Difference equations and their applications.
(Разностные уравнения и их приложения.)

*(Russian)*Zbl 0669.39001
Kiev: Naukova Dumka. 280 p. R. 3.30 (1986).

This monograph acquaints the reader with new and at first glance not quite typical properties of nonlinear solutions of difference equations. These properties very often allow the application of difference equations for simulating involved oscillating processes in those cases where the use of ordinary differential equations is difficult. The exposition of the subject in the book is based on the contemporary theory of one- dimensional dynamical systems.

The first chapter is devoted to this theory. This chapter investigates the problems of the existence of periodic trajectories and their bifurcation, spectral decomposition of the set of nonwandering points, strange attractors, stability of trajectories and stability of whole dynamical systems. The second chapter describes the asymptotic properties of the solutions of difference equations \(x(t+1)=f(x(t))\), \(t\in \mathbb R^+\). The stability and bifurcation of these solutions are considered also. In particular, the solutions of relaxational and turbulent types and their special properties are examined in greater detail.

The third chapter is devoted to differential-difference equations. It investigates the problem of how much the equations of this type imitate some properties of difference equations and what changes take place with them under regular and singular perturbations. The fourth and final chapter develops a method for the investigation of nonlinear boundary value problems for hyperbolic systems based on the reduction to difference and differential-difference equations. Some classes of problems described by chaotic dynamics are selected.

Many of the results given in the book are new and have not been published elsewhere. This book is especially interesting for specialists in differential equations applying their results to some practical problems in the natural sciences and technology.

The first chapter is devoted to this theory. This chapter investigates the problems of the existence of periodic trajectories and their bifurcation, spectral decomposition of the set of nonwandering points, strange attractors, stability of trajectories and stability of whole dynamical systems. The second chapter describes the asymptotic properties of the solutions of difference equations \(x(t+1)=f(x(t))\), \(t\in \mathbb R^+\). The stability and bifurcation of these solutions are considered also. In particular, the solutions of relaxational and turbulent types and their special properties are examined in greater detail.

The third chapter is devoted to differential-difference equations. It investigates the problem of how much the equations of this type imitate some properties of difference equations and what changes take place with them under regular and singular perturbations. The fourth and final chapter develops a method for the investigation of nonlinear boundary value problems for hyperbolic systems based on the reduction to difference and differential-difference equations. Some classes of problems described by chaotic dynamics are selected.

Many of the results given in the book are new and have not been published elsewhere. This book is especially interesting for specialists in differential equations applying their results to some practical problems in the natural sciences and technology.

##### MSC:

39A10 | Additive difference equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39A28 | Bifurcation theory for difference equations |

39A30 | Stability theory for difference equations |

39A33 | Chaotic behavior of solutions of difference equations |