[en] The performance of a wakefield accelerator in a high energy collider application is analyzed. In order to carry out this task, it is necessary to construct a strawman design system (no matter how preliminary) and build a code of the systems approach. A nonlinear dynamics map built on a simple theoretical model of the wakefield generated by the laser pulse (or whatever other method) is obtained and they employ this as a base for building a system with multi-stages (and components) as a high energy collider. The crucial figures of merit for such a system other than the final energy include the emittance (that determines the luminosity). The more complex the system is, the more opportunities the system has to degrade the emittance (or entropy of the beam). Thus the map gu ides one to identify where the crucial elements lie that affect the emittance. They find that a strong focusing force of the wakefield coupled with a possible jitter of the axis (or laser aiming) of each stage and a spread in the betatron frequencies arising from different phase space positions for individual particles leads to a phase space mixing. This sensitively controls the emittance degradation. They show that in the case of a uniform plasma the effect of emittance growth is large and may cause serious problems. They discuss possibilities to avoid it and control the situation