Cosmological evolution of "hessence" dark energy and avoidance of the big rip

Recently, many dark energy models whose equation-of-state parameter can cross the phantom divide w(de)=-1 have been proposed. In a previous paper [H. Wei, R. G. Cai, and D. F. Zeng, Classical Quantum Gravity 22, 3189 (2005)], we suggest such a model named hessence, in which a noncanonical complex scalar field plays the role of dark energy. In this work, the cosmological evolution of the hessence dark energy is investigated. We consider two cases: one is the hessence field with an exponential potential, and the other is with a (inverse) power-law potential. We separately investigate the dynamical system with four different interaction forms between hessence and background perfect fluid. It is found that the big rip never appears in the hessence model, even in the most general case, beyond particular potentials and interaction forms.