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We have studied homogeneous isotropic FRW model having dynamical dark energyDBI-essence with scalar field. The existence of cosmological scaling solutionsrestricts the Lagrangian of the scalar field $\phi$. Choosing $p=X g(Xe^{\lambda \phi})$, where $X=-g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi/2$ with $g$ is any function of $X e^{\lambda \phi}$ and defining some suitabletransformations, we have constructed the dynamical system in different gravity:(i) Loop Quantum Cosmology (LQC), (ii) DGP BraneWorld and (iii) RS-II BraneWorld. We have investigated the stability of this dynamical system around thecritical point for three gravity models and investigated the scalar fielddominated attractor solution in support of accelerated universe. The role ofphysical parameters have also been shown graphically during accelerating phaseof the universe.

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