We have carried out dynamical system analysis of hessence field coupling with dark matter in $f\left(T\right)$ gravity. We have analysed the critical points due to autonomous system. The resulting autonomous system is nonlinear. So, we have applied the theory of nonlinear dynamical system. We have noticed that very few papers are devoted to this kind of study. Maximum works in literature are done treating the dynamical system as done in linear dynamical analysis, which are unable to predict correct evolution. Our work is totally different from those kinds of works. We have used nonlinear dynamical system theory, developed till date, in our analysis. This approach gives totally different stable solutions, in contrast to what the linear analysis would have predicted. We have discussed the stability analysis in detail due to exponential potential through computational method in tabular form and analysed the evolution of the universe. Some plots are drawn to investigate the behaviour of the system (this plotting technique is different from usual phase plot and that devised by us). Interestingly, the analysis shows that the universe may resemble the “cosmological constant” like evolution (i.e., $\Lambda$CDM model is a subset of the solution set). Also, all the fixed points of our model are able to avoid Big Rip singularity.