Dynamical Study of DBI-essence in Loop Quantum Cosmology and Braneworld

We have studied homogeneous isotropic FRW model having dynamical dark energy DBI-essence with scalar field. The existence of cosmological scaling solutions restricts the Lagrangian of the scalar field $\varphi$. Choosing $p=X g(X e^{\lambda \phi})$,where$X=-g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi /2$with$g$isanyfunctionof$X e^{\lambda \phi}$ and defining some suitable transformations, we have constructed the dynamical system in different gravity: (i) Loop Quantum Cosmology (LQC), (ii) DGP BraneWorld and (iii) RS-II Brane World. We have investigated the stability of this dynamical system around the critical point for three gravity models and investigated the scalar field dominated attractor solution in support of accelerated universe. The role of physical parameters have also been shown graphically during accelerating phase of the universe.