  This paper deals with various routes to hyperchaos with all three positive
Lyapunov exponents in a three-dimensional quadratic map. The map under
consideration displays strong hyperchaoticity in the sense that in a wider
range of parameter space the system showcase three positive Lyapunov exponents.
It is shown that the saddle periodic orbits eventually become repellers at this
hyperchaotic regime. By computing the distance of the repellers to the
attractors as a function of parameters, it is shown that the hyperchaotic
attractors absorb the repelling periodic orbits. First we discuss a route from
stable fixed point undergoing period-doubling bifurcations to chaos and then
hyperchaos, and role of saddle periodic orbits. We then illustrate a route from
doubling bifurcation of quasiperiodic closed invariant curves to hyperchaotic
attractors. Finally, presence of weak hyperchaotic flow like attractors are
discussed.
