A note on the dynamical features for the extended $f(P)$ cubic gravity

The paper studies the physical characteristics for the extended $f(P)$ cubic gravity from a transitive perspective based on dynamical system analysis, by considering the linear stability theory in two specific cases, corresponding to power-law $f(P)=f_0 P^{\alpha}$ and exponential $f(P)=f_0 e^{\alpha P}$ gravity types, where $f_0$ and $\alpha$ are constant parameters. In these cases we have analyzed the effects in the phase space complexity, revealing the cosmological solutions attached to the critical points. For the power-law and exponential gravity types, we have noticed the presence of two cosmological epochs associated to the critical points involved, corresponding to de-Sitter eras and quintessence-like epochs, described by a constant effective equation of state. For all of these solutions we have studied the dynamical characteristics which are associated to the stability properties, determining possible constraints to various parameters from a transient perspective. The dynamical prospects asserted that the extended $f(P)$ cubic gravity can represent a promising modified theory of gravitation, leading to the manifestation of the accelerated expansion at late time evolution.