Dynamics near the three-point heteroclinic cycles with saddle-focus

Abstract

This paper studies the bifurcation phenomena of heteroclinic cycles connecting three equilibria in a three-dimensional vector field. Based on Lin's method, we prove the existence of shift dynamics near the three-point heteroclinic cycle, showing the existence of chaotic behavior. Moreover, we present more details about the bifurcation results, such as the existence of a three-point heteroclinic cycle, two-point heteroclinic cycles, homoclinic cycles and 1-periodic orbits bifurcated from the primary three-point heteroclinic cycle. Furthermore, the coexistence of 1-periodic orbit and homoclinic cycle, and the coexistence of 1-periodic orbit and two-point heteroclinic cycle are proved respectively.