A new 4D smooth quadratic autonomous system with complex hyper-chaotic dynamics is presented.

The stability of equilibria is observed near the bifurcation points.

The Hopf bifurcation and pitchfork bifurcation are analyzed by using the center manifold theorem and bifurcation theory.

A horseshoe with two-directional expansions in the 4D hyper-chaotic system has been found, which rigorously proves the existence of hyper-chaos in theory.