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We perform a phase space analysis of a non-minimally coupled modified gravitytheory with the Lagrangian density of the form $\frac{1}{2} f_{1}(R)+[1+\lambdaf_{2}(R)]{{\cal{L}}_{m}}$, where $f_1(R)$ and $f_2(R)$ are arbitrary functionsof the curvature scalar $R$ and ${{\cal{L}}_{m}}$ is the matter Lagrangiandensity. We apply the dynamical system approach to this scenario in twoparticular models. In the first model we assume $f_1(R)=2R$ with a general formfor $f_2(R)$ and set favorable values for effective equation of state parameterwhich is related to the several epochs of the cosmic evolution and study thecritical points and their stability in each cosmic eras. In the second case, weallow the $f_1(R)$ to be an arbitrary function of $R$ and set $f_2(R)=2R$. Wefind the late time attractor solution for the model and show that this modelhas a late time accelerating epoch and an acceptable matter era.

Cosmological Dynamics of Modified Gravity With a Non-minimal Curvature-matter Coupling