It is still a challenge to accurately predict homoclinic cycles and chaos in smooth nonlinear systems, letting alone for non-smooth objects. This paper analytically investigates occurrence of homoclinic cycles in a class of three-dimensional piecewise nonlinear systems governed by a nonlinear subsystem and an affine one, which under some conditions can be transformed into a linear form. By a series of equivalent transformations, the solution of the considered systems can be obtained explicitly. Furthermore, via deriving analytic expression of Poincaré return maps, it rigorously proves that the considered system presents complicated chaotic dynamics. This approach offers a way to identify singular cycles and chaos in other piecewise systems exhibiting nonlinearities. Two examples are provided finally to numerically illustrate and verify effectiveness of our theoretical results established.