  In a flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) geometry, we
consider the expansion of the universe powered by the gravitationally induced
`adiabatic' matter creation. To demonstrate how matter creation works well with
the expanding universe, we have considered a general creation rate and analyzed
this rate in the framework of dynamical analysis. The dynamical analysis hints
the presence of a non-singular universe (without the big bang singularity) with
two successive accelerated phases, one at the very early phase of the universe
(i.e. inflation), and the other one describes the current accelerating
universe, where this early, late accelerated phases are associated with an
unstable fixed point (i.e. repeller) and a stable fixed (attractor) points,
respectively. We have described this phenomena by analytic solutions of the
Hubble function and the scale factor of the FLRW universe. Using Jacobi Last
multiplier method, we have found a Lagrangian for this matter creation rate
describing this scenario of the universe. To match with our early physics
results, we introduce an equivalent dynamics driven by a single scalar field
and discussed the associated observable parameters compared them with the
latest PLANCK data sets. Finally, introducing the teleparallel modified
gravity, we have established an equivalent gravitational theory in the
framework of matter creation.
