Valentin Afraimovich

This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.

Dynamical networks are characterized by 1) their topology (structure of the graph of interactions among the elements of a network); 2) the interactions between the elements of the network; 3) the intrinsic (local) dynamics of the elements of the network. A general approach to studying the commulative effect of all these three factors on the evolution of networks of a very general type has been developed in [1]. Besides, in this paper there were obtained sufficient conditions for a global stability (generalized strong synchronization) of networks with an arbitrary topology and the dynamics which is a composition (action of one after another) of a local dynamics of the elements of a network and of the interactions between these elements. Here we extend the results of [1] on global stability (generalized strong synchronization) to the case of a general dynamics in discrete time dynamical networks and to general dynamical networks with continuous time.