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**Local methods in nonlinear differential equations. Part I: The local method of nonlinear analysis of differential equations; Part II: The sets of analyticity of a normalizing transformation. Transl. from the Russian by William Hovingh and Courthney S. Coleman.**
*(English)*
Zbl 0674.34002

Springer Series in Soviet Mathematics. Berlin etc.: Springer-Verlag. x, 348 p. DM 188.00 (1989).

This book systematically treats the local methods developed for solving the systems of nonlinear autonomous ordinary differential equations of the form \((1)\quad dX/dt=\Phi (X),\) where the vector variable \(X=(x_ 1,...,x_ n)\) and the vector function \(\Phi\) are real or complex valued. The fundamental idea of the local methods consists of considering those solutions to (1) which lie in a neighborhood U of a stationary solution, say \(X=0\). To solve this problem by a local method one constructs first a special local coordinate transformation which changes the system (1) into an integrable system. In some simple cases such a transformation exists for the entire neighborhood U while in more complicated cases the neighborhood must be divided up into parts, each with its own local coordinates with respect to which the system is integrable. The book is a combination of two different monograms which complete each other, namely:

Part I. The local method of nonlinear analysis of differential equations.

Part II. The sets of analyticity of a normalizing transformation.

The material of the first part is organized into five chapters as follows:

1. Foundations of the local method,

2. A system of two differential equations

3. The normal form of a system on n differential equations,

4. On the Newton polyhedron,

5. Applications of the normal form in mechanics.

The first two chapters (along with Chapter V) contain detailed proofs, many figures, examples and exercices. As such they are in the form of a textbook and are understable to students who have completed two years of University. Chapter III and IV are more review and problem oriented. They can be read independently. Chapter V includes the application of local methods to two concrete problems, namely:

i) Motion of a gyroscope in a Cartan suspension,

ii) Oscillations of a satellite in the plane of an elliptical orbit.

In the second part of the book the author considers two basic problems connected with the invariant irreducible k-dimensional torus \({\mathcal T}\) of conditionally periodic solutions of an analytic system of ordinary differential equations, namely:

Pr.1. For system \(\dot Z=F(Z,E)\), where Z is a vector variable while E is a vector of parameters, in the neighborhood of the torus \({\mathcal T}\), find all invariant sets which contain \({\mathcal T},\)

Pr.2. Find the sets of analyticity of such a transformation.

In this part the author shows, for a special class of equation, how the method of normal forms yields classical results of Lyapunov concerning families of periodic orbits in the neighborhood of equilibrium points of Hamiltonian systems. He also gives some new results concerning families of quasiperiodic orbits.

The book largely dwells on the results obtained in the earlier extensive investigations by the author himself. It is suitable and useful for advanced students and researchers in pure as well as in applied mathematics.

Part I. The local method of nonlinear analysis of differential equations.

Part II. The sets of analyticity of a normalizing transformation.

The material of the first part is organized into five chapters as follows:

1. Foundations of the local method,

2. A system of two differential equations

3. The normal form of a system on n differential equations,

4. On the Newton polyhedron,

5. Applications of the normal form in mechanics.

The first two chapters (along with Chapter V) contain detailed proofs, many figures, examples and exercices. As such they are in the form of a textbook and are understable to students who have completed two years of University. Chapter III and IV are more review and problem oriented. They can be read independently. Chapter V includes the application of local methods to two concrete problems, namely:

i) Motion of a gyroscope in a Cartan suspension,

ii) Oscillations of a satellite in the plane of an elliptical orbit.

In the second part of the book the author considers two basic problems connected with the invariant irreducible k-dimensional torus \({\mathcal T}\) of conditionally periodic solutions of an analytic system of ordinary differential equations, namely:

Pr.1. For system \(\dot Z=F(Z,E)\), where Z is a vector variable while E is a vector of parameters, in the neighborhood of the torus \({\mathcal T}\), find all invariant sets which contain \({\mathcal T},\)

Pr.2. Find the sets of analyticity of such a transformation.

In this part the author shows, for a special class of equation, how the method of normal forms yields classical results of Lyapunov concerning families of periodic orbits in the neighborhood of equilibrium points of Hamiltonian systems. He also gives some new results concerning families of quasiperiodic orbits.

The book largely dwells on the results obtained in the earlier extensive investigations by the author himself. It is suitable and useful for advanced students and researchers in pure as well as in applied mathematics.

Reviewer: M.Idemen

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

70E05 | Motion of the gyroscope |

70M20 | Orbital mechanics |

70H05 | Hamilton’s equations |

### Keywords:

nonlinear autonomous ordinary differential equations; local method; Newton polyhedron; gyroscope; Oscillations of a satellite; Hamiltonian systems
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\textit{A. D. Bruno}, Local methods in nonlinear differential equations. Part I: The local method of nonlinear analysis of differential equations; Part II: The sets of analyticity of a normalizing transformation. Transl. from the Russian by William Hovingh and Courthney S. Coleman. Berlin etc.: Springer-Verlag (1989; Zbl 0674.34002)