Dynamical system analysis for extended f(P) gravity coupled with scalar field

In the present article, we have explored the physical characteristics of extended f(P) gravity through the dynamical system analysis. We choose the function f(P) in the form of a polynomial of second order, i.e. \(f(P)=\alpha P +\beta P^2\), where \(\alpha \) and \(\beta \) are constant parameters. As an additional dark energy component, we take a canonical scalar field \(\phi \), and several types of interaction between the dark components are considered. After presenting the field equation for the corresponding cosmological setup, we introduce some dimensionless variables and formulate a nonlinear autonomous system. For the different interaction scenarios, we have investigated the phase space and their physical characteristics and the dynamics of the cosmological solution associated with each critical point. For the first interaction model, the results we obtained by the analysis of phase space reveals that the cosmological solution associated with critical points exhibits two different cosmological epochs, namely the de-sitter epoch and quintessence epoch. For the second interaction model, the solutions represent the quintessence era. We have also studied the dynamical stability properties of each critical point by linear stability theory and determined possible physical constraints to the parameters. The cosmological solutions support cosmic acceleration. Moreover, an analysis of statefinder parameter {r,s} in terms of the dynamical system variable is presented. Based on the mathematical prospects, the modified theory of gravitation based on the extension of f(P) cubic gravity has the potential to manifest an accelerated expansion during late-time evolution.