coupled with a barotropic fluid in a spatially flat Friedmann-Robertson-Walker universe has been studied by many authors (see [@Copeland-IJMPD; @Copeland-scalar; @Leon-Phase], and the first section of [@Coley-Book]).

The phase-plane analysis of the cosmological autonomous system is an useful method for studying the dynamical behavior of a scalar field. One always considers the dynamical behavior of a scalar field with an exponential potential in the classical cosmology [@Copeland-exponential; @Hao-PRD1; @Hao-PRD2] or modified cosmology [@Samart-Phantom; @Li-O(N)]. And, if one considers the dynamical behavior of a scalar field coupled with a barotropic fluid, the exponential potential is also the first choice [@Ferreira-scaling; @Hoogen-scaling; @Billyard-interaction; @Fu-phantom]. The exponential potential $V$ leads to the fact that the variables $\Gamma=VV''/V'^2$ equal 1 and that $\lambda=V'/V$ is also a constant. Then the autonomous system is always two dimensional in classical cosmology [@Copeland-exponential], and three dimensional in loop quantum cosmology (LQC) [@Samart-Phantom]. Although one can also consider a more complex case with $\lambda$ being a dynamically changing quantity [@Copeland-IJMPD; @Macorra-lambda; @Nunes-lambda], the fixed point is not a real one, and this method is not exact. Recently, Zhou *et al* [@Zhou-plb; @Fang-CQG] introduced a new method by which one can make $\Gamma$ a general function of $\lambda$. Then the 