Navier–Stokes equations. The one-parameter numerical bifurcation analysis reveals a complex dynamical behaviour including multistability of equilibria, sustained oscillations, limit points of limit cycles and homoclinic bifurcations. The two-parameter numerical bifurcation analysis revealed the existence of Bogdanov–Takens and Bautin (Generalized Hopf) bifurcations.
Basic concepts of bifurcation analysis As it is stated above, in dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden ” qualitative" or topological change in its by: 4. Bifurcation Theory: An Introduction with Applications to Partial Differential Equations (Applied Mathematical Sciences)Cited by: The book is very useful as a reference because it collects and organizes the bifurcation analysis of infinite-dimensional operators. It could also be used as a text in an advanced course on bifurcation theory with an emphasis on partial differential equations." (HW Haslach, Applied Mechanics Reviews, Vol. 57 (5), September, ). Chapdevoted to the numerical analysis of bifurcations,has been changed most substantially. We have introduced bordering methods to continue fold and Hopf bifur- cations in two parameters. In this approach,the deﬁning function for the bifurcation used in the minimal augmented system is computed by solving a bordered linear system.