BIFURCATIONS OF DOUBLE HETERODIMENSIONAL CYCLES WITH THREE SADDLE POINTS

In this paper, bifurcations of double heterodimensional cycles of an “∞” shape consisting of two saddles of (1,2) type and one saddle of (2,1) type are studied in three dimensional vector field. We discuss the gaps between returning points in transverse sections by establishing a local active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the preservation of “∞”-shape double heterodimensional cycles is proved. We then get the existence of a new heteroclinic cycle consisting of two saddles of (1,2) type and one saddle of (2,1) type, which is composed of one big orbit linking p₁, p₃ and two orbits linking p₃, p₂ and p₂, p₁ respectively, and another heterodimensional cycle consisting of one saddle p₁ of (2,1) type and one saddle p₂ of (1,2) type, which is composed of one orbit starting from p₁ to p₂ and another orbit starting from p₂ to p₁. Moreover, the 1-fold and 2-fold large 1-heteroclinic cycle consisting of two saddles p₁ and p₃ of (1,2) type is also presented. As well as the coexistence of a 1-fold large 1-heteroclinic cycle and the “∞”-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.