Using a combination of analytical and numerical methods, the chaos of a new 4D quadratic autonomous system was investigated. The mechanism and sufficient conditions of system parameters for chaotic motions were investigated rigorously. By using the undetermined coefficient method, the homoclinic orbit was found and the uniform convergence of the homoclinic orbit series expansion was proved. Therefore Smale horseshoe chaos occurs for this system via Si’lnikov criterion. Numerical simulations confirmed the analytical results.