Abstract This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyperchaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent doublewing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.