We study the chaotic behavior of the T system, a three dimensional autonomous nonlinear system introduced by Tigan (2005, “Analysis of a Dynamical System Derived From the Lorenz System,” Scientific Bulletin Politehnica University of Timisoara, Tomul, 50, pp. 61–72), which has potential application in secure communications. Here, we first recount the heteroclinic orbits of Tigan and Dumitru (2008, “Analysis of a 3D Chaotic System,” Chaos, Solitons Fractals, 36, pp. 1315–1319), and then we analytically construct homoclinic orbits describing the observed Smale horseshoe chaos. In the parameter regimes identified by this rigorous Shil’nikov analysis, the occurrence of interesting behaviors thus predicted in the T system is verified by the use of numerical diagnostics.