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Highlights • The dynamic responses of a bit-rock system with both the linear and nonlinear interaction models are verified as similar. Hence, the linear model is suggested to predict the system dynamics due to its higher computing efficiency. • An isola of period-doubling bifurcations is observed during two-parameter continuation, and the isola is formed via two turning points which lie on its left and right endpoints. • As the increase of either excitation amplitude or static force, the ROP of the stable period-one motion can be promoted almost linearly. While, the increase of excitation frequency will lead to a nonlinear decrease of the ROP. Graphical abstract Abstract This paper presents a comprehensive numerical study of a higher order drifting oscillator that has been used to model vibro-impact drilling dynamics in previous publications by our research group [1,2,3,4,5,6,7,8,9]. We focus on the study of the bit-rock interactions, for which both linear and nonlinear models of the drilled medium are considered. Our investigation employed a numerical approach based on direct numerical integration via a newly developed MATLAB-based computational tool, ABESPOL (Chong et al., 2017) [10], and based on path-following methods implemented via a software package for continuation and bifurcation analysis, COCO (Continuation Core) (Dankowicz and Schilder, 2013) [11]. The analysis considered the excitation frequency, amplitude of excitation and the static force as the main control parameters, while the rate of penetration (ROP) was chosen as the main system output so as to assess the performance of the system when linear and nonlinear bit-rock impact models are used. Furthermore, our numerical investigation reveals a rich system dynamics, owing to the presence of codimension-one bifurcations of limit cycles that influence the system behaviour dramatically, as well as multistability phenomenon and chaotic motion. [ABSTRACT FROM AUTHOR]