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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
