Pathways to hyperchaos in a three-dimensional quadratic map

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 showcases 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, the presence of weak hyperchaotic flow-like attractors is discussed.