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**Ordinary differential equations with applications.
2nd ed.**
*(English)*
Zbl 1120.34001

Texts in Applied Mathematics 34. New York, NY: Springer (ISBN 0-387-30769-9/hbk). xix, 636 p. (2006).

The first edition of this textbook on modern qualitative theory of ordinary differential equations [New York, NY: Springer (1999)] has been reviewed in Zbl 0937.34001. The second edition contains rewritten sections, corrections and new material. Also there are more than 160 new exercises and most of them with multiple parts.

The most substantial additions and revisions are the following ones:

The introductory Section 1.9.3 on contraction is rewritten to include a discussion of the continuity of fiber contractions and a more informative first application of the fiber contraction theorem, which is the proof of the smoothness of the solution of the corresponding functional equation.

Section 3.1 on the Euler-Lagrange equation is rewritten and expanded to include a more detailed discussion of Hamilton theory, a presentation of Noether’s Theorem, and several new exercises on the calculus of variations; Section 3.2 on classical mechanics has been revised by including more details; the application (Section 3.5) of Floquet theory to the stability of the inverted pendulum is rewritten to incorporate a more elegant dimensionless model.

A new Section 4.3.3 introduces the Lie derivative and applies it to prove the Hartman-Grobman theorem for flows; multidimensional continuation theory for periodic orbits in the presence of first integrals is discussed in the new Section 5.3.8, the basic result on the continuation of manifolds of periodic orbits in the presence of first integrals in involutions is proved, and the Lie derivative is used again to characterize commuting flows; and the subject of dynamic bifurcation theory is introduced in a new Section 8.4, where the fundamental idea of delayed bifurcation is presented with applications to the pichfork bifurcation and bursting.

The list of references increased from 191 to 240 sources. The number of pictures increased to 73.

The most substantial additions and revisions are the following ones:

The introductory Section 1.9.3 on contraction is rewritten to include a discussion of the continuity of fiber contractions and a more informative first application of the fiber contraction theorem, which is the proof of the smoothness of the solution of the corresponding functional equation.

Section 3.1 on the Euler-Lagrange equation is rewritten and expanded to include a more detailed discussion of Hamilton theory, a presentation of Noether’s Theorem, and several new exercises on the calculus of variations; Section 3.2 on classical mechanics has been revised by including more details; the application (Section 3.5) of Floquet theory to the stability of the inverted pendulum is rewritten to incorporate a more elegant dimensionless model.

A new Section 4.3.3 introduces the Lie derivative and applies it to prove the Hartman-Grobman theorem for flows; multidimensional continuation theory for periodic orbits in the presence of first integrals is discussed in the new Section 5.3.8, the basic result on the continuation of manifolds of periodic orbits in the presence of first integrals in involutions is proved, and the Lie derivative is used again to characterize commuting flows; and the subject of dynamic bifurcation theory is introduced in a new Section 8.4, where the fundamental idea of delayed bifurcation is presented with applications to the pichfork bifurcation and bursting.

The list of references increased from 191 to 240 sources. The number of pictures increased to 73.

Reviewer: Alexander Grin (Grodno)

### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |