  The bouncing evolution of an universe in Loop Quantum Cosmolgy can be
described very well by a set of effective equations, involving a function $sin
\; x$. Recently, we have generalised these effective equations to $(d + 1)$
dimensions and to any function $f(x) \;$. Depending on $f(x) \;$ in these
models inspired by Loop Quantum Cosmolgy, a variety of cosmological evolutions
are possible, singular as well as non singular. In this paper, we study them in
detail. Among other things, we find that the scale factor $a(t) \; \propto \;
t^{ \frac {2 q} {(2 q - 1) \; (1 + w) d}} \;$ for $f(x) = x^q \;$, and find
explicit Kasner--type solutions if $w = 2 q - 1 \;$ also. A result which we
find particularly fascinating is that, for $f(x) = \sqrt{x} \;$, the evolution
is non singular and the scale factor $a(t)$ grows exponentially at a rate set,
not by a constant density, but by a quantum parameter related to the area
quantum.
