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

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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.