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On November 30, 2020, at the age of 78, Vadim S. Anishchenko suddenly died. He was Head of the Department of Radiophysics and Nonlinear Dynamics, Saratov State University (SSU), Doctor of Physics and Mathematics, Professor, Honored Scientist of the Russian Federation, Laureate of the International Science Prize of the Alexander von Humboldt Foundation, Honorary Worker of Education of the Russian Federation, Honorary Professor of SSU, world-class specialist in the field of nonlinear dynamics, founder of one of the leading scientific schools in radiophysics and nonlinear dynamics. The memories of people who knew Vadim S. Anishchenko closely at different periods of his life are published, as a tribute to the memory of an outstanding personality.

Background and Objectives: As external and mutual synchronisation effects are conventional, the study of these phenomena in networks of nonlocally coupled chaotic maps is of much interest. In this paper we study the effects of synchronization in a two-layer network of nonlocally coupled discrete-time systems. Each layer represents a ring of nonlocally coupled logistic maps in the chaotic regime. Depending on the initial conditions and parameter values, they can demonstrate various spatiotemporal patterns, including amplitude and phase chimera structures. The network equations are being solved numerically for periodic boundary conditions and randomly distributed initial conditions. We consider interaction between identical and nonidentical ensembles for dissipative and inertial intercoupling. Materials and Methods: The analysis is carried out with the use of a set of programs in C++ which was developed for modelling dynamic systems with complex intercoupling definition. Synchronization effect is estimated by calculating root-mean-square deviations between the symmetric elements of the rings. The first characteristic to consider is the time deviation which is used to determine the clusters that respond to the impact and synchronize faster as the intercoupling strength grows. The second quantity is the time and ensemble deviation and it is used for quantifying the synchronization effect and estimating the synchronization region in dependence of the interlayer coupling. Results: The numerical research has shown that the effects of external and mutual synchronisation are clearly visible in the case of dissipative intercoupling. The best results were obtained for identical ensembles, however, nonidentical ensembles can be synchronized as well with a given accuracy. In the case of external synchronisation of nonidentical ensembles for dissipative intercoupling we have ascertained that amplitude chimera structures synchronize faster than the phase ones. Both systems are much harder or impossible to synchronize for the case of inertial intercoupling. Conclusion: The numerical results obtained in this paper allow to understand the course of synchronization effects of chimera states appearing in a two-layer network of nonlocally coupled chaotic oscillators for dissipative and inertial intercoupling.

In this paper, we investigate the characteristics of quasihyperbolic attractors and quasiattractors in Invertible dissipative maps of the plane. The criteria which allow one to diagnose the indicated types of attractors in numerical experiments are formulated.