Abstract
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 <i>p</i><sub>1</sub>, <i>p</i><sub>3</sub> and two orbits linking <i>p</i><sub>3</sub>, <i>p</i><sub>2</sub> and <i>p</i><sub>2</sub>, <i>p</i><sub>1</sub> respectively, and another heterodimensional cycle consisting of one saddle <i>p</i><sub>1</sub> of (2,1) type and one saddle <i>p</i><sub>2</sub> of (1,2) type, which is composed of one orbit starting from <i>p</i><sub>1</sub> to <i>p</i><sub>2</sub> and another orbit starting from <i>p</i><sub>2</sub> to <i>p</i><sub>1</sub>. Moreover, the 1-fold and 2-fold large 1-heteroclinic cycle consisting of two saddles <i>p</i><sub>1</sub> and <i>p</i><sub>3</sub> 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.
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