Physical interpretation and theory of existence of cluster structures in lattices of dynamical systems

Abstract

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.