Stability, structures and chaos in nonlinear synchronization networks. Ed. by A. V. Gaponov-Grekhov and M. I. Rabinovich.

*(English)*Zbl 0845.93003
World Scientific Series on Nonlinear Science. Series A. 6. Singapore: World Scientific. xii, 246 p. (1994).

The articles of this volume will not be indexed individually.

According to the authors: “Synchronization is usually understood as the capacity of objects of different nature to acquire a common operation regime. The fact that various objects seek to achieve order and harmony in their behaviour, which is the characteristic of synchronization, seems to be a manifestation of the natural tendency to self-organization existing in nature”. The reviewed book is a research monography with results reached mainly by the experimental way and demonstrated by colour plates of symmetric patterns in the plane. The basic ordinary differential equation of the considered systems is \(\varphi_{n,m}{''} + \lambda \varphi_{n,m}' + \sin \varphi_{n, m} = \gamma - \delta_1 \sin \varphi_{n,m - 1} - \delta_2 \sin \varphi_{n,m - 1}\). Computer experiments with mathematical models were performed with a multiprocessor computation complex. In the case of discretized systems, some analytical results are reached, too. From the last words of the authors: “Only the first attempts have been taken to develop the theory of nonlinear synchronization networks. We hope that the diversity, fascination and freshness of the problems coming into play will excite the curiosity of the readers and will justify the author’s optimism as to the strong possibilities and perspectives offered by further investigations”.

According to the authors: “Synchronization is usually understood as the capacity of objects of different nature to acquire a common operation regime. The fact that various objects seek to achieve order and harmony in their behaviour, which is the characteristic of synchronization, seems to be a manifestation of the natural tendency to self-organization existing in nature”. The reviewed book is a research monography with results reached mainly by the experimental way and demonstrated by colour plates of symmetric patterns in the plane. The basic ordinary differential equation of the considered systems is \(\varphi_{n,m}{''} + \lambda \varphi_{n,m}' + \sin \varphi_{n, m} = \gamma - \delta_1 \sin \varphi_{n,m - 1} - \delta_2 \sin \varphi_{n,m - 1}\). Computer experiments with mathematical models were performed with a multiprocessor computation complex. In the case of discretized systems, some analytical results are reached, too. From the last words of the authors: “Only the first attempts have been taken to develop the theory of nonlinear synchronization networks. We hope that the diversity, fascination and freshness of the problems coming into play will excite the curiosity of the readers and will justify the author’s optimism as to the strong possibilities and perspectives offered by further investigations”.

Reviewer: A.Vaněček (Praha)

##### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93A14 | Decentralized systems |

93A13 | Hierarchical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |