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Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. (English) Zbl 0808.34001
Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press,. xiii, 388 p. (1994).
This textbook covers a two semester course for advanced undergraduates and beginning graduate students in mathematics and physics. The first six chapters present classical results about linear differential equations, stability, two dimensional dynamics (PoincarĂ©-Bendixson theorem, Dulac’s criterion) and about the dynamics near hyperbolic stationary and periodic solutions. In chapter 7 perturbation methods are applied to various equations with small nonlinearities in order to describe the asymptotic behavior (frequency locking, parametric excitation, hysteresis, relaxation oscillations). The chapters 8-10 deal with local bifurcations of differential equations and maps (saddlenode, transcritical bifurcation, pitchfork, Hopf bifurcation, period doubling, subharmonic bifurcation) on one resp. two dimensional phase spaces. Chapter 11 provides results about chaotic behavior of maps on intervals (period three implies chaos, Sharkovskii’s theorem, period doubling cascades, intermittency), and in chapter 12 homoclinic bifurcations are studied. In this very readable and useful book, the emphasis is on understanding and ability to apply theory to examples rather than rigorous mathematical developments. Therefore, a great number of exercises and examples is presented. Some of the theorems are stated without proofs (Grobman- Hartman theorem stable manifold theorem, Peixoto’s theorem about structurally stable two dimensional vector fields) or the proofs are sketched without going into the technical details (stable manifold theorem, Feigenbaum’s theorem).
Reviewer: L.Recke (Berlin)

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37E99 Low-dimensional dynamical systems
37B99 Topological dynamics
34A26 Geometric methods in ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C23 Bifurcation theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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