Nonlinear difference equations. Theory with applications to social science models.

*(English)*Zbl 1020.39007
Mathematical Modelling: Theory and Applications. 15. Dordrecht: Kluwer Academic Publishers. xv, 388 p. (2003).

After short preliminaries concerning the basic conceptions of discrete dynamical systems the dynamics on the real line, in particular, the stability of equilibria, limit cycles and one-parameter bifurcations are treated. It follows the generalization to vector difference equations with emphasis on semiconjugacy, chaotic maps, polymodal systems, complex threshold systems and ejector cycles. Higher-order scalar difference equations are investigated with respect to persistent oscillations, absorbing intervals, semi-permanences and global attractivity.

In the second half of the book the foregoing theory is applied to a series of non-random models from economics and other social sciences. The reader find both the description and derivation of each model, and its detailed qualitative analysis, which yields a better understanding of the fundamental mechanisms of the real phenomena.

The book presents the best known results up to quite recent ones and, in particular, those of the author himself. Besides of the theorems with rigorous proofs it contains many examples, figures, remarks and historical notes. It can be recommended both to researchers and students in mathematics and social sciences.

In the second half of the book the foregoing theory is applied to a series of non-random models from economics and other social sciences. The reader find both the description and derivation of each model, and its detailed qualitative analysis, which yields a better understanding of the fundamental mechanisms of the real phenomena.

The book presents the best known results up to quite recent ones and, in particular, those of the author himself. Besides of the theorems with rigorous proofs it contains many examples, figures, remarks and historical notes. It can be recommended both to researchers and students in mathematics and social sciences.

Reviewer: Lothar Berg (Rostock)

##### MSC:

39A12 | Discrete version of topics in analysis |

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

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

39A10 | Additive difference equations |

91B28 | Finance etc. (MSC2000) |

91D10 | Models of societies, social and urban evolution |

39A11 | Stability of difference equations (MSC2000) |