The dissertation is devoted to the study of collective dynamics in complex models of coupled oscillators given by the systems of ordinary differential equations with parameters. By collective dynamics we understand different types of interaction between different elements that have their individual dynamics. In particular, such collective dynamics in phase models as full or partial phase and frequency synchronization, anti–phase modes, regimes of the uniform phase distribution, traveling waves, cluster modes, slow switching regimes between clusters, chaotic synchronization, chimera states, heteroclinic chimeras are being described. Also, we study such regimes as winner–take–all, winnerless competition, competition for synchronization, the opposition between conformists and contrarians. Oscillator models with different individual dynamics of elements, different architecture of relationships, as well as different types of interaction between elements are being considered. We study well–known models, as well as propose and analyze the new ones. Models are constructed taking into account certain natural processes and each mathematical result has a specific natural interpretation. We prove the existence, stability, multistability and bifurcation transitions in the systems of globally coupled phase oscillators, systems with the central element, systems with circulant connection, systems of indistinguishable elements, modular networks, oscillatory networks with adaptation, attractive and repulsive systems that model the interaction of groups of conformists and contrarians. New types of bifurcations of heteroclinic cycles appearance, as well as saddle-node bifurcations on an invariant torus have been identified and described. The coexistence of conservative and dissipative dynamics in complex systems with a circulant skew-symmetric connection is shown. The relationship of the infinite-dimensional circulant systems with the nonlinear Schrödinger equation is presented. For systems of identical elements, the results of the relationship between the symmetries of the network and the existence of invariant manifolds, invariant regions, and cluster modes are obtained. Conditions of extreme sensitivity to natural frequency perturbations and the appearance of phase-unlocked trajectories in the systems of non-identical oscillators are described. It is proved that the systems with even and odd interaction functions are gradient or divergence-free. It is demonstrated when the above systems are time-reversible and integrated. Minimal networks of phase oscillators with stable chimeric states are presented.
