The book is written for students of universities and postgraduate students specializing in the field of nonlinear dynamics, as well as specialists in various fields of mechanical engineering. It is devoted to the development of original methods, and outlines the results of analytical studies of dynamical chaos, synchronization, and dynamic structures in lattices of coupled rotators. It presents its findings within the context of the phase space of models and by involving methods of the qualitative theory of differential equations, the theory of bifurcations, and qualitative numerical methods.

Consider a model shown in Fig. 3.1. An asynchronous motor is coupled to the elastic shaft with an unbalanced disk (rotator) and balancing sleeve (oscillator) elastically coupled to the disk that can oscillate along the shaft.

This paper studies the conditions for existence and stability of stationary cluster structures in lattices of diffusively coupled dynamical systems within the framework of a new interpretation of cluster synchronization as classical synchronization of cluster oscillators (C-oscillators). The study of existence of cluster attractors is based on the linear chains of cluster oscillators, defining possible types of cluster structures in chains. First, we present interval estimates for the range of coupling strengths in which cluster attractors can exist. Then we formulate and prove the basic theorems about the local stability of the various cluster structures. The presented methodology can be extended to study cluster structures on lattices of different geometry and forms such as linear cluster structures in two-dimensional lattices, layered cluster structures in three-dimensional lattices and cluster structures in ring-shaped systems. Crown

This study discusses the development of a technique for analysis of the dynamical regimes of complex mechanical systems consisting of a rotor motor coupled to a system with multi-degrees-of-freedom. To understand the possible qualitatively different dynamical regimes in such systems, a simple mechanical system is considered of the "rotator-oscillator" type with a finite power source. This system has four degrees-of-freedom and is defined in four-dimensional cylindrical phase space with 12 parameters. Near the main resonance the original system is reduced to the Lorenz system with four parameters defined in a three-dimensional Cartesian phase space. This is done with the help of a special change of variables, parameters, and employing an averaging method. Studying the latter system, the existence of one of the chaotic attractors, namely of Lorenz attractor is established. Also established is the Feigenbaum attractor and the alternation. Chaotic limit sets define chaotic behavior of the instantaneous frequency of rotation of the asynchronous motor. The Poincare mappings are presented to show the correspondence of the original 4 dof and averaged 3 dof systems. The qualitative rotational characteristics for different values of the system parameters are obtained. In particular, the system can possess normal Sommerfeld effect, doubled Sommerfeld effect and a so-called scattering of the torque curve. The scattering of the torque curve (which is a known effect in micro-electronics) is likely to be a new effect in mechanics. In contrast to the Sommerfeld effect, when frequency or amplitude jumps occur instantaneously (once the unstable point of the characteristic is reached), the jump to a next stable point may take a certain time, even infinite one. Such chaotic mistuning of the motor frequency would result in random vibrations leading to system wear and damage.

The alternative theory of existence of cluster structures in lattices of dynamical systems (oscillators) is proposed. This theory is based on representation of structures as a result of classical (full) synchronization of structural objects called cluster oscillators (C-oscillators). Different types of C-oscillators, whose number is defined by the geometrical properties of lattices (dimensions and forms) are introduced. The completeness of all types of C-oscillators for lattices of different dimensions is proven. This fact provides a full set of types of cluster structures that can be realized in a given lattice. The solution is derived without the necessity to verify the existence of invariant (cluster) manifolds. The principles of coupling of C-oscillators into the cluster structures and principles of transformations of such structures are described. Having interpreted the processes of structuring in terms of the classical synchronization of C-oscillators, one can solve the problem of fusion of lattices with pre-described properties at the engineering level.