In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen’s system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all the regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifuraction as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is analyzed. Numerical simulations are employed to corroborate the predictions from the resulting normal form. They reveal the existence of stable periodic attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations. Numerical diagnostics are also used to characterize the solutions in the latter cases. Future work will consider other possible routes into the chaotic regimes, including: (a) further bifurcations of the post-supercritical Hopf limit cycle attractors, (b) double Hopf bifurcations, leading into periodic, two-period quasiperiodic, or aperiodic (bounded or unbounded) dynamics, and (c) secondary bifurcations of two-periodic quasiperiodic attractors via torus doubling or breakdown.