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Dynamic systems. Theory and numerics. (Dynamische Systeme. Theorie und Numerik.) (German) Zbl 1213.37002
Heidelberg: Spektrum Akademischer Verlag (ISBN 978-3-8274-2447-1/hbk; 978-3-8274-2448-8/ebook). xii, 436 p. (2011).
The reviewed monograph is based on an advanced course on nonlinear differential equations, dynamic systems and bifurcation theory intended mainly for engineer-researchers. But it will be interesting also for mathematicians especially with respect to numerical methods in bifurcation theory and nonlinear oscillations. Chapter 1 contains the basic principles of functional analysis as the foundation of dynamical systems theory. Chapter 2 gives, besides solving methods for ODE systems, elements of the qualitative theory and stability of solutions, Grobman-Hartman theorem. Chapter 3 discusses the bifurcation theory; basic subjects: saddle-node, transcritic and pitchfork bifurcations, dynamic Poincaré-Andronov-Hopf bifurcation. In Chapter 4 the analytic bifurcation theory is presented: construction and investigation of the Lyapounov-Schmidt branching equation, abstract Hopf bifurcation theorem, nonlinear oscillations in autonomous systems. Chapter 5 is devoted to iterative methods and the convergence investigation for the equilibrium positions and their bifurcations, to parametrization and stability of bifurcating solutions. Chapter 6 contains the stability analysis of periodic solutions, particulary in autonomous systems, and parameter continuation questions in bifurcation analysis. Chapter 7 is devoted to quasiperiodic solutions and invariant tori together with, in the last issue 7.3, discretization of 2-tori via Galerkin and Galerkin-Fourier methods. Every chapter is equipped with numerous well selected exercises.

37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
65P30 Numerical bifurcation problems
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