The manufacturing process has been a heavily studied area over the past century. The study completed herein has established a foundation for the future of manufacturing research. The next step of this industry is to become proficient at the micro and nano scale levels of manufacturing. In order to accomplish this goal, the modeling of machining system needs to be completely understood throughout the entire process. In effort to attack this problem, this study will focus on the boundaries present in machining systems; and will define and interpret the associated phenomena. This particular focus is selected since nearly all manufacturing related studies concentrate on continuous processes; which by definition considers only one particular operation. There is a need to understand the phenomena corresponding to interactions of multiple processes of manufacturing systems. As a means to this end, the nonlinear phenomena associated in the continuous domains of machining systems will be modeled as linear to ensure the boundary interactions are clearly observed. Interference of additional nonlinearities is not the focus of this research. In this dissertation, the mechanical model for a widely accepted machine-tool system is presented. The state and continuous domains are defined with respect to the boundaries in this system (contact and frictional force acting at the point of tool and work-piece contact). The switching sets defining plane boundaries for the continuous systems of this machine-tool will be defined and studied herein. The forces and force products, at the point of switching from one continuous system to another, govern the pass-ability of the machine-tool through the respective boundary. The forces and force product components at the switching points are derived according to discontinuous systems theory Luo [1]. Mapping definitions and notations are developed through the switching sets for each of the boundaries. A mapping structure and notation for periodic interrupted cutting, non-cutting and chip seizure motions are defined. The interruption of the chip flow for a machining system will be investigated through a range of system parameters. The prediction of interrupted periodic cutting, non-cutting and chip seizure motion will be completed via closed form solutions for this machine-tool. The state of this system is defined to utilize the theory of Luo [1]. This is necessary to properly handle the frictional force boundary at the chip/tool interface, the onset of cutting boundary and the contact boundary between the tool and work-pieces. The predictions by this method will be verified via numerical simulation and comparison to existing research. A goal of this research is to illustrate the effects of the dynamical systems interacting at the frictional force (chip/tool) boundary and the chip onset of growth and vanishing boundary. The parameter space for this machine-tool model is studied through numerical and analytical predictions, which provide limits on the existence of interrupted periodic cutting, non-cutting and chip seizure motions.