Bifurcations of systems with structurally unstable homoclinic orbits and moduli of ω-equivalence

Bifurcations of both two-dimensional diffeomorphisms with a homoclinic tangency and three-dimensional flows with a homoclinic loop of an equilibrium state of saddle-focus type are studied in one- and two-parameter families. Due to the well-known impossibility of a complete study of such bifurcations, the problem is restricted to the study of the bifurcations of the so-called low-round periodic orbits. In this connection, the idea of taking ω-moduli (continuous invariants of the topological conjugacy on the nonwandering set) as the main control parameters (together with the standard splitting parameter) is proposed. On this way, new bifurcational effects are found which do not occur at a one-parameter analysis. In particular, the density of cusp-bifurcations is revealed.